OFFSET
1,3
COMMENTS
This array is used to compute A269254: A269254(n) = least k such that A(n,k) is a prime, or -1 if no such k exists.
For detailed theory, see [Hone]. - L. Edson Jeffery, Feb 09 2018
The array can be extended to k<0 with A(n, k) = -A(n, -k-1) for all k in Z. - Michael Somos, Jun 19 2023
LINKS
Robert Price, Table of n, a(n) for n = 1..5050
Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.
FORMULA
A(n,0) = 1, A(n,1) = n + 1, A(n,k) = n*A(n,k-1) - A(n,k-2), n >= 1, k >= 2.
G.f. for row n: (1 + x)/(1 - n*x + x^2), n >= 1.
A(n, k) = B(-n, k) where B = A299045. - Michael Somos, Jun 19 2023
EXAMPLE
Array begins:
1 2 1 -1 -2 -1 1 2 1 -1
1 3 5 7 9 11 13 15 17 19
1 4 11 29 76 199 521 1364 3571 9349
1 5 19 71 265 989 3691 13775 51409 191861
1 6 29 139 666 3191 15289 73254 350981 1681651
1 7 41 239 1393 8119 47321 275807 1607521 9369319
1 8 55 377 2584 17711 121393 832040 5702887 39088169
1 9 71 559 4401 34649 272791 2147679 16908641 133121449
1 10 89 791 7030 62479 555281 4935050 43860169 389806471
1 11 109 1079 10681 105731 1046629 10360559 102558961 1015229051
MATHEMATICA
(* Array: *)
Grid[Table[LinearRecurrence[{n, -1}, {1, 1 + n}, 10], {n, 10}]]
(* Array antidiagonals flattened (gives this sequence): *)
PROG
(PARI) {A(n, k) = sum(j=0, k, (-1)^(j\2)*binomial(k-(j+1)\2, j\2)*n^(k-j))}; /* Michael Somos, Jun 19 2023 */
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
STATUS
approved