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%I #38 Jan 22 2023 20:33:31
%S 0,0,1,5,15,36,76,148,273,485,839,1424,2384,3952,6505,10653,17383,
%T 28292,45964,74580,120905,195885,317231,513600,831360,1345536,2177521,
%U 3523733,5701983,9226500,14929324,24156724,39087009,63244757,102332855
%N a(n) = A000045(n+6) - (n^2 + 4*n + 8).
%C Given on p. 2 of Freixas, and proved as Theorem 3.2.
%C Partial sums of A001891. - _Bill McEachen_, Jan 20 2023
%C Original name was: The number of nonisomorphic complete simple games with n voters of two different types. - _Charles R Greathouse IV_, Jan 22 2023
%H G. C. Greubel, <a href="/A163250/b163250.txt">Table of n, a(n) for n = 0..1000</a>
%H Josep Freixas, Xavier Molinero, and Salvador Roura, <a href="http://arxiv.org/abs/0907.3853">A Fibonacci sequence for linear structures with two types of components</a>, arXiv:0907.3853 [math.CO], Jul 22 2009.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,1,2,-1).
%F a(n) = F(n+6) - (n^2 + 4*n + 8), where F(n) are the Fibonacci numbers.
%F From _R. J. Mathar_, Jul 27 2009: (Start)
%F a(n) = 4*a(n-1) -5*a(n-2) +a(n-3) +2*a(n-4) -a(n-5).
%F G.f.: x^2*(1+x)/((1-x-x^2)*(1-x)^3). (End)
%F 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
%F a(n) = A053808(n-2) for n >= 2. - _Georg Fischer_, Oct 28 2018
%F a(n) = (n-1)^2 + a(n-1) + a(n-2), n>2 (conjectured). - _Bill McEachen_, Jan 20 2023
%p 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
%t 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 *)
%o (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
%o (Magma) [Fibonacci(n+6)-(n^2+4*n+8): n in [0..40]]; // _Vincenzo Librandi_, Sep 22 2017
%o (GAP) List([0..35],n->Fibonacci(n+6)-(n^2+4*n+8)); # _Muniru A Asiru_, Oct 28 2018
%o (Sage) f=fibonacci; [f(n+6) -(n^2+4*n+8) for n in (0..40)] # _G. C. Greubel_, Jul 06 2019
%Y Cf. A000045, A053808, A001891.
%K easy,nonn
%O 0,4
%A _Jonathan Vos Post_, Jul 23 2009
%E More terms from _R. J. Mathar_, Jul 27 2009
%E New name using given formula from _Joerg Arndt_, Jan 21 2023
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