A015552 a(n) = 6*a(n-1) + 7*a(n-2), a(0) = 0, a(1) = 1.

0, 1, 6, 43, 300, 2101, 14706, 102943, 720600, 5044201, 35309406, 247165843, 1730160900, 12111126301, 84777884106, 593445188743, 4154116321200, 29078814248401, 203551699738806, 1424861898171643, 9974033287201500, 69818233010410501, 488727631072873506

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• easy

Comments

Number of walks of length n between any two distinct nodes of the complete graph K_8. Example: a(2)=6 because the walks of length 2 between the nodes A and B of the complete graph ABCDEFGH are: ACB, ADB, AEB, AFB, AGB and AHB. - Emeric Deutsch, Apr 01 2004

General form: k=7^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

The ratio a(n+1)/a(n) converges to 7 as n approaches infinity. - Felix P. Muga II, Mar 09 2014

Examples

			G.f. = x + 6*x^2 + 43*x^3 + 300*x^4 + 2101*x^5 + 14706*x^6 + 102943*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.
• F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014.
• Index entries for linear recurrences with constant coefficients, signature (6,7).

Crossrefs

Cf. A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Programs

Magma

[Round(7^n/8): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011

Maple

seq(round(7^n/8),n=0..25); # Mircea Merca, Dec 28 2010

Mathematica

k=0;lst={k};Do[k=7^n-k;AppendTo[lst, k], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
Table[(7^n - (-1)^n)/8, {n,0,30}] (* G. C. Greubel, Dec 30 2017 *)

PARI

{a(n) = if ( n<0, 0, (7^n - (-1)^n) / 8)};

Sage

[lucas_number1(n,6,-7) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009

Formula

a(n) = 6*a(n-1) + 7*a(n-2).

From Emeric Deutsch, Apr 01 2004: (Start)

G.f.: x/(1-6*x-7*x^2).

a(n) = 7^(n-1) - a(n-1). (End)

a(n) = (7^n - (-1)^n)/8. - Rolf Pleisch, Jul 06 2009

a(n) = round(7^n/8). - Mircea Merca, Dec 28 2010

E.g.f. exp(3*x)*sinh(4*x)/4. - Elmo R. Oliveira, Aug 17 2024