A024305 a(n) = 2*(n+1) + 3*n + ... + (k+1)*(n+2-k), where k = floor((n+1)/2).

4, 6, 17, 22, 43, 52, 86, 100, 150, 170, 239, 266, 357, 392, 508, 552, 696, 750, 925, 990, 1199, 1276, 1522, 1612, 1898, 2002, 2331, 2450, 2825, 2960, 3384, 3536, 4012, 4182, 4713, 4902, 5491, 5700, 6350, 6580, 7294, 7546, 8327, 8602, 9453, 9752, 10676, 11000, 12000

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Crossrefs

Bisection: 2*A051925(n).

Cf. A024854, A023855, A023856, A023857, A024868.

Programs

Magma

b:= func< n | (1-(-1)^n)/2 >;
[(2*n^3 + 3*(6 +b(n))*n^2 + 2*(14 +9*b(n))*n + 27*b(n))/24 : n in [1..50]] // G. C. Greubel, Jul 12 2022

Maple

seq(sum((i+1)*(k-i+2), i=1..ceil(k/2)), k=1..70); # Wesley Ivan Hurt, Sep 20 2013

Mathematica

Table[Ceiling[n/2]*(-2*Ceiling[n/2]^2+3n*Ceiling[n/2]+9n+14)/6,{n,100}] (* Wesley Ivan Hurt, Sep 20 2013 *)

SageMath

def b(n): return (1-(-1)^n)/2
[(2*n^3 + 3*(6 +b(n))*n^2 + 2*(14 +9*b(n))*n + 27*b(n))/24 for n in (1..50)] # G. C. Greubel, Jul 12 2022

Formula

From Vladeta Jovovic, Jan 01 2003: (Start)

a(n) = (1/48)*(4*n^3 + (3*(-1)^(n+1) + 39)*n^2 + (18*(-1)^(n+1) + 74)*n + 27*(-1)^(n+1) + 27).

a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7).

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

a(n) = Sum_{i=1..ceiling(n/2)} (i+1)*(n-i+2) = ceiling(n/2)*(-2*ceiling(n/2)^2 + 3n*ceiling(n/2) + 9*n + 14)/6. - Wesley Ivan Hurt, Sep 20 2013

E.g.f.: (1/24)*( x*(69 + 24*x + 2*x^2)*cosh(x) + (27 + 48*x + 27*x^2 + 2*x^3)*sinh(x) ). - G. C. Greubel, Jul 12 2022

Extensions

Name simplified by Jon E. Schoenfield, Jun 12 2019