A330640 a(n) is the number of partitions of n with Durfee square of size <= 2.

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 40, 51, 67, 83, 105, 127, 156, 185, 222, 259, 305, 351, 407, 463, 530, 597, 676, 755, 847, 939, 1045, 1151, 1272, 1393, 1530, 1667, 1821, 1975, 2147, 2319, 2510, 2701, 2912, 3123, 3355, 3587, 3841, 4095, 4372, 4649, 4950, 5251, 5577, 5903, 6255, 6607, 6986

Offset: 0

Author

Omar E. Pol, Dec 22 2019

Keywords

• nonn
• easy

Comments

This is an easy sequence since A006918 is the partial sums of A008805 (triangular numbers repeated).

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Crossrefs

Cf. A000041, A006918, A008805, A028310, A115994, A115720.

Programs

PARI

Vec((1 - x - x^2 + 2*x^3 - x^5 + x^6) / ((1 - x)^4*(1 + x)^2) + O(x^60)) \\ Colin Barker, Dec 31 2019

Formula

a(n) = A028310(n), 0 <= n <= 2.

a(n) = A028310(n) + A006918(n-3), n >= 3.

Or without A028310:

a(0) = 1, a(1) = 1, a(2) = 2.

a(n) = n + A006918(n-3), n >= 3.

From Colin Barker, Dec 31 2019: (Start)

G.f.: (1 - x - x^2 + 2*x^3 - x^5 + x^6) / ((1 - x)^4*(1 + x)^2).

a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n>6.

a(n) = (3 - 3*(-1)^n + (49+3*(-1)^n)*n - 6*n^2 + 2*n^3) / 48.

(End)