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A008999 a(n) = 2*a(n-1) + a(n-4).

%I A008999 #48 Oct 21 2022 22:05:17
%S A008999 1,2,4,8,17,36,76,160,337,710,1496,3152,6641,13992,29480,62112,130865,
%T A008999 275722,580924,1223960,2578785,5433292,11447508,24118976,50816737,
%U A008999 107066766,225581040,475281056,1001378849
%N A008999 a(n) = 2*a(n-1) + a(n-4).
%H A008999 Vincenzo Librandi, <a href="/A008999/b008999.txt">Table of n, a(n) for n = 0..1000</a>
%H A008999 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=479">Encyclopedia of Combinatorial Structures 479</a>
%H A008999 Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Janjic/janjic73.html">Binomial Coefficients and Enumeration of Restricted Words</a>, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.
%H A008999 Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv:1406.3081 [math.CO], 2014
%H A008999 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,0,1).
%F A008999 G.f.: 1/(1-2*x-x^4). - _Philippe Deléham_, Dec 02 2006
%F A008999 a(n) = Sum_{m=0..n} Sum_{j=0..(n-m)/3} binomial(n-m+(-3)*j,j)*binomial(n-3*j,m). - _Vladimir Kruchinin_, May 23 2011
%F A008999 O.g.f.: exp( Sum {n>=1} ( (1 + sqrt(1 + x^2))^n + (1 - sqrt(1 + x^2))^n ) * x^n/n ). Cf. A008998. - _Peter Bala_, Dec 22 2014
%p A008999 A008999 := proc(n) option remember; if n <= 3 then 2^n else 2*A008999(n-1)+A008999(n-4); fi; end;
%t A008999 LinearRecurrence[{2,0,0,1},{1,2,4,8},40] (* _Harvey P. Dale_, May 09 2012 *)
%t A008999 CoefficientList[Series[1/(1-2x-x^4),{x,0,40}],x] (* _Vincenzo Librandi_, May 09 2012 *)
%o A008999 (Maxima) a(n):=sum(sum(binomial(n-m+(-3)*j,j)*binomial(n-3*j,m),j,0,(n-m)/3),m,0,n); /* _Vladimir Kruchinin_, May 23 2011 */
%o A008999 (Magma) I:=[1, 2, 4, 8]; [n le 4 select I[n] else 2*Self(n-1)+Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, May 09 2012
%o A008999 (PARI) my(x='x+O('x^40)); Vec(1/(1-2*x-x^4)) \\ _G. C. Greubel_, Jun 12 2019
%o A008999 (Sage) (1/(1-2*x-x^4)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Jun 12 2019
%o A008999 (GAP) a:=[1,2,4,8];; for n in [5..40] do a[n]:=2*a[n-1]+a[n-4]; od; a; # _G. C. Greubel_, Jun 12 2019
%Y A008999 Cf. A008998.
%K A008999 nonn,easy
%O A008999 0,2
%A A008999 _N. J. A. Sloane_