OFFSET
0,4
COMMENTS
Essentially, repeated values of A152749.
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
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1).
FORMULA
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.
a(n) = Sum_{i=ceiling(n/2)..n-1} i * (i+1 mod 2).
a(n) = Sum_{i=1..floor(n/2)} (n-i) * (n-i+1 mod 2).
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
From Wesley Ivan Hurt, Apr 22 2016, Apr 23 2016: (Start)
a(12n) = 2*A268351(n).
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
EXAMPLE
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.
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.
MAPLE
MATHEMATICA
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}]
Table[Total@ Map[First, IntegerPartitions[n, {2}] /. {k_, _} /; OddQ@ k -> Nothing], {n, 0, 57}] (* Michael De Vlieger, Apr 20 2016, Version 10.2 *)
PROG
(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]]
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Apr 20 2016
STATUS
approved