Diagonal Sums in the Pascal Pyramid, II: Applications

Hacène Belbachir and Abdelghani Mehdaoui

USTHB Faculty of Mathematics RECITS Laboratory BP 32 El Alia 16111 Bab Ezzouar Algiers Algeria

     hacenebelbachir@gmail.com
      mehabdelghani@gmail.com

László Szalay University of Sopron Institute of Mathematics Bajcsy-Zsilinszky utca 4 H-9400 Sopron Hungary

    szalay.laszlo@uni-sopron.hu

Abstract

In a previous paper, we derived a theorem which describes, in the language of linear recurrences, the sum of diagonal elements lying along a finite ray of a certain type crossing the 3-dimensional Pascal pyramid. The corresponding generating function was also determined. In this paper, we apply these results to prove several (more precisely

24) recurrence relations previously conjectured, or not given in the On-Line Encyclopedia of Integer Sequences. Moreover, we provide many (exactly 75) new summatory identities linked to sequences listed in the Encyclopedia.

1 Introduction

Let (F_{n})_{n=0}^{∞} denote the Fibonacci sequence given by the initial values F_{0} = 0 and F_{1} = 1, and by F_{n} = F_{n−1} + F_{n−2} for n ≥ 2. Probably Binet [4] was the first to publish the identity

F_{n+1} = \binom{n}{0} + \binom{n − 1}{1} + \binom{n − 2}{2} + ⋯, (n ≥ 0)

about the sum of elements of ascending diagonals, which motivated the paper [2] to investigate diagonals of arbitrary direction similarly generating finite sums. It turned out that such sums can also be described by suitable linear recurrences. In this paper, we recall the two main theorems and some corollaries of [3] which study the analogous question in the 3D Pascal pyramid (in short: PP3D), and we apply the results to many sequences appearing in the On-Line Encyclopedia of Integer Sequences (OEIS) [4].

It is known that trinomial coefficients \binom{n}{i, j, k} = \frac{n!}{i!·j!·k!} (where i, j, k are non-negative integers and i + j + k = n) appear in the expansion of (x + y + z)^{n} as

(x + y + z)^{n} = ∑_{i+j+k=n} \binom{n}{i, j, k}x^{i} y^{j} z^{k}  (1)

(see, for example, Feinberg [5] or Anatriello and Vincenzi [1]).

We adapted the trinomial coefficients to the 3D coordinate-system — more precisely, to the vertices of the lattice ℤ^{3}. The location of \binom{n}{i, j, k} is the point (i, j, −n) (recall that n = i + j + k).

Take two arbitrary vertices P_{0}(x_{0}, y_{0}, z_{0}) ∈ ℤ^{3} and P_{1}(x_{1}, y_{1}, z_{1}) ∈ ℤ^{3} of the pyramid (that is, such that the conditions x_{i} ≥ 0, y_{i} ≥ 0, z_{i} ≤ 0, x_{i} + y_{i} + z_{i} ≤ 0 are fulfilled) such that if r = z_{1} − z_{0} holds, then

α_{1} = x_{1} − x_{0} > 0,  α_{2} = y_{1} − y_{0} > 0,  α_{1} + α_{2} + r > 0.

We also require that x_{0} + y_{0} + z_{0} < 0, excluding the case when the vector \overrightarrow{v} = \overrightarrow{P_{0}P_{1}} is on the plane x + y + z = 0.

Along the straight line determined by the points P_{0} and P_{1} we translate −\overrightarrow{v} from P_{0} as many times as possible: let x_{0} = s_{1}α_{1} + ℓ_{1} with 0 ≤ ℓ_{1} < α_{1}, and let x_{1} = s_{2}α_{2} + ℓ_{2} with 0 ≤ ℓ_{2} < α_{2}. Put s = min{s_{1}, s_{2}}, and then set

θ_{1} = x_{0} − sα_{1}, θ_{2} = y_{0} − sα_{2}.

The point we obtain, say P, has coordinates (θ_{1}, θ_{2}, z_{0}−sr). Now we raise the point P parallel to the z-axis until it reaches the plane x + y + z = 0. Then we consider the uppermost ray defined by the quintuple (α_{1}, α_{2}, r, θ_{1}, θ_{2}). Note that, seemingly, the conditions α_{1} > 0 and α_{2} > 0 together would not cover all the possible directions in PP3D. Nevertheless, here they do because of the symmetry of rotation around the vertical axis (before fitting PP3D to the coordinate system).

