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.

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, 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, 311, 86, 22, 6, 2, 1, 0, 1, 10, 1870, 2752, 1152, 342, 86, 22, 6, 2, 1, 0

Offset

0,8

References

Greg Dresden and Yichen Wang, Sums and convolutions of k-bonacci and k-Lucas numbers, draft 2021.

Links

Table of n, a(n) for n=0..77.

Russell Jay Hendel, Sums of Squares: Methods for Proving Identity Families, arXiv:2103.16756 [math.NT], 2021

Formula

Russell Jay Hendel gives the following representation, valid for k >= 2:

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

Example

Array starts:
  n = 0  1  2  3   4   5    6     7     8     9      10
------------------------------------------------------------
[k=0] 0, 1, 1, 1,  1,  1,   1,    1,    1,     1,     1, ...  [A057427]
[k=1] 0, 1, 2, 3,  4,  5,   6,    7,    8,     9,    10, ...  [A001477]
[k=2] 0, 1, 2, 6, 15, 40, 104,  273,  714,  1870,  4895, ...  [A001654]
[k=3] 0, 1, 2, 6, 22, 71, 240,  816, 2752,  9313, 31514, ...  [A107239]
[k=4] 0, 1, 2, 6, 22, 86, 311, 1152, 4288, 15952, 59216, ...
[k=5] 0, 1, 2, 6, 22, 86, 342, 1303, 5024, 19424, 75120, ...
[k=6] 0, 1, 2, 6, 22, 86, 342, 1366, 5335, 20960, 82464, ...
[k=7] 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21591, 85600, ...
[k=8] 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 86871, ...
[k=9] 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 87382, ...
[...]
[ oo] 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 87382, ...  [A047849]
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.

Maple

F := (k, n) -> (F(k, n) := `if`(n<2, n, add(F(k, n-j), j = 1..min(k, n)))):
A := (k, n) -> add(F(k, m)^2, m = 0..n):
seq(seq(A(k, n-k), k=0..n), n = 0..11);
# The following two functions implement Russell Jay Hendel's formula for k >= 2:
T := (k, n) -> (n + 3)*(k - n) - 4:
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)
- add(T(k, j-1)*F(k, n+j)^2, j = 1..k) + (k - 2))/(2*k - 2):
seq(lprint([k], seq(H(k, n), n = 0..11)), k=2..9); # Peter Luschny, Apr 07 2021

Mathematica

A343138[k_, len_] := Take[Accumulate[LinearRecurrence[PadLeft[{1}, k, 1], PadLeft[{1}, k], len + k]^2], -len - 2];
A343138[0, len_] := Table[Boole[n != 0], {n, 0, len}];
A343138[1, len_] := Table[n, {n, 0, len}];
(* Table: *) Table[A343138[k, 12], {k, 0, 9}]
(* Sequence / descending antidiagonals: *)
Table[Table[Take[A343138[j, 12], {k + 1 - j, k + 1 - j}], {j, 0, k}], {k, 0, 10}] // Flatten (* Georg Fischer, Apr 08 2021 *)

Crossrefs

Cf. A092921, A057427, A001477, A001654, A107239, A047849, A343125.

Sequence in context: A360763 A332670 A118344 * A119270 A267109 A341524

Adjacent sequences: A343135 A343136 A343137 * A343139 A343140 A343141

Keyword

nonn,tabl,easy

Author

Peter Luschny, Apr 06 2021

Status

approved