A115341 -id:A115341 - cp's OEIS frontend

A015585 a(n) = 9a(n-1) + 10a(n-2).

0, 1, 9, 91, 909, 9091, 90909, 909091, 9090909, 90909091, 909090909, 9090909091, 90909090909, 909090909091, 9090909090909, 90909090909091, 909090909090909, 9090909090909091, 90909090909090909, 909090909090909091, 9090909090909090909, 90909090909090909091

Offset: 0

Olivier Gérard

Number of walks of length n between any two distinct nodes of the complete graph K_11. Example: a(2)=9 because the walks of length 2 between the nodes A and B of the complete graph ABCDEFGHIJK are: ACB, ADB, AEB, AFB, AGB, AHB, AIB, AJB and AKB. - Emeric Deutsch, Apr 01 2004
Beginning with n=1 and a(1)=1, these are the positive integers whose balanced base-10 representations (A097150) are the first n digits of 1,-1,1,-1,.... Also, a(n) = (-1)^(n-1)*A014992(n) = |A014992(n)| for n >= 1. - Rick L. Shepherd, Jul 30 2004
General form: k=10^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A093134, A015565, A015577. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

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.
Index entries for linear recurrences with constant coefficients, signature (9,10).

Cf. A014992 (q-integers for q=-10), A097150.
Cf. A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A093134, A015565, A015577. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008
Cf. A094028, A098610, A098611.

[Round(10^n/11): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011

k=0;lst={k};Do[k=10^n-k;AppendTo[lst, k], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
LinearRecurrence[{9,10},{0,1},30] (* Harvey P. Dale, Aug 08 2014 *)

a(n)=10^n\/11 \\ Charles R Greathouse IV, Jun 24 2011

[lucas_number1(n,9,-10) for n in range(0, 19)] # Zerinvary Lajos, Apr 26 2009

[abs(gaussian_binomial(n,1,-10)) for n in range(0,19)] # Zerinvary Lajos, May 28 2009

a(n) = 9*a(n-1) + 10*a(n-2).
From Emeric Deutsch, Apr 01 2004: (Start)
a(n) = 10^(n-1) - a(n-1).
G.f.: x/(1 - 9x - 10x^2). (End)
From Henry Bottomley, Sep 17 2004: (Start)
a(n) = round(10^n/11).
a(n) = (10^n - (-1)^n)/11.
a(n) = A098611(n)/11 = 9*A094028(n+1)/A098610(n). (End)
E.g.f.: exp(-x)*(exp(11*x) - 1)/11. - Elmo R. Oliveira, Aug 17 2024

Extended by T. D. Noe, May 23 2011

A109501 Number of closed walks of length n on the complete graph on 7 nodes from a given node.

Original entry on oeis.org

1, 0, 6, 30, 186, 1110, 6666, 39990, 239946, 1439670, 8638026, 51828150, 310968906, 1865813430, 11194880586, 67169283510, 403015701066, 2418094206390, 14508565238346, 87051391430070, 522308348580426, 3133850091482550, 18803100548895306, 112818603293371830

Offset: 0

Mitch Harris, Jun 30 2005

G. C. Greubel, Table of n, a(n) for n = 0..1000
Ji Young Choi, A Generalization of Collatz Functions and Jacobsthal Numbers, J. Int. Seq., Vol. 21 (2018), Article 18.5.4.
Christopher R. Kitching, Henri Kauhanen, Jordan Abbott, Deepthi Gopal, Ricardo Bermúdez-Otero, and Tobias Galla, Estimating transmission noise on networks from stationary local order, arXiv:2405.12023 [cond-mat.stat-mech], 2024. See p. 48.
Index entries for linear recurrences with constant coefficients, signature (5,6).

Cf. A109502.

Cf. sequences with the same recurrence form: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A015540. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

[(6^n + 6*(-1)^n)/7: n in [0..30]]; // G. C. Greubel, Dec 30 2017

k=0;lst={k};Do[k=6^n-k;AppendTo[lst, k], {n, 1, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
CoefficientList[Series[(1 - 5*x)/(1 - 5*x - 6*x^2), {x, 0, 50}], x] (* or *) LinearRecurrence[{5,6}, {1,0}, 30] (* G. C. Greubel, Dec 30 2017 *)

for(n=0,30, print1((6^n + 6*(-1)^n)/7, ", ")) \\ G. C. Greubel, Dec 30 2017

G.f.: (1 - 5*x)/(1 - 5*x - 6*x^2).
a(n) = (6^n + 6*(-1)^n)/7.
a(n) = 6^(n-1) - a(n-1), a(0) = 1. - Jon E. Schoenfield, Feb 09 2015
a(n) = 5*a(n-1) + 6*a(n-2). - G. C. Greubel, Dec 30 2017
E.g.f.: exp(-x)*(exp(7*x) + 6)/7. - Elmo R. Oliveira, Aug 17 2024

