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A126116 a(n) = a(n-1) + a(n-3) + a(n-4), with a(0)=a(1)=a(2)=a(3)=1.

%I A126116 #66 Jul 20 2025 10:46:07
%S A126116 1,1,1,1,3,5,7,11,19,31,49,79,129,209,337,545,883,1429,2311,3739,6051,
%T A126116 9791,15841,25631,41473,67105,108577,175681,284259,459941,744199,
%U A126116 1204139,1948339,3152479,5100817,8253295,13354113,21607409,34961521
%N A126116 a(n) = a(n-1) + a(n-3) + a(n-4), with a(0)=a(1)=a(2)=a(3)=1.
%C A126116 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.
%C A126116 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
%D A126116 S. Wolfram, A New Kind of Science. Champaign, IL: Wolfram Media, pp. 82-92, 2002
%H A126116 Seiichi Manyama, <a href="/A126116/b126116.txt">Table of n, a(n) for n = 0..4786</a>
%H A126116 K. T. Atanassov, D. R. Deford, A. G. Shannon, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/52-5/Atanassov.pdf">Pulsated Fibonacci recurrences</a>, Fibonacci Quarterly, Vol. 52, No. 5, Dec. 2014, pp. 22-27.
%H A126116 Kelley L. Ross, <a href="http://www.friesian.com/golden.htm">The Golden Ratio and The Fibonacci Numbers</a>
%H A126116 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldenRatio.html">Golden Ratio</a>
%H A126116 Wikipedia, <a href="http://en.wikipedia.org/wiki/Golden_ratio">Golden Ratio</a>
%H A126116 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,1).
%F A126116 From _R. J. Mathar_, Jul 22 2010: (Start)
%F A126116 G.f.: (1-x)*(1+x+x^2)/((1-x-x^2)*(1+x^2)).
%F A126116 a(n) = ( (-1)^floor(n/2) * A010684(n) + 2*A000032(n))/5.
%F A126116 a(2*n) = A061646(n). (End)
%F A126116 From _Johannes W. Meijer_, Aug 05 2011: (Start)
%F A126116 a(n) = F(n-1) + A070550(n-4) with F(n) = A000045(n).
%F A126116 a(n) = F(n-1) + F(floor((n-4)/2) + 1)*F(ceiling((n-4)/2) + 2). (End)
%F A126116 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
%F A126116 (-1)^n * a(-n) = a(n) = F(n) - A070550(n - 6). - _Michael Somos_, Feb 05 2012
%F A126116 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
%F A126116 a(n) = A293411(n)-A293411(n-1). - _R. J. Mathar_, Jul 20 2025
%e A126116 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 + ...
%p A126116 # From _R. J. Mathar_, Jul 22 2010: (Start)
%p A126116 A010684 := proc(n) 1+2*(n mod 2) ; end proc:
%p A126116 A000032 := proc(n) coeftayl((2-x)/(1-x-x^2),x=0,n) ; end proc:
%p A126116 A126116 := proc(n) ((-1)^floor(n/2)*A010684(n)+2*A000032(n))/5 ; end proc: seq(A126116(n),n=0..80) ; # (End)
%p A126116 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
%t A126116 LinearRecurrence[{1,0,1,1},{1,1,1,1},50] (* _Harvey P. Dale_, Nov 08 2011 *)
%o A126116 (PARI) Vec((x-1)*(1+x+x^2)/((x^2+x-1)*(x^2+1)) + O(x^50)) \\ _Altug Alkan_, Dec 25 2015
%o A126116 (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
%o A126116 (Sage) ((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
%o A126116 (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
%Y A126116 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.
%K A126116 nonn
%O A126116 0,5
%A A126116 Luis A Restrepo (luisiii(AT)mac.com), Mar 05 2007
%E A126116 Edited by _Don Reble_, Mar 09 2007