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A030191 Scaled Chebyshev U-polynomial evaluated at sqrt(5)/2. 32
1, 5, 20, 75, 275, 1000, 3625, 13125, 47500, 171875, 621875, 2250000, 8140625, 29453125, 106562500, 385546875, 1394921875, 5046875000, 18259765625, 66064453125, 239023437500, 864794921875, 3128857421875, 11320312500000, 40957275390625, 148184814453125 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of (s(0), s(1), ..., s(2n+4)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,....,2n+4, s(0) = 1, s(2n+4) = 5. - Herbert Kociemba, Jun 14 2004

Binomial transform of A002878 . - Philippe Deléham, Oct 04 2005

Diagonal of square array A216219. - Philippe Deléham, Mar 15 2013

REFERENCES

A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=5, q=-5.

W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eqs. (38) and (45), lhs, m=5.

LINKS

Table of n, a(n) for n=0..25.

Index to sequences with linear recurrences with constant coefficients, signature (5,-5).

Index entries for sequences related to Chebyshev polynomials.

FORMULA

a(n)=(sqrt(5))^n*U(n, sqrt(5)/2), g.f.: 1/(5*(x^2-x+1/5)), a(2*k+1)=5^(k+1)*F(2*k+2), F(n) = Fibonacci (A000045), a(2*k)=5^k*L(2*k+1), L(n) = Lucas (A000032)

a(n-1)=sum(k=0, n, C(n, k)*F(2*k)) - Benoit Cloitre, Jun 21 2003

a(n) = 5*a(n-1)-5*a(n-2). - Benoit Cloitre, Oct 23 2003

a(n-1)=((5/2+sqrt(5)/2)^n-(5/2-sqrt(5)/2)^n)/sqrt(5) is the 2nd binomial transform of Fib(n), the first binomial transform of Fib(2n) and its n-th term is the n-th term of the third binomial transform of Fib(3n) divided by 2^n. - Paul Barry, Mar 23 2004

a(n)=Sum_{k, 0<=k<=n}5^k*A109466(n,k) . - Philippe Deléham, Nov 28 2006

a(n) = 5*A039717(n), n>0. - Philippe Deléham, Mar 12 2013

a(n) = A216219(n,n+3) = A216219(n,n+4) = A216219(n+3,n) = A216219(n+4,n). - Philippe Deléham, Mar 15 2013

G.f.: 1/(1-5*x/(1+x/(1-x))). - Philippe Deléham, Mar 15 2013

MATHEMATICA

Table[MatrixPower[{{2, 1}, {1, 3}}, n][[1]][[2]], {n, 0, 44}] [(* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)

PROG

(Sage) [lucas_number1(n, 5, 5) for n in xrange(1, 22)]# [Zerinvary Lajos, Apr 22 2009]

CROSSREFS

Sequence in context: A092490 A094828 A093131 * A224422 A000344 A061278

Adjacent sequences:  A030188 A030189 A030190 * A030192 A030193 A030194

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang

STATUS

approved

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Last modified December 19 12:22 EST 2014. Contains 252224 sequences.