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 A272104 #48 Apr 30 2025 14:53:21
%S A272104 0,0,0,2,2,4,4,10,10,14,14,24,24,30,30,44,44,52,52,70,70,80,80,102,
%T A272104 102,114,114,140,140,154,154,184,184,200,200,234,234,252,252,290,290,
%U A272104 310,310,352,352,374,374,420,420,444,444,494,494,520,520,574,574,602
%N A272104 Sum of the even numbers among the larger parts of the partitions of n into two parts.
%C A272104 Essentially, repeated values of A152749.
%C A272104 Sum of the lengths of the distinct rectangles with even length and integer width such that L + W = n, W <= L. For example, a(10) = 14; the rectangles are 2 X 8 and 4 X 6, so 8 + 6 = 14. - _Wesley Ivan Hurt_, Nov 04 2017
%H A272104 Colin Barker, <a href="/A272104/b272104.txt">Table of n, a(n) for n = 0..1000</a>
%H A272104 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H A272104 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,2,-2,0,0,-1,1).
%F A272104 a(n) = (1 + 3*(2n-3-(-1)^n)/2 + 3*(2n-3-(-1)^n)^2/8 + (2n-1-(-1)^n) * (-1)^((2n+1-(-1)^n)/4)/2) / 8.
%F A272104 a(n) = Sum_{i=ceiling(n/2)..n-1} i * (i+1 mod 2).
%F A272104 a(n) = Sum_{i=1..floor(n/2)} (n-i) * (n-i+1 mod 2).
%F A272104 a(2n+1) = a(2n+2) = A152749(n) = 2*A001318(n).
%F A272104 G.f.: 2*x^3*(1-x+x^2)*(1+x+x^2) / ((1-x)^3*(1+x)^2*(1+x^2)^2). - _Colin Barker_, Apr 20 2016
%F A272104 From _Wesley Ivan Hurt_, Apr 22 2016, Apr 23 2016: (Start)
%F A272104 a(2n+2)-a(2n) = A109043(n) = 2*A026741(n).
%F A272104 a(4n) = A049450(n) = 2*A000326(n),
%F A272104 a(8n) = A126964(n) = 2*A049452(n),
%F A272104 a(12n) = 2*A268351(n).
%F A272104 a(n+1) = A001318(n) - A272212(n+1). (End)
%F A272104 E.g.f.: ((2 + 3*x*(1 + x))*cosh(x) - 2*(cos(x) + x*cos(x) + x*sin(x)) + (-1 + 3*(-1 + x)*x)*sinh(x))/16. - _Ilya Gutkovskiy_, Apr 29 2016
%e A272104 a(5) = 4; the partitions of 5 into 2 parts are (4,1),(3,2) and the sum of the larger even parts is 4.
%e A272104 a(6) = 4; the partitions of 6 into 2 parts are (5,1),(4,2),(3,3) and the sum of the larger even parts is also 4.
%p A272104 A272104:=n->(1+3*(2*n-3-(-1)^n)/2+3*(2*n-3-(-1)^n)^2/8+(2*n-1-(-1)^n)*(-1)^((2*n+1-(-1)^n)/4)/2)/8: seq(A272104(n), n=0..100);
%t A272104 Table[(1 + 3(2n-3-(-1)^n)/2 + 3(2n-3-(-1)^n)^2/8 + (2n-1-(-1)^n) * (-1)^((2n+1-(-1)^n)/4)/2) / 8, {n, 0, 50}]
%t A272104 Table[Total@ Map[First, IntegerPartitions[n, {2}] /. {k_, _} /; OddQ@ k -> Nothing], {n, 0, 57}] (* _Michael De Vlieger_, Apr 20 2016, Version 10.2 *)
%o A272104 (Magma) [(1+3*(2*n-3-(-1)^n)/2+3*(2*n-3-(-1)^n)^2/8+(2*n-1-(-1)^n)*(-1)^((2*n+1-(-1)^n) div 4)/2)/8 : n in [0..50]];
%o A272104 (PARI) concat(vector(3), Vec(2*x^3*(1-x+x^2)*(1+x+x^2)/((1-x)^3*(1+x)^2*(1+x^2)^2) + O(x^50))) \\ _Colin Barker_, Apr 20 2016
%Y A272104 Cf. A000326, A001318, A026741, A049450, A049452, A109043, A126964, A152749, A268351, A272212.
%K A272104 nonn,easy
%O A272104 0,4
%A A272104 _Wesley Ivan Hurt_, Apr 20 2016