0,5

This sequence has the same growth rate as the Fibonacci sequence, since x^4 - x^3 - x - 1 has the real roots phi and -1/phi.

The Ca1 sums, see A180662 for the definition of these sums, of triangle A035607 equal the terms of this sequence without the first term. - Johannes W. Meijer, Aug 05 2011

S. Wolfram, A New Kind of Science. Champaign, IL: Wolfram Media, pp. 82-92, 2002

Seiichi Manyama, Table of n, a(n) for n = 0..4786

K. T. Atanassov, D. R. Deford, A. G. Shannon, Pulsated Fibonacci recurrences, Fibonacci Quarterly, Vol. 52, No. 5, Dec. 2014, pp. 22-27.

Kelley L. Ross, The Golden Ratio and The Fibonacci Numbers

Eric Weisstein's World of Mathematics, Golden Ratio

Wikipedia, Golden Ratio

Index entries for linear recurrences with constant coefficients, signature (1,0,1,1).

From R. J. Mathar, Jul 22 2010: (Start)

G.f.: (1-x)*(1+x+x^2)/((1-x-x^2)*(1+x^2)).

a(n) = ( (-1)^floor(n/2) * A010684(n) + 2*A000032(n))/5.

a(2*n) = A061646(n). (End)

From Johannes W. Meijer, Aug 05 2011: (Start)

a(n) = F(n-1) + A070550(n-4) with F(n) = A000045(n).

a(n) = F(n-1) + F(floor((n-4)/2) + 1)*F(ceiling((n-4)/2) + 2). (End)

a(n) = (1/5)*((sqrt(5)-1)*(1/2*(1+sqrt(5)))^n - (1+sqrt(5))*(1/2*(1-sqrt(5)))^n + sin((Pi*n)/2) - 3*cos((Pi*n)/2)). - Harvey P. Dale, Nov 08 2011

(-1)^n * a(-n) = a(n) = F(n) - A070550(n - 6). - Michael Somos, Feb 05 2012

a(n)^2 + 3*a(n-2)^2 + 6*a(n-5)^2 + 3*a(n-7)^2 = a(n-8)^2 + 3*a(n-6)^2 + 6*a(n-3)^2 + 3*a(n-1)^2. - Greg Dresden, Jul 07 2021

a(n) = A293411(n)-A293411(n-1). - R. J. Mathar, Jul 20 2025

G.f. = 1 + x + x^2 + x^3 + 3*x^4 + 5*x^5 + 7*x^6 + 11*x^7 + 19*x^8 + 31*x^9 + ...

A010684 := proc(n) 1+2*(n mod 2) ; end proc:

A000032 := proc(n) coeftayl((2-x)/(1-x-x^2), x=0, n) ; end proc:

A126116 := proc(n) ((-1)^floor(n/2)*A010684(n)+2*A000032(n))/5 ; end proc: seq(A126116(n), n=0..80) ; # R. J. Mathar, Jul 22 2010

# Alternative:

with(combinat): A126116 := proc(n): fibonacci(n-1) + fibonacci(floor((n-4)/2)+1)* fibonacci(ceil((n-4)/2)+2) end: seq(A126116(n), n=0..38); # Johannes W. Meijer, Aug 05 2011

LinearRecurrence[{1, 0, 1, 1}, {1, 1, 1, 1}, 50] (* Harvey P. Dale, Nov 08 2011 *)

(PARI) Vec((x-1)*(1+x+x^2)/((x^2+x-1)*(x^2+1)) + O(x^50)) \\ Altug Alkan, Dec 25 2015

(Magma) [n le 4 select 1 else Self(n-1) + Self(n-3) + Self(n-4): n in [1..50]]; // Vincenzo Librandi, Dec 25 2015

(SageMath) ((1-x)*(1+x+x^2)/((1-x-x^2)*(1+x^2))).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Jul 15 2019

(GAP) a:=[1, 1, 1, 1];; for n in [5..50] do a[n]:=a[n-1]+a[n-3]+a[n-4]; od; a; # G. C. Greubel, Jul 15 2019

Cf. Fibonacci numbers A000045; Lucas numbers A000032; tribonacci numbers A000213; tetranacci numbers A000288; pentanacci numbers A000322; hexanacci numbers A000383; 7th-order Fibonacci numbers A060455; octanacci numbers A079262; 9th-order Fibonacci sequence A127193; 10th-order Fibonacci sequence A127194; 11th-order Fibonacci sequence A127624, A128429.

Sequence in context: A175196 A077858 A327461 * A161423 A133846 A056208

Adjacent sequences: A126113 A126114 A126115 * A126117 A126118 A126119

Luis A Restrepo (luisiii(AT)mac.com), Mar 05 2007

Edited by Don Reble, Mar 09 2007

approved