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A083099
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a(0) = 0, a(1) = 1; for n>1, a(n) = 2a(n-1)+6a(n-2).
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23
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0, 1, 2, 10, 32, 124, 440, 1624, 5888, 21520, 78368, 285856, 1041920, 3798976, 13849472, 50492800, 184082432, 671121664, 2446737920, 8920205824, 32520839168, 118562913280, 432250861568, 1575879202816, 5745263575040
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OFFSET
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0,3
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COMMENTS
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a(n+1) = a(n)+A083098(n+1). A083098(n+1)/a(n) converges to sqrt(7).
The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the denominators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 7 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(7). - Cino Hilliard (hillcino368(AT)gmail.com), Sep 25 2005
Pisano period lengths: 1, 1, 2, 1, 12, 2, 7, 1, 6, 12, 60, 2,168, 7, 12, 1,288, 6, 18, 12,... - R. J. Mathar, Aug 10 2012
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REFERENCES
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John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.
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LINKS
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Table of n, a(n) for n=0..24.
Index to sequences with linear recurrences with constant coefficients, signature (2,6).
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FORMULA
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G.f.: x/(1-2x-6x^2).
E.g.f. : dif(exp(x)sinh(sqrt(7)x)/sqrt(7), x); a(n-1)=sum{k=0..n, binomial(n, 2k+1)7^k}. - Paul Barry, Sep 29 2004
a(n)=-(1/14)*[1-sqrt(7)]^n*sqrt(7)+(1/14)*[1+sqrt(7)]^n*sqrt(7), with n>=0 - Paolo P. Lava, Jun 10 2008
Simplified formula: ((1+sqrt7)^n-(1-sqrt7)^n)/sqrt28. Offset 1. a(3)=10 [From Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009]
G.f.: G(0)*x/(2*(1-x)), where G(k)= 1 + 1/(1 - x*(7*k-1)/(x*(7*k+6) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
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MATHEMATICA
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CoefficientList[Series[1/(1-2x-6x^2), {x, 0, 25}], x]
Expand[Table[((1 + Sqrt[7])^n - (1 - Sqrt[7])^n)7/(14Sqrt[7]), {n, 0, 25}]] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 22 2007
LinearRecurrence[{2, 6}, {0, 1}, 25] (* Sture Sjöstedt, Dec 06 2011 *)
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PROG
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(Sage) [lucas_number1(n, 2, -6) for n in xrange(0, 25)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
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CROSSREFS
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The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.
Sequence in context: A131068 A034555 A084154 * A032095 A151019 A004028
Adjacent sequences: A083096 A083097 A083098 * A083100 A083101 A083102
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KEYWORD
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easy,nonn
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AUTHOR
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Mario Catalani (mario.catalani(AT)unito.it), Apr 22 2003
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STATUS
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approved
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