Thus the direction (α_{1}, α_{2}, r, θ_{1}, θ_{2}) defines diagonal rays in the Pascal pyramid. Such a ray contains the elements

A_{k} = \binom{n − rk}{θ_{1} + α_{1}k, θ_{2} + α_{2}k, η_{k}}x^{θ_{1}+α_{1}k}y^{θ_{2}+α_{2}k}z^{η_{k}},  (2)

where k = 0, …, ⌊\frac{n−θ_{1}−θ_{2}}{α_{1}+α_{2}+r}⌋ and η_{k} = n−θ_{1} −θ_{2} −(α_{1} +α_{2} +r)k. Here x, y and z are nonzero real parameters. What most interests us is the sum

T_{n}^{(α_{1},α_{2},r,θ_{1},θ_{2})} = ∑_{k=0}^{⌊ \frac{n−θ_{1}−θ_{2}}{α_{1}+α_{2}+r} ⌋} A_{k},

but computing it is a hard problem in general.

Belbachir, Mehdaoui, and Szalay [3] considered only

T_{n}^{(r)} = ∑_{k=0}^{⌊ \frac{n}{r+2} ⌋} \binom{n − rk}{k, k, n − (r + 2)k}x^{k}y^{k}z^{n−(2+r)k}

corresponding to the particular case (α_{1}, α_{2}, r, θ_{1}, θ_{2}) = (1, 1, r, 0, 0). They also assumed r ≥ −1 to ensure the finiteness of the rays. Here we mention that Shannon [7] found a connection between such sums and ternary recurrences. In fact, our approach is more general.

In the next section, we summarize the main result of [3] and its consequences.

2 Previous results

Assume t = xy. For notational convenience, set T_{n} = T_{n}^{(r)}.

Theorem 1. The terms of the sequence (T_{n}) given by

T_{n} = ∑_{k=0}^{⌊n/(r+2)⌋} \binom{n − rk}{k, k, n − (r + 2)k}t^{k}z^{n−(r+2)k}  (3)

satisfy the linear recurrence relation

nT_{n} = (2n − 1)zT_{n−1} − (n − 1)z^{2}T_{n−2} + 2(2n − (r + 2))tT_{n−r−2}.  (4)

Observe that the last subscript which appears in the right-hand side of (4) is n − r − 2. If r = −1 or r = 0, then T_{n−r−2} can be joined to T_{n−1} or T_{n−2}, respectively. This is the Morgan-Voyce phenomenon, and under such conditions we obtain the following corollaries.

Corollary 2. If r = 0, then the recurrence relation (4) simplifies to

nT_{n} = (2n − 1)zT_{n−1} + (n − 1)(4t − z^{2})T_{n−2}.  (5)

Corollary 3. Assume r = −1. Then (4) provides

nT_{n} = (2n − 1)(2t + z)T_{n−1} − (n − 1)z^{2}T_{n−2}.  (6)

The comparison of the two formulae (5) and (6) makes it possible to establish equalities between two sums belonging to the same recurrence. This idea leads to the following result.

Corollary 4. We have the identity

∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n − 2k}(t_{1}τ)^{k} (τ + t_{1})^{n−2k} = ∑_{k}^{n} \binom{n + k}{k, k, n − k}t_{1}^{k} (τ − t_{1})^{n−k},  (7)

where τ ≠ 0 and τ ≠ ±t_{1}.

Finally, we were able to determine the generating function of sequence (3) [3].

Theorem 5. The generating function of variable u of the sequence given by (3) is

g(u) = \frac{1}{\sqrt{(zu − 1)^{2} − 4tu^{r+2}}} .  (8)

Now we list several examples of the type of Corollary 4 we found in the OEIS [7]. In the sequel, keeping the notation of [7] we use (a_{n}) for the sequences (instead of (T_{n})).

3 List of new identities, and a combinatorial interpretation of Theorem 1

3.1 Supplement of OEIS

We use the generating function for the identification of sequences since a sequence in the OEIS is usually defined by its generating function or the generating function is often given as a feature of the sequence. In other words, the generating function is the connection between the sequence, its recurrence relation, and the corresponding sum(s).

We split our results into four blocks. The first one contains the sequences for which the appropriate recurrence relations were given in the OEIS with the note “conjecture”. The application of Theorem 5 (identification) and Theorem 1 provide the proofs automatically. So we showed

Theorem 6. Each sequence of the first column of Table 1 satisfies the linear recurrence rule located in the same row of the second column.

