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A004254
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a(n) = 5*a(n-1) - a(n-2), a(0) = 0, a(1) = 1.
(Formerly M3930)
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36
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0, 1, 5, 24, 115, 551, 2640, 12649, 60605, 290376, 1391275, 6665999, 31938720, 153027601, 733199285, 3512968824, 16831644835, 80645255351, 386394631920, 1851327904249, 8870244889325, 42499896542376, 203629237822555
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Nonnegative values of y satisfying x^2 - 21*y^2 = 4; values of x are in A003501. - W. Lang, Nov 29 2002
a(n) is equal to the permanent of the (n-1) X (n-1) Hessenberg matrix with 5's along the main diagonal, i's along the superdiagonal and the subdiagonal (i is the imaginary unit), and 0's everywhere else. [From John M. Campbell, June 9 2011]
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REFERENCES
| E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pps. 231-242.
F. A. Haight, On a generalization of Pythagoras' theorem, pp. 73-77 of J. C. Butcher, editor, A Spectrum of Mathematics. Auckland University Press, 1971.
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=5, q=-1.
A. F. Horadam, Pell Identities, Fib. Quart., Vol. 9, No. 3, 1971, pps. 245-252.
W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eq.(44), lhs, m=7.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
F. M. van Lamoen, Square wreaths around hexagons, Forum Geometricorum, 6 (2006) 311-325.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..200
Tanya Khovanova, Recursive Sequences
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.
F. M. van Lamoen, Article in Forum Geometricorum
Index entries for sequences related to linear recurrences with constant coefficients
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
| G.f.: x/(1-5*x+x^2).
a(n)=((5+sqrt(21))/2)^n-((5-sqrt(21))/2)^n)/sqrt(21). - Barry E. Williams, Aug 29 2000
a(n)= S(2*n-1, sqrt(7))/sqrt(7) = S(n-1, 5); S(n, x)=U(n, x/2), Chebyshev polynomials of 2nd kind, A049310.
(A003501)=sqrt{[21*(a(n))^2]+4}.
a(n)=sum(k=0..n-1, binomial(n+k, 2*k+1)*2^k ) - Paul Barry, Nov 30 2004
[A004253(n), a(n)] = [1,3; 1,4]^n * [1,0] - Gary W. Adamson, Mar 19 2008
a(n+1) = Sum_{k, 0<=k<=n} A101950(n,k)*4^k. - DELEHAM Philippe, Feb 10 2012
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EXAMPLE
| a(n+1)=sum{k=0..n, Gegenbauer_C(n-k,k+1,2)}. [From Paul Barry, Apr 21 2009]
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MAPLE
| A004254:=1/(1-5*z+z**2); [S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| a[n_]:=(MatrixPower[{{1, 3}, {1, 4}}, n].{{1}, {1}})[[2, 1]]; Table[a[n], {n, 0, 40}] [From Vladimir Orlovsky, Feb 19 2010]
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PROG
| (PARI) a(n)=if(n<0, 0, subst(4*poltchebi(n+1)-10*poltchebi(n), x, 5/2)/21)
(PARI) a(n)=if(n<0, 0, imag((5+quadgen(84))^n)/2^(n-1))
(Sage) [lucas_number1(n, 5, 1) for n in range(27)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
(MAGMA) [ n eq 1 select 0 else n eq 2 select 1 else 5*Self(n-1)-Self(n-2): n in [1..30] ]; // Vincenzo Librandi, Aug 19 2011
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CROSSREFS
| Partial sums of A004253.
Cf. A000027, A001906, A001353, A003501, A030221. a(n) = sqrt((A003501(n)^2 - 4)/21).
First differences of a(n) are in A004253, partial sums in A089817.
Cf. A004253.
INVERT transformation yields A001109. [From R. J. Mathar, Sep 11 2008]
Sequence in context: A140766 A026388 A057969 * A086347 A200739 A026707
Adjacent sequences: A004251 A004252 A004253 * A004255 A004256 A004257
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KEYWORD
| easy,nonn,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 31 2000
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