Corrected by Franklin T. Adams-Watters, Sep 18 2006

Edited by Jon E. Schoenfield, Feb 09 2015

A015577 a(n+1) = 8a(n) + 9a(n-1), a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 8, 73, 656, 5905, 53144, 478297, 4304672, 38742049, 348678440, 3138105961, 28242953648, 254186582833, 2287679245496, 20589113209465, 185302018885184, 1667718169966657, 15009463529699912, 135085171767299209, 1215766545905692880, 10941898913151235921

Offset: 0

Olivier Gérard

Binomial transform is A011557, with a leading zero. - Paul Barry, Jul 09 2003
Number of walks of length n between any two distinct nodes of the complete graph K_10. Example: a(2) = 8 because the walks of length 2 between the nodes A and B of the complete graph ABCDEFGHIJ are: ACB, ADB, AEB, AFB, AGB, AHB, AIB and AJB. - Emeric Deutsch, Apr 01 2004
The ratio a(n+1)/a(n) converges to 9 as n approaches infinity. - Felix P. Muga II, Mar 09 2014

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.
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Index entries for linear recurrences with constant coefficients, signature (8,9).

Cf. A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A093134, A015565. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

[Round(9^n/10): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011

seq(round(9^n/10),n=0..25); # Mircea Merca, Dec 28 2010

k=0;lst={k};Do[k=9^n-k;AppendTo[lst, k], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
Table[(9^n - (-1)^n)/10, {n,0,30}] (* or *) LinearRecurrence[{8,9}, {0,1}, 30] (* G. C. Greubel, Jan 06 2018 *)

a[0]:0$
a[n]:=9^(n-1)-a[n-1]$
A015577(n):=a[n]$
makelist(A015577(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

A015577_vec(N=20)=Vec(O(x^N)+1/(1-8*x-9*x^2), -N-1) \\ M. F. Hasler, Jun 14 2008, edited Oct 25 2019

for(n=0,30, print1((9^n - (-1)^n)/10, ", ")) \\ G. C. Greubel, Jan 06 2018

apply( {A015577(n)=9^n\/10}, [0..25]) \\ M. F. Hasler, Oct 25 2019

[lucas_number1(n,8,-9) for n in range(0, 19)] # Zerinvary Lajos, Apr 25 2009

From Paul Barry, Jul 09 2003: (Start)
G.f.: x/((1+x)*(1-9*x)).
E.g.f. exp(4*x)*sinh(5*x)/5.
a(n) = (9^n - (-1)^n)/10. (End)
a(n) = 9^(n-1)-a(n-1). - Emeric Deutsch, Apr 01 2004
a(n) = round(9^n/10). - Mircea Merca, Dec 28 2010

Extended by T. D. Noe, May 23 2011

A093134 A Jacobsthal trisection.

Original entry on oeis.org

1, 0, 8, 56, 456, 3640, 29128, 233016, 1864136, 14913080, 119304648, 954437176, 7635497416, 61083979320, 488671834568, 3909374676536, 31274997412296, 250199979298360, 2001599834386888, 16012798675095096, 128102389400760776, 1024819115206086200, 8198552921648689608

Offset: 0

Paul Barry, Mar 23 2004

Counts closed walks at a vertex of the complete graph on 9 nodes K_9.

Second binomial transform is A047855.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,8).

Other sequences with a(n+1) = 8^n - a(n) are A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A015565. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Cf. A047855.

[(8^n/9+8*(-1)^n/9): n in [0..20]]; // Vincenzo Librandi, Oct 11 2011

k=0;lst={1, k};Do[k=8^n-k;AppendTo[lst, k], {n, 1, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
Table[(8^n + 8*(-1)^n)/9, {n,0,30}] (* or *) LinearRecurrence[{7,8}, {1,0}, 30] (* G. C. Greubel, Jan 06 2018 *)

for(n=0,30, print1((8^n + 8*(-1)^n)/9, ", ")) \\ G. C. Greubel, Jan 06 2018

G.f.: (1-7*x)/(1 - 7*x - 8*x^2).
a(n) = (8^n + 8*(-1)^n)/9.
a(n) = 8*A001045(3*n-3)/3.
From Elmo R. Oliveira, Aug 17 2024: (Start)
E.g.f.: exp(-x)*(exp(9*x) + 8)/9.
a(n) = 7*a(n-1) + 8*a(n-2) for n > 1. (End)

A015592 a(n) = 10a(n-1) + 11a(n-2).