Table 2 gives the sequences for which the recurrence relations were not given in the OEIS.

Theorem 7. The sequences given in Table 2 satisfy the corresponding recurrence relations.

The next theorem, thanks to Theorem 1 and Corollary 4, collects at least two new trinomial sums for some sequences. Note that the parameter r of the direction (α_{1}, α_{2}, r) = (1, 1, r) appears, for example, in the upper index of the trinomial coefficient in (3). Hence one can easily conclude the value of r from the corresponding sum.

Theorem 8. Columns 2–4 of Table 3 provide new sum identities for the sequences of the first column.

Finally, we found a few new identities for certain sequences when only one sum exists or we could add only one new sum to the detected one(s).

Theorem 9. The sums in Table 4 give new descriptions for the corresponding sequences.

3.2 Powers of integers

Corollary 10. Formula (3) simplifies to

T_{n} = ∑_{k=0}^{n} \binom{n + k}{k, k, n − k} (−1)^{n−k} = 1  (9)

if r = −1, t = 1, and z = −1.

OEIS code Recurrence relation conjectured in the OEIS
A084768 na_{n} = 7(2n − 1)a_{n−1} − (n − 1)a_{n−2}
A084769 na_{n} = 9(2n − 1)a_{n−1} − (n − 1)a_{n−2}
A084771 na_{n} = 5(2n − 1)a_{n−1} − 9(n − 1)a_{n−2}
A098270 na_{n} = 10(2n − 1)a_{n−1} − 4(n − 1)a_{n−2}
A098333 na_{n} = (2n − 1)a_{n−1} − 13(n − 1)a_{n−2}
A098334 na_{n} = (2n − 1)a_{n−1} − 17(n − 1)a_{n−2}
A098336 na_{n} = 2(2n − 1)a_{n−1} − 12(n − 1)a_{n−2}
A098337 na_{n} = 2(2n − 1)a_{n−1} − 20(n − 1)a_{n−2}
A098338 na_{n} = 3(2n − 1)a_{n−1} − 13(n − 1)a_{n−2}
A098340 na_{n} = 3(2n − 1)a_{n−1} − 21(n − 1)a_{n−2}
A098341 na_{n} = 3(2n − 1)a_{n−1} − 25(n − 1)a_{n−2}
A098411 na_{n} = 8(2n − 1)a_{n−1} − 48(n − 1)a_{n−2}
A098439 na_{n} = (2n − 1)a_{n−1} + 47(n − 1)a_{n−2}
A098456 na_{n} = 2(2n − 1)a_{n−1} + 64(n − 1)a_{n−2}
A098479 na_{n} = (2n − 1)a_{n−1} − (n − 1)a_{n−2} + 2(2n − 3)a_{n−3}
A098480 na_{n} = (2n − 1)a_{n−1} − (n − 1)a_{n−2} + 4(2n − 3)a_{n−3}
A106186 na_{n} = 2(2n − 1)a_{n−1} − 4(n − 1)a_{n−2} + 8(2n − 3)a_{n−3}
A115864 na_{n} = 8(2n − 1)a_{n−1} − 16(n − 1)a_{n−2}
A116092 na_{n} = −4(2n − 1)a_{n−1} − 64(n − 1)a_{n−2}
A116093 na_{n} = −2(2n − 1)a_{n−1} − 12(n − 1)a_{n−2}
A122868 na_{n} = 3(2n − 1)a_{n−1} + 3(n − 1)a_{n−2}

Table 1: Conjectured recurrence relations we proved.

OEIS code Recurrence relation we discovered
A113179 na_{n} = 2(2n − 1)a_{n−1} − 4(n − 1)a_{n−2} + 4(2n − 3)a_{n−3}
A248168 na_{n} = 7(2n − 1)a_{n−1} − 33(n − 1)a_{n−2}
A258723 na_{n} = 6(2n − 1)a_{n−1} − 48(n − 1)a_{n−2}

Table 2: Recurrence relation discovered.

