A015540 a(n) = 5*a(n-1) + 6*a(n-2), a(0) = 0, a(1) = 1.

0, 1, 5, 31, 185, 1111, 6665, 39991, 239945, 1439671, 8638025, 51828151, 310968905, 1865813431, 11194880585, 67169283511, 403015701065, 2418094206391, 14508565238345, 87051391430071, 522308348580425, 3133850091482551, 18803100548895305, 112818603293371831

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• easy

Comments

Number of walks of length n between any two distinct vertices of the complete graph K_7. Example: a(2)=5 because the walks of length 2 between the vertices A and B of the complete graph ABCDEFG are ACB, ADB, AEB, AFB and AGB. - Emeric Deutsch, Apr 01 2004

General form: k=6^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500. -Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Pisano period lengths: 1, 1, 2, 2, 2, 2, 14, 2, 2, 2, 10, 2, 12, 14, 2, 2, 16, 2, 18, 2, ... - R. J. Mathar, Aug 10 2012

Sum_{i=0..m} (-1)^(m+i)*6^i, for m >= 0, gives all terms after 0. - Bruno Berselli, Aug 28 2013

The ratio a(n+1)/a(n) converges to 6 as n approaches infinity. Also A053524, A080424, A051958. - Felix P. Muga II, Mar 09 2014

Examples

			G.f. = x + 5*x^2 + 31*x^3 + 185*x^4 + 1111*x^5 + 6665*x^6 + 39991*x^7 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Jean-Paul Allouche, Jeffrey Shallit, Zhixiong Wen, Wen Wu, Jiemeng Zhang, Sum-free sets generated by the period-k-folding sequences and some Sturmian sequences, arXiv:1911.01687 [math.CO], 2019.
• Ji Young Choi, A Generalization of Collatz Functions and Jacobsthal Numbers, J. Int. Seq., Vol. 21 (2018), Article 18.5.4.
• F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.
• Index entries for linear recurrences with constant coefficients, signature (5,6).

Crossrefs

Partial sums are in A033116. Cf. A014987.

Programs

Magma

[Floor(6^n/7-(-1)^n/7): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011

Maple

seq(round(6^n/7),n=0..25); # Mircea Merca, Dec 28 2010

Mathematica

k=0; lst={k}; Do[k = 6^n-k; AppendTo[lst, k], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
CoefficientList[Series[x / ((1 - 6 x) (1 + x)), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 26 2014 *)
LinearRecurrence[{5,6},{0,1},30] (* Harvey P. Dale, May 12 2015 *)

PARI

my(x='x+O('x^30)); concat([0], Vec(x/((1-6*x)*(1+x)))) \\ G. C. Greubel, Jan 24 2018

PARI

a(n) = round(6^n/7); \\ Altug Alkan, Sep 05 2018

Sage

[lucas_number1(n,5,-6) for n in range(21)] # Zerinvary Lajos, Apr 24 2009

Formula

a(n) = 5*a(n-1) + 6*a(n-2).

From Paul Barry, Apr 20 2003: (Start)

a(n) = (6^n - (-1)^n)/7.

G.f.: x/((1-6*x)*(1+x)).

E.g.f.: (exp(6*x) - exp(-x))/7. (End)

a(n) = 6^(n-1) - a(n-1). - Emeric Deutsch, Apr 01 2004

a(n+1) = Sum_{k=0..n} binomial(n-k, k)*5^(n-2*k)*6^k. - Paul Barry, Jul 29 2004

a(n) = round(6^n/7). - Mircea Merca, Dec 28 2010

a(n) = (-1)^(n-1)*Sum_{k=0..n-1} A135278(n-1,k)*(-7)^k = (6^n - (-1)^n)/7 = (-1)^(n-1)*Sum_{k=0..n-1} (-6)^k. Equals (-1)^(n-1)*Phi(n,-6), where Phi is the cyclotomic polynomial when n is an odd prime. (For n > 0.) - Tom Copeland, Apr 14 2014