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

0,4

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).

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

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 *)

Prog

(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
(Magma) [Fibonacci(n+6)-(n^2+4*n+8): n in [0..40]]; // Vincenzo Librandi, Sep 22 2017
(GAP) List([0..35], n->Fibonacci(n+6)-(n^2+4*n+8)); # Muniru A Asiru, Oct 28 2018
(Sage) f=fibonacci; [f(n+6) -(n^2+4*n+8) for n in (0..40)] # G. C. Greubel, Jul 06 2019

Crossrefs

Cf. A000045, A053808, A001891.

Sequence in context: A046776 A360486 A144898 * A053808 A111926 A369804

Adjacent sequences: A163247 A163248 A163249 * A163251 A163252 A163253

Keyword

easy,nonn

Author

Jonathan Vos Post, Jul 23 2009

Extensions

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

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

Status

approved