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A069403
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a(n) = 2*Fibonacci(2n+1)-1.
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9
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1, 3, 9, 25, 67, 177, 465, 1219, 3193, 8361, 21891, 57313, 150049, 392835, 1028457, 2692537, 7049155, 18454929, 48315633, 126491971, 331160281, 866988873, 2269806339, 5942430145, 15557484097, 40730022147, 106632582345
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Half the number of n X 3 binary arrays with a path of adjacent 1's and a path of adjacent 0's from top row to bottom row.
Indices of A017245=9n+7=7,16,25,34, for submitted A153819=16,34,88,. A153819(n)=9a(n)+7=18*F(2n+1)-2;F(n)=Fibonacci=A000045,2's=A007395. Other recurrence: a(n)=4a(n-1)-4a(n-2)+a(n-3). [From Paul Curtz (bpcrtz(AT)free.fr), Jan 02 2009]
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LINKS
| J. Hietarinta and C.-M. Viallet, Singularity confinement and chaos in discrete systems, Physical Review Letters 81 (1998), pp. 326-328.
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FORMULA
| a(0) = 1, a(1) = 3, a(2) = 9, a(3) = 25; a(n) = 3 a(n-1) - 3 a(n-3) + a(n-4).
a(n) = 3*a(n-1) - a(n-2) + 1 for n>1, a(1) = 3, a(0) = 0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 02 2006
a(n)=4*a(n-1)-4*a(n-2)+a(n-3). G.f.: (1-x+x^2)/((1-x)(1-3x+x^2)). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 23 2009]
a(n)= 1+2*sum(fibonacci(2*k),k=0..n) = 1+2*A027941(n). [From Gary Detlefs (gdetlefs(At)aol.com), Dec 7 2010
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MATHEMATICA
| a[n_] := a[n] = 3 a[n - 1] - 3 a[n - 3] + a[n - 4]; a[0] = 1; a[1] = 3; a[2] = 9; a[3] = 25; Table[ a[n], {n, 0, 27}]
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PROG
| (MAGMA) [2*Fibonacci(2*n+1)-1: n in [0..30]]; // Vincenzo Librandi, Apr 18 2011
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CROSSREFS
| Cf. A084707.
Cf. 1 X n A000225, 2 X n A016269, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.
Equals A052995 - 1. Bisection of A001595, A062114, A066983.
Sequence in context: A106514 A156561 A085327 * A094292 A201533 A000242
Adjacent sequences: A069400 A069401 A069402 * A069404 A069405 A069406
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KEYWORD
| nonn
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AUTHOR
| R. H. Hardin (rhhardin(AT)att.net), Mar 22 2002.
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EXTENSIONS
| Simpler definition from Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 19 2003
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