A020876 a(n) = ((5+sqrt(5))/2)^n + ((5-sqrt(5))/2)^n.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

Examples

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Santiago Alzate, Oscar Correa, and Rigoberto Flórez, Fibonacci identities from Jordan Identities, arXiv:2009.02639 [math.NT], 2020.
• Noah Giansiracusa, Fibonacci, Golden Ratio, and Vector Bundles, Mathematics (2021) Vol. 9, Issue 4, 426.
• J. P. McSorley, Counting structures in the Moebius ladder, Discrete Math., 184 (1998), 137-164.
• Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019).
• Index entries for linear recurrences with constant coefficients, signature (5,-5).

Crossrefs

Cf. A020876, A093131, A005248.

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

Programs

Magma

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

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 *)

Sage

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

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

From R. J. Mathar, Feb 06 2010: (Start)

a(n)= 5*a(n-1) - 5*a(n-2).

G.f.: (2-5*x)/(1-5*x+5*x^2). (End)

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

Lim_{k->infinity} a(n+k)/a(k) = (A020876(n) + A093131(n)*sqrt(5))/2.

Lim_{k->infinity} A020876(n)/A093131(n) = 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

From Kai Wang, Dec 22 2019: (Start)

a((2*m+1)*k)/a(k) = Sum_{i=0..m-1} (-1)^(i*(k+1))*a(2*(m-i)*k) + 5^(m*k).

A093131(m+r)*A093131(n+s) + A093131(m+s)*A093131(n+r) = (2*a(m+n+r+s) - 5^(n+s)*a(m-n)*a(r-s))/5.

a(m+r)*a(n+s) - a(m+s)*a(n+r) = 5^(n+s+1)*A093131(m-n)*A093131(r-s).

a(m+r)*a(n+s) + a(m+s)*a(n+r) = 2*a(m+n+r+s) + 5^(n+s)*a(m-n)*a(r-s).

a(m+r)*a(n+s) - 5*A093131(m+s)*A093131(n+r) = 5^(n+s)*a(m-n)*a(r-s).

a(m+r)*a(n+s) + 5*A093131(m+s)*A093131(n+r) = 2*a(m+n+r+s)+ 5^(n+s+1)*A093131(m-n)*A093131(r-s).

A093131(m-n) = (A093131(m)*a(n) - a(m)*A093131(n))/(2*5^n).

A093131(m+n) = (A093131(m)*a(n) + a(m)*A093131(n))/2.

a(n)^2 - a(n+1)*a(n-1) = -5^n.

a(n)^2 - a(n+r)*a(n-r) = -5^(n-r+1)*A093131(r)^2.

a(m)*a(n+1) - a(m+1)*a(n) = -5^(n+1)*A093131(m-n).

a(m+n) - 5^(n)*a(m-n) = 5*A093131(m)*A093131(n).

a(m+n) + 5^(n)*a(m-n) = a(m)*a(n).

a(m-n) = (a(m)*a(n) - 5*A093131(m)*A093131(n))/(2*5^n).

a(m+n) = (a(m)*a(n) + 5*A093131(m)*A093131(n))/2. (End)

E.g.f.: 2*exp(5*x/2)*cosh(sqrt(5)*x/2). - Stefano Spezia, Dec 27 2019

Extensions

Definition simplified by N. J. A. Sloane, Aug 08 2014