OEIS code Sum 1 Sum 2 Sum 3
A012000 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} (−3)^{k} 2^{n−2k}  ∑_{k}^{n} \binom{n+k}{k, k, n−k} (−1)^{k} 4^{n−k} ∑_{k}^{n} \binom{n+k}{k, k, n−k} 3^{k} (−4)^{n−k}
A084771 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 4^{k} 5^{n−2k}  ∑_{k}^{n} \binom{n+k}{k, k, n−k} 3^{n−k}  ∑_{k}^{n} \binom{n+k}{k, k, n−k} 4^{k} (−3)^{n−k}
A084772 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 5^{k} 6^{n−2k}  ∑_{k}^{n} \binom{n+k}{k, k, n−k} 4^{n−k}  ∑_{k}^{n} \binom{n+k}{k, k, n−k} 5^{k} (−4)^{n−k}
A084773 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 8^{k} 6^{n−2k}  ∑_{k}^{n} \binom{n+k}{k, k, n−k} 2^{k}2^{n−k}  ∑_{k}^{n} \binom{n+k}{k, k, n−k} 4^{k} (−2)^{n−k}
A084774 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 10^{k} 7^{n−2k}  ∑_{k}^{n} \binom{n+k}{k, k, n−k} 2^{k}3^{n−k}  ∑_{k}^{n} \binom{n+k}{k, k, n−k} 5^{k} (−3)^{n−k}
A098269 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 15^{k} 8^{n−2k}  ∑_{k}^{n} \binom{n+k}{k, k, n−k} 3^{k}2^{n−k}  ∑_{k}^{n} \binom{n+k}{k, k, n−k} 5^{k} (−2)^{n−k}
A098270 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 24^{k} 10^{n−2k}  ∑_{k}^{n} \binom{n+k}{k, k, n−k} 4^{k}2^{n−k}  ∑_{k}^{n} \binom{n+k}{k, k, n−k} 6^{k} (−2)^{n−k}
A098341 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} (−4)^{k} 3^{n−2k}  ∑_{k}^{n} \binom{n+k}{k, k, n−k} (−1)^{k}5^{n−k} ∑_{k}^{n} \binom{n+k}{k, k, n−k} 4^{k} (−5)^{n−k}
A098659 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 6^{k} 7^{n−2k}  ∑_{k}^{n} \binom{n+k}{k, k, n−k} 5^{n−k}  ∑_{k}^{n} \binom{n+k}{k, k, n−k} 6^{k} (−5)^{n−k}
A115864 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 12^{k} 8^{n−2k}  ∑_{k}^{n} \binom{n+k}{k, k, n−k} 2^{k}4^{n−k}  ∑_{k}^{n} \binom{n+k}{k, k, n−k} 6^{k} (−4)^{n−k}
A115865 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 27^{k} 12^{n−2k}  ∑_{k}^{n} \binom{n+k}{k, k, n−k} 3^{k}6^{n−k}  ∑_{k}^{n} \binom{n+k}{k, k, n−k} 9^{k} (−6)^{n−k}
A116091 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} (−3)^{k} (−2)^{n−2k} ∑_{k}^{n} \binom{n+k}{k, k, n−k} (−3)^{k}4^{n−k} ∑_{k}^{n} \binom{n+k}{k, k, n−k} (−4)^{n−k}
A116092 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} (−12)^{k} (−4)^{n−2k} ∑_{k}^{n} \binom{n+k}{k, k, n−k} (−6)^{k}8^{n−k} ∑_{k}^{n} \binom{n+k}{k, k, n−k} 2^{k} (−8)^{n−k}
A069835 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 3^{k} 4^{n−2k}  already known  ∑_{k}^{n} \binom{n+k}{k, k, n−k} 3^{k} (−2)^{n−k}
A084768 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 12^{k} 7^{n−2k}  already known  ∑_{k}^{n} \binom{n+k}{k, k, n−k} 4^{k} (−1)^{n−k}
A084769 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 20^{k} 9^{n−2k}  already known  ∑_{k}^{n} \binom{n+k}{k, k, n−k} 5^{k} (−1)^{n−k}
A098332 already known  ∑_{k}^{n} \binom{n+k}{k, k, n−k} (−1)^{k}3^{n−k} ∑_{k}^{n} \binom{n+k}{k, k, n−k} 2^{k}(−3)^{n−k}

Table 3: Sums discovered.

