This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A343125 #64 Nov 25 2021 02:04:25
%S A343125 0,4,1,8,6,2,12,11,8,3,16,16,14,10,4,20,21,20,17,12,5,24,26,26,24,20,
%T A343125 14,6,28,31,32,31,28,23,16,7,32,36,38,38,36,32,26,18,8,36,41,44,45,44,
%U A343125 41,36,29,20,9,40,46,50,52,52,50,46,40,32,22,10
%N A343125 Triangle T(k, n) = (n+3)*(k-n) - 4, k >= 2, 1 <= n <= k-1, read by rows.
%C A343125 T(k, n) is even if k is odd.
%C A343125 T(k, n) = T(k, n+1) for n = k/2 - 2 if k >= 6 is even.
%C A343125 T(k, n) = T(k, n+2) for n = (k-1)/2 - 2 if k >= 7 is odd.
%C A343125 For fixed n, T(k, n) is linear in k.
%C A343125 The T(k, j) contribute coefficients to a closed formula for the sum of the first n+1 squares of the k-generalized Fibonacci numbers, F(k, j) = A092921(k, j). See A343138 for sums of squares of F(k, j). See the Formula section for closed formula. Although other sequences occur in coefficients in the closed formula for sums of squares, they are linear in nature. All coefficient sequences are mentioned in the arXiv link. The closed formula generalizes results of Schumacher (see References) for the cases k=3 and k=4 with a uniform proof method (see arXiv link).
%D A343125 Raphael Schumacher, How to Sum the Squares of the Tetranacci Numbers and the Fibonacci m-step Numbers, Fibonacci Quarterly, 57, (2019), 168-175.
%D A343125 Raphael Schumacher, Explicit Formulas for Sums Involving the Squares of the First n Tribonacci Numbers, Fibonacci Quarterly, 58 (2020), 194-202.
%H A343125 Russell Jay Hendel, <a href="https://arxiv.org/abs/2103.16756">Sums of Squares: Methods for Proving Identity Families</a>, arXiv:2103.16756 [math.NT], 2021.
%H A343125 Proof Wiki, <a href="https://proofwiki.org/wiki/Sum_of_Sequence_of_Squares_of_Fibonacci_Numbers">Sum of Sequence of Squares of Fibonacci Numbers</a>
%F A343125 Let F(k, n) = A092921(k, n), the k-generalized Fibonacci numbers. Let A(k, n) = A343138(k, n) = Sum_{m=0..n} F(k, m)^2, the sum of the first m+1 k-generalized Fibonacci numbers. Then, for k >= 2, a closed formula for A(k, n) is:
%F A343125 A(k, n) = (1/(2*k-2)) * (Sum_{j=0..k-2, m=j+1..k-1} 2*(j+1)*(m-k+1) * F(k, n+j) * F(k, n+m)) - (k-2)*F(k, n)^2 - Sum_{j=1..k}(T(k, j) * F(k, n+j)^2) + (k-2)).
%F A343125 From _G. C. Greubel_, Nov 22 2021: (Start)
%F A343125 T(2*n-2, n) = A028557(n-2), n >= 2.
%F A343125 T(4*n-6, n) = 2*A140672(n-2), n >= 2. (End)
%e A343125 Triangle T(k, n) begins:
%e A343125 k \ n| 1 2 3 4 5 6 7 8 9 10 11
%e A343125 ------+----------------------------------
%e A343125 2 | 0
%e A343125 3 | 4 1
%e A343125 4 | 8 6 2
%e A343125 5 | 12 11 8 3
%e A343125 6 | 16 16 14 10 4
%e A343125 7 | 20 21 20 17 12 5
%e A343125 8 | 24 26 26 24 20 14 6
%e A343125 9 | 28 31 32 31 28 23 16 7
%e A343125 10 | 32 36 38 38 36 32 26 18 8
%e A343125 11 | 36 41 44 45 44 41 36 29 20 9
%e A343125 12 | 40 46 50 52 52 50 46 40 32 22 10
%e A343125 .
%e A343125 The following are the closed formulas for k = 3, 4 for A(k, n) = Sum_{m=0..n} F(k, m)^2, with F(k, n) = A092921(k, n), the k-generalized Fibonacci numbers, and A(k, n) = A343138(k, n), the sum of squares of F(k, n). These formulas are derived from the closed formula in the formula section. Of course further simplifications are possible. For k = 2, T(2, 1) = 0 so illustrations start with k = 3.
%e A343125 k | Formula
%e A343125 --+--------------------------------------------------------
%e A343125 3 | Sum_{m=0..n} F(3,m)^2 = (1/4)*(2*F(3,n)*F(3,n+2) + 4*F(3,n+1)*F(3,n+2) - (k - 2)*F(3,n)^2 - T(3,1)*F(3,n+1)^2 - T(3,2)*F(3,n+2)^2 + 1).
%e A343125 4 | Sum_{m=0..n} F(3,m)^2 = (1/6)*(-2*F(4,n)*F(4,n+1) + 2*F(4,n)*F(4,n+3) + 4*F(4,n+1)*F(4,n+3) + 6*F(4,n+2)*F(4,n+3) - (k-2)*F(4,n)^2 - T(4,1)*F(4,n+1)^2 - T(4, 2)*F(4,n+2)^2 - T(4,3)*F(4,n+3)^2 + 2).
%p A343125 T := (k, n) -> (n + 3)*(k - n) - 4:
%p A343125 seq(print(seq(T(k, n), n=1..k-1)), k = 2..12); # _Peter Luschny_, Apr 02 2021
%t A343125 Table[(n + 3) (k - n) - 4, {k, 2, 12}, {n, k - 1}] // Flatten (* _Michael De Vlieger_, Apr 06 2021 *)
%o A343125 (PARI) T(k,n)=(n + 3)*(k - n) - 4
%o A343125 for(k = 2,12,for(n = 1,k - 1, print1(T(k,n),", ")))
%o A343125 (Sage) flatten([[(n+3)*(k-n) -4 for n in (1..k-1)] for k in (2..15)]) # _G. C. Greubel_, Nov 22 2021
%Y A343125 Cf. A028557, A085697, A092921, A140672, A343138.
%K A343125 easy,nonn,tabl
%O A343125 2,2
%A A343125 _Russell Jay Hendel_, Apr 06 2021