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A007805
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a(n)=F(6n+3)/2, where F=A000045 (the Fibonacci sequence).
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20
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1, 17, 305, 5473, 98209, 1762289, 31622993, 567451585, 10182505537, 182717648081, 3278735159921, 58834515230497, 1055742538989025, 18944531186571953, 339945818819306129, 6100080207560938369
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Hypotenuse (z) of Pythagorean triples (x,y,z) with |2x-y|=1.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..100
Tanya Khovanova, Recursive Sequences
Index to sequences with linear recurrences with constant coefficients, signature (18,-1).
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FORMULA
| G.f.: (1-x)/(1-18*x+x^2). a(n)=18*a(n-1)-a(n-2), n>1. a(0)=1, a(1)=17.
a(n)=A001076(2n+1).
a(n+1)=9*a(n)+4*(5*a(n)^2-1)^0.5 - Richard Choulet (richardchoulet(AT)yahoo.fr), Aug 30 2007, Dec 28 2007
a(n) = ((2+sqrt(5))^(2*n+1)-(2-sqrt(5))^(2*n+1))/(2*sqrt(5)). - Dean Hickerson (dean.hickerson(AT)yahoo.com), Dec 09 2002
a(n) ~ (1/10)*sqrt(5)*(sqrt(5) + 2)^(2*n+1) - Joe Keane (jgk(AT)jgk.org), May 15 2002
For all elements x of the sequence, 5*x^2 - 1 is a square, A075796(n+1)^2. Lim. n->Inf. a(n)/a(n-1) = 8*phi + 5 = 9 + 4*Sqrt(5) - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 13 2002
Let q(n, x)=sum(i=0, n, x^(n-i)*binomial(2*n-i, i)); then a(n)=q(n, 16). - Benoit Cloitre (benoit7848c(AT)orange.fr), Dec 06 2002
a(n) = 19*a(n-1)- 19*a(n-2) + a(n-3); f(x) = (sqrt(5)/10)*((2+sqrt(5))*(9+4*sqrt(5))^(x-1) - (2-sqrt(5))*(9-4*sqrt(5))^(x-1)) - Antonio A. Olivares (olivares14031(AT)yahoo.com), May 15 2008
a(n) = 17a(n-1) + 17a(n-2) - a(n-3) - Antonio A. Olivares (olivares14031(AT)yahoo.com), Jun 19 2008
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MATHEMATICA
| f[n_]:=IntegerQ[Sqrt[5*n^2-1]]; Select[Range[0, 9! ], f[ # ]&] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Feb 19 2010]
LinearRecurrence[{18, -1}, {1, 17}, 50] (* Sture Sjöstedt, Nov 29 2011 *)
Table[Fibonacci[6n+3]/2, {n, 0, 20}] (* From Harvey P. Dale, Dec 17 2011 *)
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CROSSREFS
| Row 18 of array A094954.
Row 2 of array A188647.
Sequence in context: A163049 A083453 A090437 * A158585 A201232 A156085
Adjacent sequences: A007802 A007803 A007804 * A007806 A007807 A007808
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KEYWORD
| nonn,nice,easy
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AUTHOR
| James A. Raymond (raymond(AT)unlv.edu), Clark Kimberling (ck6(AT)evansville.edu)
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EXTENSIONS
| Better description and more terms from Michael Somos
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