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 A097451 #32 Jun 05 2025 20:38:11
%S A097451 1,0,1,1,2,1,3,2,5,4,7,6,11,9,15,14,22,20,31,29,43,41,58,57,80,78,106,
%T A097451 107,142,143,188,191,247,253,321,332,418,432,537,561,690,721,880,924,
%U A097451 1118,1178,1412,1493,1781,1884,2231,2370,2789,2965,3472,3698,4309,4596
%N A097451 Number of partitions of n into parts congruent to {2, 3, 4} mod 6.
%C A097451 Number of partitions of n in which no part is 1, no part appears more than twice and no two parts differ by 1. Example: a(6)=3 because we have [6],[4,2] and [3,3]. - _Emeric Deutsch_, Feb 16 2006
%C A097451 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A097451 G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, Exercise 7.9.
%H A097451 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A097451 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A097451 Euler transform of period 6 sequence [ 0, 1, 1, 1, 0, 0, ...].
%F A097451 G.f.: 1/Product_{j>=0} ((1-x^(2+6j))(1-x^(3+6j))(1-x^(4+6j))). - _Emeric Deutsch_, Feb 16 2006
%F A097451 Expansion of psi(x^3) / f(-x^2) in powers of x where psi(), f() are Ramanujan theta functions. - _Michael Somos_, Sep 24 2013
%F A097451 Expansion of q^(-7/24) * eta(q^6)^2 / (eta(q^2) * eta(q^3)) in powers of q. - _Michael Somos_, Sep 24 2013
%F A097451 a(n) ~ exp(Pi*sqrt(n/3)) / (4*3^(3/4)*n^(3/4)). - _Vaclav Kotesovec_, Aug 30 2015
%F A097451 Expansion of f(-x, -x^5) / f(-x, -x^2) in powers of x where f(, ) is Ramanujan's general theta function. - _Michael Somos_, Oct 06 2015
%e A097451 a(8)=5 because we have [8],[44],[422],[332] and [2222].
%e A097451 G.f. = 1 + x^2 + x^3 + 2*x^4 + x^5 + 3*x^6 + 2*x^7 + 5*x^8 + 4*x^9 + ...
%e A097451 G.f. = q^7 + q^55 + q^79 + 2*q^103 + q^127 + 3*q^151 + 2*q^175 + 5*q^199 + ...
%p A097451 g:=1/product((1-x^(2+6*j))*(1-x^(3+6*j))*(1-x^(4+6*j)),j=0..15): gser:=series(g,x=0,75): seq(coeff(gser,x,n),n=0..67); # _Emeric Deutsch_, Feb 16 2006
%t A097451 a[ n_] := SeriesCoefficient[ 1 / Product[ 1 - Boole[ OddQ[ Quotient[ k + 1, 3]]] x^k, {k, n}], {x, 0, n}]; (* _Michael Somos_, Sep 24 2013 *)
%t A097451 a[ n_] := SeriesCoefficient[ QPochhammer[ -x^3, x^3] QPochhammer[ x^6] / QPochhammer[ x^2], {x, 0, n}]; (* _Michael Somos_, Sep 24 2013 *)
%o A097451 (Haskell)
%o A097451 a097451 n = p a047228_list n where
%o A097451 p _ 0 = 1
%o A097451 p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
%o A097451 -- _Reinhard Zumkeller_, Nov 16 2012
%o A097451 (PARI) {a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - ( (k+1)\3 % 2) * x^k, 1 + x * O(x^n)), n))}; /* _Michael Somos_, Sep 24 2013 */
%o A097451 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A)), n))}; /* _Michael Somos_, Sep 24 2013 */
%Y A097451 Cf. A070047.
%Y A097451 Cf. A047228, A056970, A096981, A098884.
%K A097451 easy,nonn
%O A097451 0,5
%A A097451 _Vladeta Jovovic_, Aug 23 2004
%E A097451 More terms from _Emeric Deutsch_, Feb 16 2006