A039825 a(n) = floor((n^2 + n + 8) / 4).

2, 3, 5, 7, 9, 12, 16, 20, 24, 29, 35, 41, 47, 54, 62, 70, 78, 87, 97, 107, 117, 128, 140, 152, 164, 177, 191, 205, 219, 234, 250, 266, 282, 299, 317, 335, 353, 372, 392, 412, 432, 453, 475, 497, 519, 542, 566, 590, 614, 639

Offset: 1

Author

Olivier Gérard

Keywords

• nonn
• easy

Comments

Number of different coefficient values in expansion of Product_{i=1..n} (1 + q^2 + q^4 + ... + q^(2i)).

The given terms have a second difference that is periodic with the period 1, 0, 0, 1, ... of length 4, an implicit recurrence. - John W. Layman, Jan 23 2001

Conjecturally, apart from the first term, the sequence terms are the exponents in the expansion of Sum_{n >= 1} q^(3*n) * (Product_{k >= 2*n} 1 - q^k) = q^3 - q^5 - q^7 + q^9 + q^12 - q^16 - q^20 + + - - .... Cf. A054925. - Peter Bala, Apr 13 2025

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Crossrefs

Cf. A039823, A054925.

Programs

Magma

[Floor((n^2+n+8)/4): n in [1..50]]; // Bruno Berselli, Jul 25 2012

Formula

O.g.f.: -x*(2*x^4 - 4*x^3 + 4*x^2 - 3*x + 2)/((x-1)^3*(x^2+1)). - R. J. Mathar, Dec 05 2007

a(n) = A039823(n) + 1. - Bruno Berselli, Jul 25 2012

a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5). - Wesley Ivan Hurt, May 08 2022