A173511 a(n) = 4*n^2 + floor(n/2).

0, 4, 17, 37, 66, 102, 147, 199, 260, 328, 405, 489, 582, 682, 791, 907, 1032, 1164, 1305, 1453, 1610, 1774, 1947, 2127, 2316, 2512, 2717, 2929, 3150, 3378, 3615, 3859, 4112, 4372, 4641, 4917, 5202, 5494, 5795, 6103, 6420, 6744, 7077, 7417, 7766, 8122

Offset: 0

Author

Reinhard Zumkeller, Feb 20 2010

Keywords

• nonn
• easy

Examples

			a(6) = 147; 4(6)^2 + floor(6/3) = 144 + 3 = 147.
		

Links

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

Crossrefs

Cf. A002943, A007494, A007742, A110654, A157474, A173512.

Programs

Maple

A173511:=n->4*n^2 + floor(n/2); seq(A173511(k), k=0..100); # Wesley Ivan Hurt, Nov 01 2013

Mathematica

Table[4n^2 + Floor[n/2], {n,0,100}] (* Wesley Ivan Hurt, Nov 01 2013 *)
LinearRecurrence[{2,0,-2,1},{0,4,17,37},50] (* Harvey P. Dale, Nov 23 2019 *)

PARI

a(n) = 4*n^2 + n\2 \\ Charles R Greathouse IV, Jun 11 2015

Python

def A173511(n): return (n**2<<2)+(n>>1) # Chai Wah Wu, Jan 18 2023

Formula

a(n) = floor((2*n + 1/8)^2).

a(n+1) - a(n) = A173512(n).

a(n) = A002943(n) - A007494(n) = A007742(n) - A110654(n).

a(2*n) = A157474(n) for n>0.

From - R. J. Mathar, Feb 21 2010: (Start)

a(n)= 2*a(n-1) -2*a(n-3) +a(n-4).

G.f.: -x*(4+9*x+3*x^2)/((1+x)*(x-1)^3). (End)

E.g.f.: (x*(8*x + 9)*cosh(x) + (8*x^2 + 9*x - 1)*sinh(x))/2. - Stefano Spezia, Apr 24 2024