This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A039825 #39 Apr 13 2025 07:54:23
%S A039825 2,3,5,7,9,12,16,20,24,29,35,41,47,54,62,70,78,87,97,107,117,128,140,
%T A039825 152,164,177,191,205,219,234,250,266,282,299,317,335,353,372,392,412,
%U A039825 432,453,475,497,519,542,566,590,614,639
%N A039825 a(n) = floor((n^2 + n + 8) / 4).
%C A039825 Number of different coefficient values in expansion of Product_{i=1..n} (1 + q^2 + q^4 + ... + q^(2i)).
%C A039825 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
%C A039825 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
%H A039825 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-4,4,-3,1).
%F A039825 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
%F A039825 a(n) = A039823(n) + 1. - _Bruno Berselli_, Jul 25 2012
%F A039825 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
%o A039825 (Magma) [Floor((n^2+n+8)/4): n in [1..50]]; // _Bruno Berselli_, Jul 25 2012
%Y A039825 Cf. A039823, A054925.
%K A039825 nonn,easy
%O A039825 1,1
%A A039825 _Olivier Gérard_