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URL: https://oeis.org/A020876

⇱ A020876 - OEIS

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A020876
a(n) = ((5+sqrt(5))/2)^n + ((5-sqrt(5))/2)^n.
16
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
0,1
COMMENTS
Number of no-leaf edge-subgraphs in Moebius ladder M_n.
LINKS
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.
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
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
(SageMath) [lucas_number2(n, 5, 5) for n in range(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
Appears in A109106. - Johannes W. Meijer, Jul 01 2010
Sequence in context: A149947 A149948 A093129 * A228343 A149949 A149950
KEYWORD
nonn,easy
EXTENSIONS
Definition simplified by N. J. A. Sloane, Aug 08 2014
STATUS
approved