Original entry on oeis.org

0, 1, 10, 111, 1220, 13421, 147630, 1623931, 17863240, 196495641, 2161452050, 23775972551, 261535698060, 2876892678661, 31645819465270, 348104014117971, 3829144155297680, 42120585708274481, 463326442791019290, 5096590870701212191, 56062499577713334100

Offset: 0

Olivier Gérard

Number of walks of length n between any two distinct nodes of the complete graph K_12. Example: a(2)=10 because the walks of length 2 between the nodes A and B of the complete graph ABCDEFGHIJKL are ACB, ADB, AEB, AFB, AGB, AHB, AIB, AJB, AKB and ALB. - Emeric Deutsch, Apr 01 2004

General form: k=11^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A093134, A015565, A015577, A015585. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (10,11).

Cf. A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A093134, A015565, A015577, A015585. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

[-(1/12)*(-1)^n+(1/12)*11^n: n in [0..20]]; // Vincenzo Librandi, Oct 11 2011

k=0;lst={k};Do[k=11^n-k;AppendTo[lst, k], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)

[lucas_number1(n,10,-11) for n in range(0, 18)] # Zerinvary Lajos, Apr 26 2009

a(n) = 11^(n-1) - a(n-1). G.f.: x/(1 - 10x - 11x^2). - Emeric Deutsch, Apr 01 2004
From Elmo R. Oliveira, Aug 17 2024: (Start)
E.g.f.: exp(5*x)*sinh(6*x)/6.
a(n) = (11^n - (-1)^n)/12. (End)

A015609 a(n) = 11a(n-1) + 12a(n-2).

Original entry on oeis.org

0, 1, 11, 133, 1595, 19141, 229691, 2756293, 33075515, 396906181, 4762874171, 57154490053, 685853880635, 8230246567621, 98762958811451, 1185155505737413, 14221866068848955, 170662392826187461

Offset: 0

Olivier Gérard

Number of walks of length n between any two distinct nodes of the complete graph K_13. Example: a(2)=11 because the walks of length 2 between the nodes A and B of the complete graph ABCDEFGHIJKLM are ACB, ADB, AEB, AFB, AGB, AHB, AIB, AJB, AKB, ALB and AMB. - Emeric Deutsch, Apr 01 2004

General form: k=12^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A093134, A015565, A015577, A015585. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (11,12).

Cf. A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A093134, A015565, A015577, A015585, A015592. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

[(1/13)*(12^n-(-1)^n): n in [0..20]]; // Vincenzo Librandi, Oct 11 2011

CoefficientList[Series[x/(1-11*x-12*x^2), {x, 0, 50}], x] (* or *) LinearRecurrence[{11,12}, {0,1}, 30] (* G. C. Greubel, Dec 30 2017 *)

x='x+O('x^30); concat([0], Vec(x/(1-11*x-12*x^2))) \\ G. C. Greubel, Dec 30 2017

[lucas_number1(n,11,-12) for n in range(0, 18)] # Zerinvary Lajos, Apr 27 2009

[abs(gaussian_binomial(n,1,-12)) for n in range(0,18)] # Zerinvary Lajos, May 28 2009

From Emeric Deutsch, Apr 01 2004: (Start)
a(n) = 12^(n-1) - a(n-1).
G.f.: x/(1 - 11*x - 12*x^2). (End)
E.g.f.: exp(-x)*(exp(13*x) - 1)/13. - Stefano Spezia, Mar 11 2020

cp's OEIS Frontend

A015585 a(n) = 9*a(n-1) + 10*a(n-2).

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A109501 Number of closed walks of length n on the complete graph on 7 nodes from a given node.

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A015577 a(n+1) = 8*a(n) + 9*a(n-1), a(0) = 0, a(1) = 1.

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A093134 A Jacobsthal trisection.

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A015592 a(n) = 10*a(n-1) + 11*a(n-2).

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A015609 a(n) = 11*a(n-1) + 12*a(n-2).

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Mathematica

A015585 a(n) = 9a(n-1) + 10a(n-2).

A015577 a(n+1) = 8a(n) + 9a(n-1), a(0) = 0, a(1) = 1.

A015592 a(n) = 10a(n-1) + 11a(n-2).

A015609 a(n) = 11a(n-1) + 12a(n-2).