A035679 Number of partitions of n into parts 8k+1 and 8k+2 with at least one part of each type.

0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 7, 7, 10, 10, 13, 13, 16, 16, 22, 23, 30, 31, 38, 39, 46, 47, 58, 61, 75, 78, 93, 96, 111, 114, 134, 141, 167, 176, 204, 213, 242, 251, 286, 301, 346, 365, 416, 436, 489, 509, 570, 599, 676, 714, 802, 844, 937, 980, 1083, 1138, 1265

Offset: 1

Author

Olivier Gérard

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..5000

Programs

Maple

b:= proc(n, i, t, s) option remember; `if`(n=0, t*s, `if`(i<1, 0,
       b(n, i-1, t, s)+(h-> `if`(h in {1, 2}, add(b(n-i*j, i-1,
      `if`(h=1, 1, t), `if`(h=2, 1, s)), j=1..n/i), 0))(irem(i, 8))))
    end:
a:= n-> b(n$2, 0$2):
seq(a(n), n=1..75);  # Alois P. Heinz, Sep 04 2020

Mathematica

b[n_, i_, t_, s_] := b[n, i, t, s] = If[n == 0, t s, If[i < 1, 0, b[n, i - 1, t, s] + Function[h, If[h == 1 || h == 2, Sum[b[n - i j, i - 1, If[h == 1, 1, t], If[h == 2, 1, s]], {j, 1, n/i}], 0]][Mod[i, 8]]]];
a[n_] := b[n, n, 0, 0];
Array[a, 75] (* Jean-François Alcover, Oct 31 2020, after Alois P. Heinz *)