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 A307370 #20 Feb 22 2024 09:09:24
%S A307370 0,0,0,1,2,4,4,6,7,7,10,10,11,12,15,13,17,16,19,18,22,19,25,22,26,24,
%T A307370 30,25,32,28,34,30,37,31,40,34,41,36,45,37,47,40,49,42,52,43,55,46,56,
%U A307370 48,60,49,62,52,64,54,67,55,70,58,71,60,75,61,77,64,79,66
%N A307370 Number of integer partitions of n with 2 distinct parts, none appearing more than twice.
%C A307370 The Heinz numbers of these partitions appear to be given by A296205.
%H A307370 Colin Barker, <a href="/A307370/b307370.txt">Table of n, a(n) for n = 0..1000</a>
%H A307370 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (-1,0,1,2,1,0,-1,-1).
%F A307370 From _Colin Barker_, Apr 08 2019: (Start)
%F A307370 G.f.: x^3*(1 + 3*x + 6*x^2 + 7*x^3 + 6*x^4 + 4*x^5) / ((1 - x)^2*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).
%F A307370 a(n) = -a(n-1) + a(n-3) + 2*a(n-4) + a(n-5) - a(n-7) - a(n-8) for n>8. (End)
%F A307370 a(n) = (27*n + 3*(n - 7)*(-1)^n - 53 - 6*A056594(n) + 8*A061347(n))/24 for n > 0. - _Stefano Spezia_, Feb 20 2024
%e A307370 The a(3) = 1 through a(10) = 10 partitions:
%e A307370 (21) (31) (32) (42) (43) (53) (54) (64)
%e A307370 (211) (41) (51) (52) (62) (63) (73)
%e A307370 (221) (411) (61) (71) (72) (82)
%e A307370 (311) (2211) (322) (332) (81) (91)
%e A307370 (331) (422) (441) (433)
%e A307370 (511) (611) (522) (442)
%e A307370 (3311) (711) (622)
%e A307370 (811)
%e A307370 (3322)
%e A307370 (4411)
%t A307370 Table[Length[Select[IntegerPartitions[n],Length[Union[#]]==2&&Max@@Length/@Split[#]<=2&]],{n,0,30}]
%o A307370 (PARI) concat([0,0,0], Vec(x^3*(1 + 3*x + 6*x^2 + 7*x^3 + 6*x^4 + 4*x^5) / ((1 - x)^2*(1 + x)^2*(1 + x^2)*(1 + x + x^2)) + O(x^40))) \\ _Colin Barker_, Apr 08 2019
%Y A307370 Cf. A000041, A000726, A002133, A004709, A006918, A007774, A117485, A296205, A325168.
%Y A307370 Cf. A056594, A061347.
%K A307370 nonn,easy
%O A307370 0,5
%A A307370 _Gus Wiseman_, Apr 05 2019