A272212 Sum of the odd numbers among the larger parts of the partitions of n into two parts.

0, 0, 1, 0, 3, 3, 8, 5, 12, 12, 21, 16, 27, 27, 40, 33, 48, 48, 65, 56, 75, 75, 96, 85, 108, 108, 133, 120, 147, 147, 176, 161, 192, 192, 225, 208, 243, 243, 280, 261, 300, 300, 341, 320, 363, 363, 408, 385, 432, 432, 481, 456, 507, 507, 560, 533, 588, 588

Offset: 0

Author

Wesley Ivan Hurt, Apr 22 2016

Keywords

• nonn
• easy

Comments

Sum of the lengths of the distinct rectangles with odd length and integer width such that L + W = n, W <= L. For example, a(10) = 21; the rectangles are 1 X 9, 3 X 7 and 5 X 5, so 9 + 7 + 5 = 21. - Wesley Ivan Hurt, Nov 18 2017

Examples

			a(5) = 3; the partitions of 5 into two parts are (4,1),(3,2) and the sum of the odd numbers among the larger parts is 3.
a(6) = 8; the partitions of 6 into two parts are (5,1),(4,2),(3,3) and the sum of the odd numbers among the larger parts is 5+3 = 8.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for sequences related to partitions
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1).

Crossrefs

Cf. A001318, A272104.

Programs

Magma

[(6*n^2-6*n+1+(10*n-5)*(-1)^n-(4*n-2-2*(-1)^n)*(-1)^((2*n+1-(-1)^n) div 4))/32: n in [0..100]];

Maple

A272212:=n->(6*n^2-6*n+1+(10*n-5)*(-1)^n-(4*n-2-2*(-1)^n)*(-1)^((2*n+1-(-1)^n)/4))/32: seq(A272212(n), n=0..100);

Mathematica

Table[(6n^2-6n+1+(10n-5)(-1)^n-(4n-2-2(-1)^n)(-1)^((2n+1-(-1)^n)/4))/32, {n,0,100}]
Table[Total@ Flatten[First /@ IntegerPartitions[n, {2}] /. k_ /; EvenQ@ k -> Nothing], {n, 0, 60}] (* Michael De Vlieger, Apr 26 2016, Version 10.2 *)
f[n_] := Sum[(n - i) Mod[n - i, 2], {i, Floor[n/2]}]; Array[f, 58, 0] (* Robert G. Wilson v, Dec 11 2017 *)
CoefficientList[ Series[x^2 (1 +x +x^2) (1 -2x +4x^2 -2x^3 +x^4)/((1 -x)^3 (1 +x)^2 (1 +x^2)^2), {x, 0, 57}], x] (* Robert G. Wilson v, Dec 13 2017 *)
Table[Total[Select[IntegerPartitions[n,{2}][[All,1]],OddQ]],{n,0,60}] (* Harvey P. Dale, Jun 29 2018 *)

PARI

concat(vector(2), Vec(x^2*(1+x+x^2)*(1-2*x+4*x^2-2*x^3+x^4)/((1-x)^3*(1+x)^2*(1+x^2)^2) + O(x^50))) \\ Colin Barker, Apr 23 2016

PARI

a(n)=([0,1,0,0,0,0,0,0,0; 0,0,1,0,0,0,0,0,0; 0,0,0,1,0,0,0,0,0; 0,0,0,0,1,0,0,0,0; 0,0,0,0,0,1,0,0,0; 0,0,0,0,0,0,1,0,0; 0,0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,0,1; 1,-1,0,0,-2,2,0,0,1]^n*[0;0;1;0;3;3;8;5;12])[1,1] \\ Charles R Greathouse IV, Apr 29 2016

Formula

a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5) - a(n-8) + a(n-9) for n > 8.

a(n) = (6*n^2 - 6*n + 1 + (10*n-5)*(-1)^n - (4*n - 2 - 2*(-1)^n)*(-1)^((2*n+1 - (-1)^n)/4))/32.

G.f.: x^2*(1 + x + x^2)*(1 - 2*x + 4*x^2 - 2*x^3 + x^4) / ((1-x)^3*(1+x)^2*(1+x^2)^2). - Colin Barker, Apr 22 2016

a(n+1) = A001318(n) - A272104(n+1). - Wesley Ivan Hurt, Apr 22 2016

E.g.f.: ((-5*(1 + 2*x))*exp(-x) + (1 + 6*x^2)*exp(x) + 4*(1 + x)*cos(x) + 4*x*sin(x))/32. - Ilya Gutkovskiy, Apr 27 2016

a(n) = Sum_{i=1..floor(n/2)} (n-i) * ((n-i) mod 2). - Wesley Ivan Hurt, Dec 06 2017