A343138 - OEIS

%I #20 Apr 08 2021 11:03:13

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

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

%H 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 _Russell Jay Hendel_ gives the following representation, valid for k >= 2:

%F 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 Array starts:

%e   n = 0  1  2  3   4   5    6     7     8     9      10

%e ------------------------------------------------------------

%e [k=0] 0, 1, 1, 1,  1,  1,   1,    1,    1,     1,     1, ...  [A057427]

%e [k=1] 0, 1, 2, 3,  4,  5,   6,    7,    8,     9,    10, ...  [A001477]

%e [k=2] 0, 1, 2, 6, 15, 40, 104,  273,  714,  1870,  4895, ...  [A001654]

%e [k=3] 0, 1, 2, 6, 22, 71, 240,  816, 2752,  9313, 31514, ...  [A107239]

%e [k=4] 0, 1, 2, 6, 22, 86, 311, 1152, 4288, 15952, 59216, ...

%e [k=5] 0, 1, 2, 6, 22, 86, 342, 1303, 5024, 19424, 75120, ...

%e [k=6] 0, 1, 2, 6, 22, 86, 342, 1366, 5335, 20960, 82464, ...

%e [k=7] 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21591, 85600, ...

%e [k=8] 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 86871, ...

%e [k=9] 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 87382, ...

%e [...]

%e [ oo] 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 87382, ...  [A047849]

%e 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 F := (k, n) -> (F(k, n) := `if`(n<2, n, add(F(k, n-j), j = 1..min(k, n)))):

%p A := (k, n) -> add(F(k, m)^2, m = 0..n):

%p seq(seq(A(k, n-k), k=0..n), n = 0..11);

%p # The following two functions implement _Russell Jay Hendel_'s formula for k >= 2:

%p T := (k, n) -> (n + 3)*(k - n) - 4:

%p 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 - add(T(k, j-1)*F(k, n+j)^2, j = 1..k) + (k - 2))/(2*k - 2):

%p seq(lprint([k], seq(H(k, n), n = 0..11)), k=2..9); # _Peter Luschny_, Apr 07 2021

%t A343138[k_, len_] := Take[Accumulate[LinearRecurrence[PadLeft[{1}, k, 1], PadLeft[{1}, k], len + k]^2], -len - 2];

%t A343138[0, len_] := Table[Boole[n != 0], {n, 0, len}];

%t A343138[1, len_] := Table[n, {n, 0, len}];

%t (* Table: *) Table[A343138[k, 12], {k, 0, 9}]

%t (* Sequence / descending antidiagonals: *)

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

%K nonn,tabl,easy

%O 0,8

%A _Peter Luschny_, Apr 06 2021