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 A035939 #43 Feb 16 2025 08:32:37
%S A035939 1,1,2,2,3,4,6,7,10,12,16,19,25,30,38,46,57,68,84,99,121,143,172,202,
%T A035939 242,283,336,392,462,537,630,729,851,982,1140,1312,1518,1741,2006,
%U A035939 2295,2635,3007,3442,3917,4470,5077,5776,6545,7429,8399,9510,10731
%N A035939 Number of partitions of n into parts not of the form 7k, 7k+3 or 7k-3. Also number of partitions such that the differences between parts at distance 2 are greater than 1.
%C A035939 Case k=3, i=3 of Gordon Theorem.
%C A035939 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A035939 G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.
%H A035939 G. C. Greubel, <a href="/A035939/b035939.txt">Table of n, a(n) for n = 0..1000</a>
%H A035939 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A035939 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A035939 Expansion of f(-x^3, -x^4) / f(-x, -x^2) in powers of x where f() is Ramanujan's two-variable theta function. - _Michael Somos_, Dec 29 2014
%F A035939 Euler transform of period 7 sequence [ 1, 1, 0, 0, 1, 1, 0, ...].- _Michael Somos_, Dec 29 2014
%F A035939 G.f.: 1 / (Product_{k>0} (1 - x^(7*k - 6)) * (1 - x^(7*k - 5)) * (1 - x^(7*k - 2)) * (1 - x^(7*k - 1))). - _Michael Somos_, Dec 29 2014
%F A035939 G.f.: (Product_{k>0} (1 + x^k)) * (Sum_{k>=0} x^(2*k^2) / (Product_{i=1..k} (1 - x^(2*i)) * (1 + x^(2*i-1)) * (1 + x^(2*i)))). - _Michael Somos_, Dec 31 2014
%F A035939 a(n) ~ 2^(1/4) * cos(Pi/14) * exp(2*Pi*sqrt(2*n/21)) / (3^(1/4) * 7^(3/4) * n^(3/4)). - _Vaclav Kotesovec_, Nov 13 2015
%e A035939 G.f. = 1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 7*x^7 + 10*x^8 + ...
%e A035939 G.f. = q^-1 + q^41 + 2*q^83 + 2*q^125 + 3*q^167 + 4*q^209 + 6*q^251 + ...
%p A035939 # See A035937 for GordonsTheorem
%p A035939 A035939_list := n -> GordonsTheorem([1, 1, 0, 0, 1, 1, 0], n):
%p A035939 A035939_list(40); # _Peter Luschny_, Jan 22 2012
%t A035939 a[ n_] := SeriesCoefficient[ 1 / Product[ (1 - x^(7 k - 1)) (1 - x^(7 k - 2)) (1 - x^(7 k - 5)) (1 - x^(7 k - 6)), {k, Ceiling[n/7]}], {x, 0, n}]; (* _Michael Somos_, Dec 29 2014 *)
%t A035939 a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x^1, x^7] QPochhammer[ x^2, x^7] QPochhammer[ x^5, x^7] QPochhammer[ x^6, x^7] ), {x, 0, n}]; (* _Michael Somos_, Dec 29 2014 *)
%o A035939 (Sage) # See A035937 for GordonsTheorem
%o A035939 def A035939_list(len) : return GordonsTheorem([1, 1, 0, 0, 1, 1, 0], len)
%o A035939 A035939_list(40) # _Peter Luschny_, Jan 22 2012
%o A035939 (PARI) {a(n) = my(A); if( n<0, 0, polcoeff( 1 / prod( k=1, n, 1 -[0, 1, 1, 0, 0, 1, 1][k%7 + 1] * x^k, 1 + x * O(x^n)), n))}; /* _Michael Somos_, Dec 29 2014 */
%Y A035939 Cf. A035937, A035938.
%K A035939 nonn,easy
%O A035939 0,3
%A A035939 _Olivier Gérard_
%E A035939 Missing a(0)=1 prepended by _Michael Somos_, Dec 29 2014
%E A035939 Name simplified by _George Beck_, Aug 27 2023