OEIS code Sum  OEIS code Sum
A001850 ∑_{k}^{n} \binom{n+k}{k, k, n−k} 2^{k} (−1)^{n−k}  A006134 ∑_{k}^{⌊n/3⌋} \binom{n−k}{k, k, n−3k} 3^{n−3k}
A006442 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 6^{k} 5^{n−2k}  A026375 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 3^{n−2k}
A059304 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 4^{k} 4^{n−2k}  A080609 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 2^{k} 4^{n−2k}
A081671 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 4^{n−2k}  A084605 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 4^{k}
A084770 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 5^{k} 2^{n−2k}  A098339 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} (−2)^{k}3^{n−2k}
A098409 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 5^{n−2k}  A098410 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 6^{n−2k}
A098411 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 4^{k}8^{n−2k}  A098430 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 16^{k}8^{n−2k}
A098443 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 5^{k}4^{n−2k}  A098444 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 5^{k}3^{n−2k}
A098453 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 4^{k}2^{n−2k}  A098455 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 10^{k}2^{n−2k}
A098456 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 17^{k}2^{n−2k}  A098658 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 9^{k}6^{n−2k}
A106186 ∑_{k}^{⌊n/3⌋} \binom{n−k}{k, k, n−3k} 4^{k}2^{n−3k}  A106258 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 6^{k}4^{n−2k}
A106259 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 12^{k}6^{n−2k}  A106260 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 20^{k}8^{n−2k}
A106261 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 30^{k}10^{n−2k}  A122868 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 3^{k}3^{n−2k}
A248168 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} 4^{k}7^{n−2k}  A258723 ∑_{k}^{⌊n/2⌋} \binom{n}{k, k, n−2k} (−3)^{k}6^{n−2k}

Table 4: Sums discovered.

Proof. Obviously, (9) satisfies the recurrence rule (6). Hence for n ≥ 2 we have

nT_{n} = (2 n − 1) T_{n−1} + (n − 1) T_{n−2},

which is equivalent to

n \underbrace{(T_{n} − T_{n−1})}_{U_{n}} = (n − 1) \underbrace{(T_{n−1} − T_{n−2})}_{U_{n−1}},

and then to

\underbrace{n U_{n}}_{V_{n}} = \underbrace{(n − 1) U_{n−1}}_{V_{n−1}} .

Consequently, V_{n} = V_{1} holds for n ≥ 2. The initial values T_{0} = T_{1} = 1 imply V_{1} = 0. Thus nU_{n} = 0 gives U_{n} = 0, and then T_{n} = T_{n−1} follows, which together with the initial values proves T_{n} = 1 for all n ≥ 0.

Corollary 11. Assume that r = −1 and z = −t hold in (3). Then we have T_{n} = t^{n}.

Proof. Combine (3), Corollary 10, and the conditions here to get

T_{n} = ∑_{k=0}^{n} \binom{n + k}{k, k, n − k}t^{k}(−t)^{n−k} = t^{n}.

Replacing r = −1 and t = −z in (9), together with Corollary 10 immediately leads to

Corollary 12.

T_{n} = ∑_{k=0}^{n} \binom{n + k}{k, k, n − k}(−z)^{k}z^{n−k} = (−z)^{n}.

3.3 A combinatorial interpretation

A combinatorial interpretation of sequence (3) corresponding to the direction (r, 1, 1) follows easily from the sum of (3).

Theorem 13. The sequence T_{n} in (3) gives the sum of products of weighs associated with the walk in the square lattice from the point (0, 0) to (n, n) when the steps allowed are in \{(2 + r, 0), (0, 2 + r), (1, 1)\}. The first and second steps of the walk have weight \sqrt{t}, while the last one has z.

Example 14. The triple (r, t, z) = (−1, 3, 2) yields a Morgan-Voyce direction, and provides the sequence A098269. It can be given by

T_{n} = ∑_{k=0}^{n} \binom{n + k}{k, k, n − k}3^{k} 2^{n−k},

the first few elements are: 1, 8, 94, 1232, 16966, 240368, 3468844, . . ..

