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A020876 a(n) = ((5+sqrt(5))/2)^n+((5-sqrt(5))/2)^n. 14
2, 5, 15, 50, 175, 625, 2250, 8125, 29375, 106250, 384375, 1390625, 5031250, 18203125, 65859375, 238281250, 862109375, 3119140625, 11285156250, 40830078125, 147724609375, 534472656250, 1933740234375, 6996337890625, 25312988281250, 91583251953125 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Number of no-leaf edge-subgraphs in Moebius ladder M_n.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

J. P. McSorley, Counting structures in the Moebius ladder, Discrete Math., 184 (1998), 137-164.

Index entries for linear recurrences with constant coefficients, signature (5,-5).

FORMULA

Also, a(n) = (sqrt(5)*phi)^n + (sqrt(5)/phi)^n, where phi = golden ratio. - N. J. A. Sloane, Aug 08 2014

Let S(n, m)=sum(k=0, n, binomial(n, k)*fibonacci(m*k)), then for n>0 a(n)= S(2*n, 2)/S(n, 2). - Benoit Cloitre, Oct 22 2003

a(n)= 5*a(n-1) -5*a(n-2). G.f.: (2-5*x)/(1-5*x+5*x^2). - R. J. Mathar, Feb 06 2010

From Johannes W. Meijer, Jul 01 2010: (Start)

Limit(a(n+k)/a(k), k=infinity) = (A020876(n) + A093131(n)*sqrt(5))/2.

Limit(A020876(n)/A093131(n), n=infinity) = sqrt(5).

(End)

Binomial transform of A005248. - Carl Najafi, Sep 10 2011

a(n) = 2*A030191(n)-5*A030191(n-1). - R. J. Mathar, Mar 02 2012

EXAMPLE

G.f. = 2 + 5*x + 15*x^2 + 50*x^3 + 175*x^4 + 625*x^5 + 2250*x^6 + ...

MAPLE

G:=(x, n)-> cos(x)^n+cos(3*x)^n:

seq(simplify(4^n*G(Pi/10, 2*n)), n=0..22); # Gary Detlefs, Dec 05 2010

MATHEMATICA

Table[Sum[LucasL[2*i] Binomial[n, i], {i, 0, n}], {n, 0, 50}] (* T. D. Noe, Sep 10 2011 *)

CoefficientList[Series[(2 - 5 x)/(1 - 5 x + 5 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 08 2014 *)

LinearRecurrence[{5, -5}, {2, 5}, 30] (* Harvey P. Dale, Mar 13 2016 *)

PROG

(Sage) [lucas_number2(n, 5, 5) for n in xrange(0, 24)] # Zerinvary Lajos, Jul 08 2008

(MAGMA) [Floor(((5+Sqrt(5))/2)^n+((5-Sqrt(5))/2)^n): n in [0..30]]; // Vincenzo Librandi, Aug 08 2014

CROSSREFS

Cf. A020876, A093131, A005248.

Appears in A109106. - Johannes W. Meijer, Jul 01 2010

Sequence in context: A149947 A149948 A093129 * A228343 A149949 A149950

Adjacent sequences:  A020873 A020874 A020875 * A020877 A020878 A020879

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Simplified definition. - N. J. A. Sloane, Aug 08 2014

STATUS

approved

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Last modified May 22 05:03 EDT 2019. Contains 323473 sequences. (Running on oeis4.)