A053573 -id:A053573 - cp's OEIS frontend

A015565 a(n) = 7a(n-1) + 8a(n-2), a(0) = 0, a(1) = 1.

0, 1, 7, 57, 455, 3641, 29127, 233017, 1864135, 14913081, 119304647, 954437177, 7635497415, 61083979321, 488671834567, 3909374676537, 31274997412295, 250199979298361, 2001599834386887, 16012798675095097, 128102389400760775, 1024819115206086201, 8198552921648689607

Offset: 0

Olivier Gérard

A linear 2nd order recurrence. A Jacobsthal number sequence.
Binomial transform of A053573 (preceded by zero). - Paul Barry, Apr 09 2003
Second binomial transform of A080424. Binomial transform of A053573, with leading zero. Binomial transform is 0,1,9,81,729,....(9^n - 0^n)/9. Second binomial transform is 0,1,11,111,1111,... (A002275: repunits). - Paul Barry, Mar 14 2004
Number of walks of length n between any two distinct nodes of the complete graph K_9. Example: a(2)=7 because the walks of length 2 between the nodes A and B of the complete graph ABCDEFGHI are: ACB, ADB, AEB, AFB, AGB, AHB and AIB. - Emeric Deutsch, Apr 01 2004
Unsigned version of A014990. - Philippe Deléham, Feb 13 2007
General form: k=8^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008
The ratio a(n+1)/a(n) converges to 8 as n approaches infinity. - Felix P. Muga II, Mar 09 2014

			G.f. = x + 7*x^2 + 57*x^3 + 455*x^4 + 3641*x^5 + 29127*x^6 + 233017*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jean-Paul Allouche, Jeffrey Shallit, Zhixiong Wen, Wen Wu, and Jiemeng Zhang, Sum-free sets generated by the period-k-folding sequences and some Sturmian sequences, arXiv:1911.01687 [math.CO], 2019.
M. Dukes and C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
Dale Gerdemann, Fractal generated from (7,8) recursion, YouTube Video, Dec 5, 2014.
Index entries for linear recurrences with constant coefficients, signature (7,8).

Cf. A002275, A014990, A053573, A080424, A082311, A082365.
Cf. A062160, A099322, A106566.
Cf. A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

[Round(8^n/9): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011

seq(round(8^n/9),n=0..25); # Mircea Merca, Dec 28 2010

k=0;lst={k};Do[k=8^n-k;AppendTo[lst, k], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
LinearRecurrence[{7,8},{0,1},30] (* Harvey P. Dale, Mar 04 2016 *)

x='x+O('x^30); concat([0], Vec(x/(1-7*x-8*x^2))) \\ G. C. Greubel, Dec 30 2017

[lucas_number1(n,7,-8) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009

From Paul Barry, Apr 09 2003: (Start)
a(n) = (8^n - (-1)^n)/9.
a(n) = J(3*n)/3 = A001045(3*n)/3. (End)
From Emeric Deutsch, Apr 01 2004: (Start)
a(n) = 8^(n-1) - a(n-1).
G.f.: x/(1-7*x-8*x^2). (End)
a(n) = Sum_{k = 0..n} A106566(n,k)*A099322(k). - Philippe Deléham, Oct 30 2008
a(n) = round(8^n/9). - Mircea Merca, Dec 28 2010
From Peter Bala, May 31 2024: (Start)
G.f: A(x) = x/(1 - x^2) o x/(1 - x^2), where o denotes the black diamond product of power series as defined by Dukes and White. Cf. A054878.
The black diamond product A(x) o A(x) is the g.f. for the number of walks of length n between any two distinct nodes of the complete graph K_81.
Row 8 of A062160. (End)
E.g.f.: exp(-x)*(exp(9*x) - 1)/9. - Elmo R. Oliveira, Aug 17 2024

A053428 a(n) = a(n-1) + 20*a(n-2), n >= 2; a(0)=1, a(1)=1.

Original entry on oeis.org

1, 1, 21, 41, 461, 1281, 10501, 36121, 246141, 968561, 5891381, 25262601, 143090221, 648342241, 3510146661, 16476991481, 86679924701, 416219754321, 2149818248341, 10474213334761, 53470578301581, 262954844996801

Offset: 0

Barry E. Williams, Jan 10 2000

Hankel transform is 1,20,0,0,0,0,0,0,0,0,0,0,... - Philippe Deléham, Nov 02 2008

Zero followed by this sequence gives the inverse binomial transform of A080424. - Paul Curtz, Jun 07 2011

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.
A. K. Whitford, Binet's Formula Generalized, Fibonacci Quarterly, Vol. 15, No. 1, 1979, pp. 21, 24, 29.
Index entries for linear recurrences with constant coefficients, signature (1,20).

Cf. A001045, A015441, A053404, A000302, A053573, A080424.

[((5^(n+1))-(-4)^(n+1)) div 9: n in [0..40]]; // Vincenzo Librandi, Jun 07 2011

Join[{a=1,b=1},Table[c=b+20*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)

a(n)=((5^(n+1))-(-4)^(n+1))/9 \\ Charles R Greathouse IV, Jun 10 2011

a(n) = ((5^(n+1)) - (-4)^(n+1))/9.

G.f.: 1/((1+4*x)*(1-5*x)). - R. J. Mathar, Nov 16 2007

More terms from James Sellers, Feb 02 2000

A191566 a(n) = 7a(n-1) + (-1)^n6*2^(n-1).

Original entry on oeis.org

1, 1, 19, 109, 811, 5581, 39259, 274429, 1921771, 13450861, 94159099, 659107549, 4613765131, 32296331341, 226074368539, 1582520481469, 11077643566891, 77543504575021, 542804532811579, 3799631728108189

Offset: 0

Paul Curtz, Jun 06 2011

A007283(n) = 3*2^n. A091629(n+1) = 6*2^n.
a(n) + a(n+2) = 10 * (b(n) = 2, 11, 83, 569, 4007, ...).
b(n+1) = 7*b(n) - (-1)^n*3*2^n.
Inverse binomial transform of A007613(n).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (5,14).

[(7^n+2*(-2)^n)/3: n in [0..30]]; // Vincenzo Librandi, Jun 07 2011

LinearRecurrence[{5,14},{1,1},40] (* Harvey P. Dale, Mar 01 2017 *)
CoefficientList[Series[(1 - 4*x)/(1 - 5*x - 14*x^2), {x, 0, 20}], x] (* Stefano Spezia, Sep 12 2018 *)

a(n)=(7^n+2*(-2)^n)/3 \\ Charles R Greathouse IV, Jun 06 2011

a(n+1) - a(n) = 18 * (0 followed by A053573(n)).
a(n) = (7^n + 2*(-2)^n)/3. - Charles R Greathouse IV, Jun 06 2011
G.f.: (1-4*x)/(1 - 5*x - 14*x^2). - Bruno Berselli, Jun 07 2011
a(n) = 5*a(n-1) + 14*a(n-2).

cp's OEIS Frontend

A015565 a(n) = 7*a(n-1) + 8*a(n-2), a(0) = 0, a(1) = 1.

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A053428 a(n) = a(n-1) + 20*a(n-2), n >= 2; a(0)=1, a(1)=1.

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A191566 a(n) = 7*a(n-1) + (-1)^n*6*2^(n-1).

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Magma

Mathematica

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Formula

A015565 a(n) = 7a(n-1) + 8a(n-2), a(0) = 0, a(1) = 1.

A191566 a(n) = 7a(n-1) + (-1)^n6*2^(n-1).