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 A343138 #20 Apr 08 2021 11:03:13
%S A343138 0,1,0,1,1,0,1,2,1,0,1,3,2,1,0,1,4,6,2,1,0,1,5,15,6,2,1,0,1,6,40,22,6,
%T A343138 2,1,0,1,7,104,71,22,6,2,1,0,1,8,273,240,86,22,6,2,1,0,1,9,714,816,
%U A343138 311,86,22,6,2,1,0,1,10,1870,2752,1152,342,86,22,6,2,1,0
%N A343138 Array A(k, n) read by descending antidiagonals: A(k, n) = Sum_{m=0..n} F(k, m)^2, where F are the k-generalized Fibonacci numbers A092921.
%D A343138 Greg Dresden and Yichen Wang, Sums and convolutions of k-bonacci and k-Lucas numbers, draft 2021.
%H A343138 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
%F A343138 _Russell Jay Hendel_ gives the following representation, valid for k >= 2:
%F A343138 A(n, k) = Sum_{m=0..n} F(k, m)^2 = (1/(2*k-2)) * (2*Sum_{j=1..k-1}(j*Sum_{m=j+1..k} (m-k+1) * F(k, n+j) * F(k, n+m)) - Sum_{j=1..k}(A343125(k, j-1) * F(k, n+j)^2) + (k - 2)). - _Peter Luschny_, Apr 07 2021
%e A343138 Array starts:
%e A343138 n = 0 1 2 3 4 5 6 7 8 9 10
%e A343138 ------------------------------------------------------------
%e A343138 [k=0] 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... [A057427]
%e A343138 [k=1] 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... [A001477]
%e A343138 [k=2] 0, 1, 2, 6, 15, 40, 104, 273, 714, 1870, 4895, ... [A001654]
%e A343138 [k=3] 0, 1, 2, 6, 22, 71, 240, 816, 2752, 9313, 31514, ... [A107239]
%e A343138 [k=4] 0, 1, 2, 6, 22, 86, 311, 1152, 4288, 15952, 59216, ...
%e A343138 [k=5] 0, 1, 2, 6, 22, 86, 342, 1303, 5024, 19424, 75120, ...
%e A343138 [k=6] 0, 1, 2, 6, 22, 86, 342, 1366, 5335, 20960, 82464, ...
%e A343138 [k=7] 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21591, 85600, ...
%e A343138 [k=8] 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 86871, ...
%e A343138 [k=9] 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 87382, ...
%e A343138 [...]
%e A343138 [ oo] 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 87382, ... [A047849]
%e A343138 Note that the first parameter in A(k, n) refers to rows, and the second parameter refers to columns, as always. The usual naming convention for the indices is not adhered to because the row sequences are the sums of the squares of the k-bonacci numbers.
%p A343138 F := (k, n) -> (F(k, n) := `if`(n<2, n, add(F(k, n-j), j = 1..min(k, n)))):
%p A343138 A := (k, n) -> add(F(k, m)^2, m = 0..n):
%p A343138 seq(seq(A(k, n-k), k=0..n), n = 0..11);
%p A343138 # The following two functions implement _Russell Jay Hendel_'s formula for k >= 2:
%p A343138 T := (k, n) -> (n + 3)*(k - n) - 4:
%p A343138 H := (k, n) -> (2*add(j*add((m-k+1)*F(k, n+j)*F(k, n+m), m = j+1..k), j = 1..k-1)
%p A343138 - add(T(k, j-1)*F(k, n+j)^2, j = 1..k) + (k - 2))/(2*k - 2):
%p A343138 seq(lprint([k], seq(H(k, n), n = 0..11)), k=2..9); # _Peter Luschny_, Apr 07 2021
%t A343138 A343138[k_, len_] := Take[Accumulate[LinearRecurrence[PadLeft[{1}, k, 1], PadLeft[{1}, k], len + k]^2], -len - 2];
%t A343138 A343138[0, len_] := Table[Boole[n != 0], {n, 0, len}];
%t A343138 A343138[1, len_] := Table[n, {n, 0, len}];
%t A343138 (* Table: *) Table[A343138[k, 12], {k, 0, 9}]
%t A343138 (* Sequence / descending antidiagonals: *)
%t A343138 Table[Table[Take[A343138[j, 12], {k + 1 - j, k + 1 - j}], {j, 0, k}], {k, 0, 10}] // Flatten (* _Georg Fischer_, Apr 08 2021 *)
%Y A343138 Cf. A092921, A057427, A001477, A001654, A107239, A047849, A343125.
%K A343138 nonn,tabl,easy
%O A343138 0,8
%A A343138 _Peter Luschny_, Apr 06 2021