A115271 Partial sums of floor((n+4)/4)^2.

1, 2, 3, 4, 8, 12, 16, 20, 29, 38, 47, 56, 72, 88, 104, 120, 145, 170, 195, 220, 256, 292, 328, 364, 413, 462, 511, 560, 624, 688, 752, 816, 897, 978, 1059, 1140, 1240, 1340, 1440, 1540, 1661, 1782, 1903, 2024, 2168, 2312, 2456, 2600, 2769, 2938

Offset: 0

Author

Paul Barry, Jan 18 2006

Keywords

• easy
• nonn

Comments

Central coefficients of number triangle A115268.

Links

• Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 2, -4, 2, 0, -1, 2, -1).

Programs

Mathematica

Accumulate[Table[Floor[(n+4)/4]^2,{n,0,50}]] (* or *) LinearRecurrence[ {2,-1,0,2,-4,2,0,-1,2,-1},{1,2,3,4,8,12,16,20,29,38},50] (* Harvey P. Dale, Jun 07 2011 *)

Formula

G.f.: (1+x^4)/((1-x)^2*(1-x^4)^2).

a(n) = sum{k=0..n, floor((k+4)/2)^2}.

a(n) = A115268(2n, n).

a(n) = 2*a(n-1)-a(n-2)+2*a(n-4)-4*a(n-5)+ 2*a(n-6)- a(n-8)+2*a(n-9)-a(n-10) with a(0)=1, a(1)=2, a(2)=3, a(3)=4, a(4)=8, a(5)=12, a(6)=16, a(7)=20, a(8)=29, a(9)=38. - Harvey P. Dale, Jun 07 2011

a(n) = (2*n^3+18*n^2+61*n+75+3*(n+3)*(-1)^n+6*(n+4-(n+2)*(-1)^n)*(-1)^((2*n-1+(-1)^n)/4))/96. - Luce ETIENNE, Mar 17 2015