A348543 Number of partitions of n into 3 parts with at least 1 odd part.

0, 0, 1, 1, 2, 2, 4, 4, 7, 6, 10, 9, 14, 12, 19, 16, 24, 20, 30, 25, 37, 30, 44, 36, 52, 42, 61, 49, 70, 56, 80, 64, 91, 72, 102, 81, 114, 90, 127, 100, 140, 110, 154, 121, 169, 132, 184, 144, 200, 156, 217, 169, 234, 182, 252, 196, 271, 210, 290, 225, 310, 240, 331, 256, 352

Offset: 1

Author

Wesley Ivan Hurt, Oct 21 2021

Keywords

• nonn
• easy

Links

• Index entries for sequences related to partitions
• Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,0,0,-1,0,-1,0,1).

Crossrefs

Cf. A069905.

Programs

Mathematica

a[n_] := Sum[1 - Mod[j + 1, 2] * Mod[i + 1, 2] * Mod[n - i - j + 1, 2], {j, 1, Floor[n/3]}, {i, j, Floor[(n - j)/2]}]; Array[a, 100] (* Amiram Eldar, Oct 22 2021 *)

Formula

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} (1-((j+1) mod 2)*((i+1) mod 2)*((n-i-j+1) mod 2)).

G.f.: -x^3*(x^6+x^5+x^4+x^3+x^2+x+1)/(x^12-x^10-x^8+x^4+x^2-1). - Alois P. Heinz, Oct 22 2021

a(n) = a(n-2)+a(n-4)-a(n-8)-a(n-10)+a(n-12). - Wesley Ivan Hurt, Nov 18 2021