login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A030191 Scaled Chebyshev U-polynomial evaluated at sqrt(5)/2. 36
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

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

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.

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

Index entries for 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/(1 - 5*x + 5*x^2).

a(2*k+1) = 5^(k+1)*Fibonacci(2*k+2).

a(2*k) = 5^k*Lucas(2*k+1).

a(n-1) = Sum_{k=0..n} C(n, k)*Fibonacci(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+sqrt(5))/2)^n - ((5-sqrt(5))/2)^n)/sqrt(5) is the 2nd binomial transform of Fibonacci(n), the first binomial transform of Fibonacci(2n) and its n-th term is the n-th term of the third binomial transform of Fibonacci(3n) divided by 2^n. - Paul Barry, Mar 23 2004

a(n) = Sum_{k-0..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

a(n) = -a(-2-n) * 5^(n+1) for all n in Z. - Michael Somos, Aug 27 2015

E.g.f.: exp((5-sqrt(5))*x/2)*((5 + sqrt(5))*exp(sqrt(5)*x) - 5 + sqrt(5))/(2*sqrt(5)). - Stefano Spezia, Dec 29 2019

EXAMPLE

G.f. = 1 + 5*x + 20*x^2 + 75*x^3 + 275*x^4 + 1000*x^5 + 3625*x^6 + ...

MAPLE

seq(coeff(series(1/(1-5*x+5*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Dec 28 2019

MATHEMATICA

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

a[ n_]:= (((5 + Sqrt[5])/2)^(n + 1) - ((5 - Sqrt[5])/2)^(n + 1)) / Sqrt[5] // Expand; (* Michael Somos, Aug 27 2015 *)

Table[If[EvenQ[n], 5^(n/2)*LucasL[n+1], 5^((n+1)/2)*Fibonacci[n+1]], {n, 0, 30}] (* G. C. Greubel, Dec 28 2019 *)

PROG

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

(PARI) {a(n) = imag((quadgen(5) + 2)^(n+1))}; /* Michael Somos, Aug 27 2015 *

(MAGMA) I:=[1, 5]; [n le 2 select I[n] else 5*(Self(n-1) - Self(n-2)): n in [1..30]]; // G. C. Greubel, Dec 28 2019

(GAP) a:=[1, 5];; for n in [3..30] do a[n]:=5*(a[n-1]-a[n-2]); od; a; # G. C. Greubel, Dec 28 2019

CROSSREFS

Cf. A000032, A000045.

Sequence in context: A092490 A094828 A093131 * A224422 A000344 A290922

Adjacent sequences:  A030188 A030189 A030190 * A030192 A030193 A030194

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 28 02:22 EST 2020. Contains 338699 sequences. (Running on oeis4.)