A369518 Array read by downward antidiagonals: A(n,k) = Sum_{j=0..k + (k mod 3) + 1} A(n-1,j) with A(0,k) = 1, n >= 0, k >= 0.

1, 1, 2, 1, 4, 6, 1, 6, 17, 23, 1, 5, 33, 80, 103, 1, 7, 24, 184, 408, 511, 1, 9, 41, 121, 1054, 2208, 2719, 1, 8, 63, 235, 643, 6196, 12486, 15205, 1, 10, 51, 411, 1363, 3571, 37244, 72992, 88197, 1, 12, 74, 309, 2625, 8057, 20543, 228092, 437821, 526018

Offset

0,3

Links

Table of n, a(n) for n=0..54.

Terence Tao, Elegant recursion for A301897, answer to question on MathOverflow (2023).

Formula

A(n,3k) = A(n,3k-1) - A(n-1,3k+2), A(n,3k+1) = A(n,3k) + A(n-1,3k+2) + A(n-1,3k+3), A(n,3k+2) = A(n, 3k+1) + A(n-1,3k+4) + A(n-1,3k+5) with A(n,0) = A(n-1,0) + A(n-1,1), A(0,k) = 1. - Mikhail Kurkov, Nov 24 2024

Example

Array begins:
==================================================
n\k|    0     1     2     3     4      5     6 ...
---+----------------------------------------------
0  |    1     1     1     1     1      1     1 ...
1  |    2     4     6     5     7      9     8 ...
2  |    6    17    33    24    41     63    51 ...
3  |   23    80   184   121   235    411   309 ...
4  |  103   408  1054   643  1363   2625  1861 ...
5  |  511  2208  6196  3571  8057  16701 11296 ...
6  | 2719 12486 37244 20543 48540 106560 69376 ...
  ...

Prog

(PARI)
A(m, n=m)={my(r=vectorv(m+1), v=vector(n+3*m+1, k, 1)); r[1] = v[1..n+1];
for(i=1, m, v=vector(#v-3, k, sum(j=1, k + (k-1)%3 + 1, v[j])); r[1+i] = v[1..n+1]); Mat(r)}
{ A(6) }

Crossrefs

Column k=0 is A301897 (with different offset).

Sequence in context: A208759 A033877 A059369 * A199530 A208765 A232335

Adjacent sequences: A369515 A369516 A369517 * A369519 A369520 A369521

Keyword

nonn,tabl

Author

Mikhail Kurkov, Jan 25 2024

Status

approved