A115266 Row sums of correlation triangle for floor((n+3)/3).

1, 2, 4, 8, 13, 20, 31, 44, 61, 84, 111, 144, 186, 234, 291, 360, 438, 528, 634, 752, 886, 1040, 1210, 1400, 1615, 1850, 2110, 2400, 2715, 3060, 3441, 3852, 4299, 4788, 5313, 5880, 6496, 7154, 7861, 8624, 9436, 10304, 11236, 12224, 13276, 14400, 15588

Offset: 0

Author

Paul Barry, Jan 18 2006

Keywords

• easy
• nonn

Comments

Row sums of number triangle A115265.

Sum of the smallest parts in all the partitions of k into 3 parts such that 3 <= k <= n+3. - Wesley Ivan Hurt, Nov 03 2021

a(n) is the number of partitions of n into parts 1, 2, and 3, where there are two sorts of part 1 and 3 (see formula by Mamuka Jibladze). - Joerg Arndt, Apr 09 2025

Links

• Index entries for linear recurrences with constant coefficients, signature (2,0,0,-3,0,3,0,0,-2,1).

Crossrefs

Cf. A115265.

Programs

Mathematica

T[n_, k_] := Sum[Boole[j <= k] * Floor[(k - j + 3)/3] * Boole[j <= n - k] * Floor[(n - k - j + 3)/3], {j, 0, n}]; a[n_] := Sum[T[n, k], {k, 0, n}]; Table[a[n], {n, 0, 46}] (* Jean-François Alcover, Jul 15 2017 *)
LinearRecurrence[{2,0,0,-3,0,3,0,0,-2,1},{1,2,4,8,13,20,31,44,61,84},50] (* Harvey P. Dale, Nov 20 2021 *)

Formula

G.f.: (1+x+x^2)^2/((1-x^3)^4*(1-x^2)).

a(n) = Sum_{k=0..n} Sum_{j=0..n} [j<=k]*floor((k-j+3)/3)*[j<=n-k]*floor((n-k-j+3)/3).

From Wesley Ivan Hurt, Nov 03 2021: (Start)

a(n) = Sum_{m=1..n+3} Sum_{k=1..floor(m/3)} Sum_{i=k..floor((m-k)/2)} k.

a(n) = 2*a(n-1)-3*a(n-4)+3*a(n-6)-2*a(n-9)+a(n-10). (End)

G.f.: 1/((1-x)^2*(1-x^2)*(1-x^3)^2). - Mamuka Jibladze, Apr 09 2025

a(n) = (2*n^4 + 40*n^3 + 276*n^2 + 792*n + 837 + 27*(-1)^n - 32*(n+5)*(n mod 3))/864. - Hoang Xuan Thanh, Jun 11 2025