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A015519
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a(n) = 2*a(n-1) + 7*a(n-2).
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21
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0, 1, 2, 11, 36, 149, 550, 2143, 8136, 31273, 119498, 457907, 1752300, 6709949, 25685998, 98341639, 376485264, 1441362001, 5518120850, 21125775707, 80878397364, 309637224677, 1185423230902, 4538307034543, 17374576685400
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n)=a(n-1)+A083100(n-2), n>1. A083100(n)/a(n+1) converges to sqrt(8). - Mario Catalani (mario.catalani(AT)unito.it), Apr 23 2003
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 8 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(8). - Cino Hilliard (hillcino368(AT)gmail.com), Sep 25 2005
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REFERENCES
| John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index to sequences with linear recurrences with constant coefficients, signature (2,7).
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FORMULA
| G.f.: x/ ( 1-2*x-7*x^2 ); a(n) = ((1+2*sqrt(2))^n-(1-2*sqrt(2))^n)*sqrt(2)/8. - Paul Barry, Jul 17 2003
E.g.f.: exp(x)*sinh(2*sqrt(2)*x)/(2*sqrt(2)). - Paul Barry, Nov 20 2003
Second binomial transform is A000129(2n)/2 (A001109). - Paul Barry, Apr 21 2004
a(n) = sum(k=0..floor((n-1)/2), comb(n-k-1, k)*(7/2)^k*2^(n-k-1) ). - Paul Barry, Jul 17 2004
a(n) = sum{k=0..n, binomial(n, 2*k+1)*8^k} - Paul Barry, Sep 29 2004
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MATHEMATICA
| a[n_]:=(MatrixPower[{{1, 4}, {1, -3}}, n].{{1}, {1}})[[2, 1]]; Table[Abs[a[n]], {n, -1, 40}] [From Vladimir Orlovsky, Feb 19 2010]
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PROG
| (Sage) [lucas_number1(n, 2, -7) for n in xrange(0, 25)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
(MAGMA) [ n eq 1 select 0 else n eq 2 select 1 else 2*Self(n-1)+7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 23 2011
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CROSSREFS
| 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: A071244 A005583 A176916 * A096977 A084098 A152819
Adjacent sequences: A015516 A015517 A015518 * A015520 A015521 A015522
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KEYWORD
| nonn,easy
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AUTHOR
| Olivier Gerard (olivier.gerard(AT)gmail.com)
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