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A006131
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a(n) = a(n-1) + 4*a(n-2).
(Formerly M3788)
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29
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1, 1, 5, 9, 29, 65, 181, 441, 1165, 2929, 7589, 19305, 49661, 126881, 325525, 833049, 2135149, 5467345, 14007941, 35877321, 91909085, 235418369, 603054709, 1544728185, 3956947021, 10135859761, 25963647845, 66507086889, 170361678269
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Length-n strings with letters {0,1,2,3,4} where no two consecutive letters are nonzero, see fxtbook link below. [Joerg Arndt, Apr 08 2011]
Equals INVERTi transform of A063727: (1, 2, 8, 24, 80, 256, 832,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 12 2010]
a(n) is equal to the permanent of the n X n Hessenberg matrix with 1's along the main diagonal, 2's along the superdiagonal and the subdiagonal, and 0's everywhere else. [From John M. Campbell, June 9 2011]
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=2, 5*a(n-2) equals the number of 5-colored compositions of n with all parts >=2, such that no adjacent parts have the same color.-Milan Janjic, Nov 26 2011
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REFERENCES
| N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. K. Whitford, Binet's formula generalized, Fib. Quart., 15 (1977), pp. 21, 24, 29.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Joerg Arndt, Fxtbook, pp.317-318
A. Bremner and N. Tzanakis, Lucas sequences whose 8th term is a square
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 437
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Index to sequences with linear recurrences with constant coefficients, signature (1,4).
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FORMULA
| G.f.: 1/(1-x-4*x^2).
a(n)=(((1+sqrt(17))/2)^(n+1) - ((1-sqrt(17))/2)^(n+1))/sqrt(17).
a(n+1)=sum(k=0, ceil(n/2), 4^k*binomial(n-k, k)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 06 2004
a(n)=sum{k=0..n, binomial((n+k)/2, (n-k)/2)(1+(-1)^(n-k))2^(n-k)/2}; - Paul Barry (pbarry(AT)wit.ie), Aug 28 2005
a(n)=A102446/2 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 09 2008
a(n)=sum(0<=k<=n, A109466(n,k)*(-4)^(n-k) ). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 26 2008]
a(n)=prod(k=1..floor((n - 1)/2), (1 + 16*cos(k*Pi/n)^2) ). [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 21 2008]
Limiting ratio is (1 + sqrt(1 + 16))/2 = 2.561552812... [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 21 2008]
The fraction b(n) = a(n)/2^n satisfies b(n) = 1/2 b(n-1) + b(n-2); g.f. 1/(1-x/2-x^2); b(n)=(( (1+sqrt(17))/4 )^(n+1) - ( (1-sqrt(17))/4 )^(n+1))*2/sqrt(17). [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Nov 30 2009]
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MAPLE
| A006131:=-1/(-1+z+4*z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| m = 16; f[n_] = Product[(1 + m*Cos[k*Pi/n]^2), {k, 1, Floor[(n - 1)/2]}]; Table[FullSimplify[ExpandAll[f[n]]], {n, 0, 15}]; N[%] [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 21 2008]
a[n_]:=(MatrixPower[{{1, 4}, {1, 0}}, n].{{1}, {1}})[[2, 1]]; Table[a[n], {n, 0, 40}] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Feb 19 2010]
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PROG
| (Sage) [lucas_number1(n, 1, -4) for n in xrange(1, 30)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
(MAGMA) [ n eq 1 select 1 else n eq 2 select 1 else Self(n-1)+4*Self(n-2): n in [1..40] ]; // Vincenzo Librandi, Aug 19 2011
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CROSSREFS
| Cf. A006130, A015440, A026581, A026583, A026597, A026599, A052923, A102446.
Cf. A063727 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 12 2010]
Sequence in context: A163779 A191013 A193487 * A049602 A119031 A034435
Adjacent sequences: A006128 A006129 A006130 * A006132 A006133 A006134
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Roger Bagula, Sep 26 2006
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