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%I #40 Apr 13 2023 16:27:36
%S 1,2,4,7,13,24,45,84,157,293,547,1021,1906,3558,6642,12399,23146,
%T 43208,80659,150571,281080,524709,979506,1828503,3413377,6371957,
%U 11894917,22204960,41451340,77379720,144449397,269652414
%N Expansion of g.f. x^3/(1 - 2*x + x^3 - x^4).
%H G. C. Greubel, <a href="/A059633/b059633.txt">Table of n, a(n) for n = 3..1003</a>
%H Kevin Beanland, Dmitriy Gorovoy, Jȩdrzej Hodor, and Daniil Homza, <a href="https://arxiv.org/abs/2211.01049">Counting Unions of Schreier Sets</a>, arXiv:2211.01049 [math.CO], 2022. See p. 4.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,1).
%F a(n) = 2*a(n-1) - a(n-3) + a(n-4).
%F Recurrence equations (A059633 is L(n)): I(n+1) = I(n) + J(n) + L(n); J(n+1) = I(n); K(n+1) = J(n) + K(n); L(n+1) = K(n); M(n+1) = L(n) + 2M(n); initial conditions: I(0) = 1; J(0) = 0; K(0) = 0; L(0) = 0; M(0) = 0. Values for n = 0 1 2 3 4 5 6 7 8 ...: I(n) = 1 1 2 3 6 11 21 39 73 ... J(n) = 0 1 1 2 3 6 11 21 39 ... K(n) = 0 0 1 2 4 7 13 24 45 ... L(n) = 0 0 0 1 2 4 7 13 24 ... M(n) = 0 0 0 0 1 4 12 31 75 ...
%F a(n) = A049856(n+2) - A049856(n+1) - A049856(n) + A049856(n-1).
%F For n >= 2, a(n+1) = Sum_{i=0..n} Fibonacci(i)*binomial(n-i, i). - _Benoit Cloitre_, Sep 21 2004
%F a(n) = Sum_{k=0..n+1} C(k+1, n-k+1)*Fibonacci(n-k+1) [offset 0]. - _Paul Barry_, Feb 23 2005
%p with(combstruct): SeqSeqSeqL := [T, {T=Sequence(S), S=Sequence(U, card > 1), U=Sequence(Z, card >1)}, unlabeled]: seq(count(SeqSeqSeqL, size=j), j=4..35); # _Zerinvary Lajos_, Apr 04 2009
%t LinearRecurrence[{2,0,-1,1},{1,2,4,7},40] (* _Harvey P. Dale_, Dec 25 2022 *)
%o (Magma) I:=[1,2,4,7]; [n le 4 select I[n] else 2*Self(n-1) - Self(n-3) + Self(n-4): n in [1..30]]; // _G. C. Greubel_, Apr 13 2023
%o (SageMath)
%o @CachedFunction
%o def a(n): # a = A059633
%o if (n<4): return (0,0,0,1,2,4,7)[n]
%o else: return 2*a(n-1) - a(n-3) + a(n-4)
%o [a(n) for n in range(3,51)] # _G. C. Greubel_, Apr 13 2023
%Y I and J are A049856 while K and L are A059633 (with some offsets).
%K nonn,easy
%O 3,2
%A James R. FitzSimons (cherry(AT)neta.com), Feb 19 2001
%E Comments and more terms from _Henry Bottomley_, Feb 21 2001
%E New description from _Vladeta Jovovic_, Jan 17 2004
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