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A081016
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(Lucas(4n+3)+1)/5, or Fibonacci(2n+1)*Fibonacci(2n+2), or A081015/5.
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9
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1, 6, 40, 273, 1870, 12816, 87841, 602070, 4126648, 28284465, 193864606, 1328767776, 9107509825, 62423800998, 427859097160, 2932589879121, 20100270056686, 137769300517680, 944284833567073, 6472224534451830
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| a(n-1) is, together with b(n) := A089508(n), n>=1, the solution to a binomial problem - see A089508.
Numbers k such that 1 - 2 x + 5 x^2 is a square [From Artur Jasinski (grafix(AT)csl.pl), Oct 26 2008]
Also solution y of diophantine equation x^2+y^2+y^2+y^2+y^2=k^2 for which x=y-1. [From Carmine Suriano (surianonoi5(AT)libero.it), Jun 23 2010]
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REFERENCES
| Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 26.
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FORMULA
| a(n) = 8a(n-1)-8a(n-2)+a(n-3)
G.f.: (1-2*x)/((1-x)*(1-7*x+x^2)).
F(1) + F(5) + F(9) +...+ F(4n+1) = F(2n)*F(2n+3) + 1.
a(n)=(1/5)+(2/5)*{[(7/2)-(3/2)*sqrt(5)]^n+[(7/2)+(3/2)*sqrt(5)]^n+(1/5)*sqrt(5)*{[(7/2)+(3/2) *sqrt(5)]^n-[(7/2)-(3/2)*sqrt(5)]^n}, with n>=0 [From Paolo P. Lava (paoloplava(AT)gmail.com), Dec 01 2008]
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MAPLE
| luc := proc(n) option remember: if n=0 then RETURN(2) fi: if n=1 then RETURN(1) fi: luc(n-1)+luc(n-2): end: for n from 0 to 25 do printf(`%d, `, (luc(4*n+3)+1)/5) od:
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MATHEMATICA
| Join[{a=1, b=6}, Table[c=7*b-1*a-1; a=b; b=c, {n, 60}]] (*From Vladimir Joseph Stephan Orlovsky, Jan 18 2011*)
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CROSSREFS
| Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers), A081015.
Partial sums of A033889. Bisection of A001654. Equals A003482 + 1.
A145995 [From Artur Jasinski (grafix(AT)csl.pl), Oct 26 2008]
A178898 [From Carmine Suriano (surianonoi5(AT)libero.it), Jun 23 2010]
Sequence in context: A135032 A122074 A123357 * A083426 A122471 A178397
Adjacent sequences: A081013 A081014 A081015 * A081017 A081018 A081019
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KEYWORD
| nonn,easy
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AUTHOR
| R. K. Guy (rkg(AT)cpsc.ucalgary.ca), Mar 01, 2003
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EXTENSIONS
| More terms and Maple code from James A. Sellers (sellersj(AT)math.psu.edu), Mar 03, 2003
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