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