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

1, 2, 4, 8, 17, 36, 76, 160, 337, 710, 1496, 3152, 6641, 13992, 29480, 62112, 130865, 275722, 580924, 1223960, 2578785, 5433292, 11447508, 24118976, 50816737, 107066766, 225581040, 475281056, 1001378849

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 479
• Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.
• Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv:1406.3081 [math.CO], 2014
• Index entries for linear recurrences with constant coefficients, signature (2,0,0,1).

Crossrefs

Cf. A008998.

Programs

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

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

Maple

A008999 := proc(n) option remember; if n <= 3 then 2^n else 2*A008999(n-1)+A008999(n-4); fi; end;

Mathematica

LinearRecurrence[{2,0,0,1},{1,2,4,8},40] (* Harvey P. Dale, May 09 2012 *)
CoefficientList[Series[1/(1-2x-x^4),{x,0,40}],x] (* Vincenzo Librandi, May 09 2012 *)

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 */

PARI

my(x='x+O('x^40)); Vec(1/(1-2*x-x^4)) \\ G. C. Greubel, Jun 12 2019

Sage

(1/(1-2*x-x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 12 2019

Formula

G.f.: 1/(1-2*x-x^4). - Philippe Deléham, Dec 02 2006

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

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