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A126116
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a(0) to a(3)=1; a(n+4) = a(n+3) + a(n+1) + a(n).
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4
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1, 1, 1, 1, 3, 5, 7, 11, 19, 31, 49, 79, 129, 209, 337, 545, 883, 1429, 2311, 3739, 6051, 9791, 15841, 25631, 41473, 67105, 108577, 175681, 284259, 459941, 744199, 1204139, 1948339, 3152479, 5100817, 8253295, 13354113, 21607409, 34961521
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| 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]
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REFERENCES
| Wolfram, S., A New Kind of Science. Champaign, IL: Wolfram Media, pp. 82-92, 2002
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LINKS
| Kelley L. Ross, The Golden Ratio and The Fibonacci Numbers
Eric Weisstein's World of Mathematics, MathWorld: Golden Ratio
Wikipedia, Golden Ratio
Index to sequences with linear recurrences with constant coefficients, signature (1,0,1,1). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 22 2010]
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FORMULA
| G.f.: (x-1) * (1+x+x^2) / ((x^2+x-1) * (x^2+1)). a(n) = ( (-1)^floor(n/2) * A010684(n) + 2*A000032(n))/5. a(2*n)=A061646(n). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 22 2010]
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(ceil((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)) [From Harvey P. Dale, Nov 08 2011]
(-1)^n * a(-n) = a(n) = F(n) - A070550(n - 6). - Michael Somos, Feb 05 2012
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EXAMPLE
| 1 + x + x^2 + x^3 + 3*x^4 + 5*x^5 + 7*x^6 + 11*x^7 + 19*x^8 + 31*x^9 + ...
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MAPLE
| Contribution from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 22 2010: (Start)
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) ; (End)
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]
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MATHEMATICA
| a=b=c=1; d=1; lst={a, b, c, d}; Do[AppendTo[lst, e=a+b+d]; a=b; b=c; c=d; d=e, {n, 0, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), May 28 2010]
LinearRecurrence[{1, 0, 1, 1}, {1, 1, 1, 1}, 50] (* From Harvey P. Dale, Nov 08 2011 *)
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CROSSREFS
| 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: A137814 A175196 A077858 * A161423 A133846 A175235
Adjacent sequences: A126113 A126114 A126115 * A126117 A126118 A126119
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KEYWORD
| nonn,changed
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AUTHOR
| Luis A Restrepo (luisiii(AT)mac.com), Mar 05 2007
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EXTENSIONS
| Edited by Don Reble (djr(AT)nk.ca), Mar 09 2007
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