A163250 a(n) = A000045(n+6) - (n^2 + 4*n + 8).

0, 0, 1, 5, 15, 36, 76, 148, 273, 485, 839, 1424, 2384, 3952, 6505, 10653, 17383, 28292, 45964, 74580, 120905, 195885, 317231, 513600, 831360, 1345536, 2177521, 3523733, 5701983, 9226500, 14929324, 24156724, 39087009, 63244757, 102332855

Offset: 0

Author

Jonathan Vos Post, Jul 23 2009

Keywords

• easy
• nonn

Comments

Given on p. 2 of Freixas, and proved as Theorem 3.2.

Partial sums of A001891. - Bill McEachen, Jan 20 2023

Original name was: The number of nonisomorphic complete simple games with n voters of two different types. - Charles R Greathouse IV, Jan 22 2023

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Josep Freixas, Xavier Molinero, and Salvador Roura, A Fibonacci sequence for linear structures with two types of components, arXiv:0907.3853 [math.CO], Jul 22 2009.
• Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Crossrefs

Cf. A000045, A053808, A001891.

Programs

GAP

List([0..35],n->Fibonacci(n+6)-(n^2+4*n+8)); # Muniru A Asiru, Oct 28 2018

Magma

[Fibonacci(n+6)-(n^2+4*n+8): n in [0..40]]; // Vincenzo Librandi, Sep 22 2017

Maple

with(numtheory): seq(coeff(series(x^2*(1+x)/((x^2+x-1)*(x-1)^3),x,n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Oct 28 2018

Mathematica

LinearRecurrence[{4,-5,1,2,-1}, {0,0,1,5,15}, 40] (* or *) Table[ Fibonacci[n+6] -(n^2+4*n+8), {n,0,40}] (* G. C. Greubel, Dec 12 2016 *)

PARI

concat([0,0], Vec(x^2*(1+x)/((1-x-x^2)*(1-x)^3) + O(x^40))) \\ G. C. Greubel, Dec 12 2016

Sage

f=fibonacci; [f(n+6) -(n^2+4*n+8) for n in (0..40)] # G. C. Greubel, Jul 06 2019

Formula

a(n) = F(n+6) - (n^2 + 4*n + 8), where F(n) are the Fibonacci numbers.

From R. J. Mathar, Jul 27 2009: (Start)

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

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

a(n) = Sum_{k=0..n-1} Sum_{i=0..n-1} i^2 * C(n-k-1,k-i). - Wesley Ivan Hurt, Sep 21 2017

a(n) = A053808(n-2) for n >= 2. - Georg Fischer, Oct 28 2018

a(n) = (n-1)^2 + a(n-1) + a(n-2), n>2 (conjectured). - Bill McEachen, Jan 20 2023

Extensions

More terms from R. J. Mathar, Jul 27 2009

New name using given formula from Joerg Arndt, Jan 21 2023