The sums of products of weights are related to the lattice paths from (0, 0) to (n, n) using only the steps (1, 0), (0, 1) and (1, 1), with the weights \sqrt{3}, \sqrt{3}, 2, respectively. In case of n = 2 we illustrate all paths from (0, 0) to (2, 2) in Figure 1, which imply

T_{2} = (\sqrt{3} × \sqrt{3} × \sqrt{3} × \sqrt{3}) + (2 × \sqrt{3} × \sqrt{3}) + (\sqrt{3} × \sqrt{3} × \sqrt{3} × \sqrt{3})

+(\sqrt{3} × 2 × \sqrt{3}) + (2 × 2) + (\sqrt{3} × \sqrt{3} × \sqrt{3} × \sqrt{3}) + (\sqrt{3} × \sqrt{3} × \sqrt{3} × \sqrt{3})

+(2 × \sqrt{3} × \sqrt{3}) + (\sqrt{3} × \sqrt{3} × 2) + (\sqrt{3} × \sqrt{3} × \sqrt{3} × \sqrt{3})
+(\sqrt{3} × \sqrt{3} × \sqrt{3} × \sqrt{3}) + (\sqrt{3} × \sqrt{3} × 2) + (\sqrt{3} × 2 × \sqrt{3})
= 94.

\sqrt{3} \sqrt{3} \sqrt{3} \sqrt{3} \sqrt{3} 2 \sqrt{3} \sqrt{3} \sqrt{3} \sqrt{3} \sqrt{3} \sqrt{3} 2 \sqrt{3} 2 2 \sqrt{3} \sqrt{3} \sqrt{3} \sqrt{3} \sqrt{3} \sqrt{3} \sqrt{3} \sqrt{3} 2 \sqrt{3} \sqrt{3} \sqrt{3} \sqrt{3} 2 \sqrt{3} \sqrt{3} \sqrt{3} \sqrt{3} \sqrt{3} \sqrt{3} \sqrt{3} \sqrt{3} \sqrt{3} \sqrt{3} 2 \sqrt{3} 2 \sqrt{3}

Figure 1: All lattice paths linked to A098269.

References

[1] G. Anatriello and V. Giovanni, Tribonacci-like sequences and generalized Pascal’s pyramids, Internat. J. Math. Ed. Sci. Tech. 45 (2014), 1220–1232.

[2] H. Belbachir, T. Komatsu, and L. Szalay, Linear recurrences associated with rays in Pascal’s triangle and combinatorial identities, Math. Slovaca 64 (2014), 287–300.

[3] H. Belbachir, A. Mehdaoui, and L. Szalay, Diagonal sums in 3D Pascal pyramid, J. Combin. Theory Ser. A 165 (2019), 106–116.

[4] J. P. M. Binet, Mémoire sur l’intégration des équations linéaires aux différences finies, d’un ordre quelconque à coefficients variables, Comptes Rendus des Séances de l’Académie des Sciences 17 (1843), 559–567.

[5] M. Feinberg, New slants, Fibonacci Quart. 2 (1964), 223–227.

[6] A. G. Shannon, Iterative formulas associated with generalized third order recurrence relations, SIAM J. Appl. Math. 23 (1972), 364–368.

[7] N. J. A. Sloane et al., The On-Line Encyclopedia of Integer Sequences. Available at

   https://oeis.org.

2010 Mathematics Subject Classification: Primary 11B65; Secondary 11B37, 05A10. Keywords: Pascal pyramid, trinomial coefficient, combinatorial identity, linear recurrence.

(Concerned with sequences \underline{A001850}, \underline{A006134}, \underline{A006442}, \underline{A012000}, \underline{A026375}, \underline{A059304}, \underline{A069835}, \underline{A080609}, \underline{A081671}, \underline{A084605}, \underline{A084768}, \underline{A084769}, \underline{A084770}, \underline{A084771}, \underline{A084772}, \underline{A084773}, \underline{A084774}, \underline{A098269}, \underline{A098270}, \underline{A098332}, \underline{A098333}, \underline{A098334}, \underline{A098336}, \underline{A098337}, \underline{A098338}, \underline{A098339}, \underline{A098340}, \underline{A098341}, \underline{A098409}, \underline{A098410}, \underline{A098411}, \underline{A098430}, \underline{A098439}, \underline{A098443}, \underline{A098444}, \underline{A098453}, \underline{A098455}, \underline{A098456}, \underline{A098479}, \underline{A098480}, \underline{A098658}, \underline{A098659}, \underline{A106186}, \underline{A106258}, \underline{A106259}, \underline{A106260}, \underline{A106261}, \underline{A113179}, \underline{A115864}, \underline{A115865}, \underline{A116091}, \underline{A116092}, \underline{A116093}, \underline{A122868}, \underline{A248168}, and \underline{A258723}.)

Received March 30 2018; revised versions received February 4 2019; February 12 2019; March 6 2019. Published in Journal of Integer Sequences, May 19 2019.

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