A357990 Square array T(n, k), n >= 0, k > 0, read by antidiagonals, where T(0, k) = 1 for k > 0 and where T(n, k) = R(n, k+1) - R(n, k) for n > 0, k > 0. Here R(n, k) = T(A053645(n), k)*k^(A290255(n) + 1).

1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 7, 1, 7, 1, 1, 3, 19, 1, 9, 1, 1, 7, 5, 37, 1, 11, 1, 1, 1, 11, 7, 61, 1, 13, 1, 1, 15, 1, 15, 9, 91, 1, 15, 1, 1, 7, 65, 1, 19, 11, 127, 1, 17, 1, 1, 17, 19, 175, 1, 23, 13, 169, 1, 19, 1, 1, 3, 43, 37, 369, 1, 27, 15, 217, 1, 21

Offset: 0

Author

Mikhail Kurkov, Nov 20 2022

Keywords

• nonn
• base
• tabl

Examples

			Square array begins:
   1,  1,   1,   1,   1,    1,    1,    1, ...
   1,  1,   1,   1,   1,    1,    1,    1, ...
   3,  5,   7,   9,  11,   13,   15,   17, ...
   1,  1,   1,   1,   1,    1,    1,    1, ...
   7, 19,  37,  61,  91,  127,  169,  217, ...
   3,  5,   7,   9,  11,   13,   15,   17, ...
   7, 11,  15,  19,  23,   27,   31,   35, ...
   1,  1,   1,   1,   1,    1,    1,    1, ...
  15, 65, 175, 369, 671, 1105, 1695, 2465, ...
		

Crossrefs

Cf. A000120, A053645, A290255, A329369.

Programs

PARI

R(n,k)=my(L=logint(n, 2), A=n - 2^L); T(A, k)*k^(L - if(A>0, logint(A, 2) + 1) + 1)
T(n,k)=if(n==0, 1, R(n, k+1) - R(n, k))

PARI

T(n, k) = my(A = 2*n+1, B, C, v1, v2); v1 = []; while(A > 0, B=valuation(A, 2); v1=concat(v1, B+1); A \= 2^(B+1)); v1 = Vecrev(v1); A = #v1; v2 = vector(A, i, 1); for(i=1, A-1, B = A-i; for(j=1, B, C = B-j+k+1; v2[j] = v2[j]*C^v1[B] - v2[j+1]*(C-1)^v1[B])); v2[1] \\ Mikhail Kurkov, Apr 30 2024

Formula

Conjecture: T(n, 1) = A329369(n).