A054878 -id:A054878 - cp's OEIS frontend

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

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)

A218034 Number of ways to seat 4 types of people in n labeled seats around a circle such that no two adjacent people are of the same type.

Original entry on oeis.org

1, 4, 12, 24, 84, 240, 732, 2184, 6564, 19680, 59052, 177144, 531444, 1594320, 4782972, 14348904, 43046724, 129140160, 387420492, 1162261464, 3486784404, 10460353200, 31381059612, 94143178824, 282429536484, 847288609440, 2541865828332, 7625597484984, 22876792454964

Offset: 0

Amritpal Singh, Oct 19 2012

Number of length-n words with 4 letters and no two adjacent identical letters (including, for n >= 2, the first and last letter). - Joerg Arndt, Oct 21 2012
a(n), for n > 1, apparently is the trace of the n-th power of the adjacency matrix of the complete 4-graph, a 4 X 4 matrix with diagonal elements all zeros and off-diagonal all ones (cf. A054878). - Tom Copeland, Nov 06 2012
The corrected formula by Geoffrey Critzer below (for a general k) is a special case of Theorem 2 in Burstein and Wilf (1997). See also Edlin and Zeilberger (2000), Corollary 5.5 in Taylor (2014), and Section 5 in Hadjicostas and Zhang (2018). - Petros Hadjicostas, Jul 09 2018

K. Böhmová, C. Dalfó, C. Huemer, On cyclic Kautz digraphs, Preprint 2016.
A. Burstein and H. S. Wilf, On cyclic strings without long constant blocks, Fibonacci Quarterly, 35 (1997), 240-247.
Cristina Dalfó, From subKautz digraphs to cyclic Kautz digraphs, arXiv:1709.01882 [math.CO], 2017.
C. Dalfó, The spectra of subKautz and cyclic Kautz digraphs, Linear Algebra and its Applications, 531 (2017), pp. 210-219.
A. E. Edlin and D. Zeilberger, The Goulden-Jackson cluster method for cyclic words, Adv. Appl. Math., 25 (2000), 228-232.
Petros Hadjicostas and Lingyun Zhang, On cyclic strings avoiding a pattern, Discrete Mathematics, 341 (2018), 1662-1674.
Jair Taylor, Counting Words with Laguerre Series, Electron. J. Combin., 21 (2014), P2.1.
Index entries for linear recurrences with constant coefficients, signature (2,3).

Cf. A092297.

nn=28; CoefficientList[Series[1+4x +12x^2/(1+x)^2/(1-4x/(1+x)),{x,0,nn}],x] (* Geoffrey Critzer, Apr 05 2014 *)

a[0]:1$ a[1]:4$ a[n]:=3^n + 3*(-1)^n$ makelist(a[n],n,0,40); /* Martin Ettl, Oct 21 2012 */

a(n) = if( n<2, [1,4][n+1], 3^n + 3*(-1)^n ); /* Joerg Arndt, Oct 21 2012 */

a(0) = 1, a(1) = 4, a(n) = 3^n + 3*(-1)^n for n >= 2.
a(n) = 4 * A054878(n) for n >= 2. - Joerg Arndt, Oct 21 2012
G.f.: (1 + 2*x + x^2 - 12*x^3)/((1 + x)*(1 - 3*x)). - Colin Barker, Oct 22 2012
Generally for such words on k letters, g.f.: 1 + k*x + (k^2-k)*x^2/(1 + x)^2/(1 - k*x/(1 + x)). - Geoffrey Critzer, Apr 05 2014 [Corrected by Petros Hadjicostas, Jul 08 2018]
a(n+m) = a(n)*a(m)/4 + a(n+1)*a(m+1)/12. - Yuchun Ji, Sep 12 2017

a(0) = 1 prepended and more terms added by Joerg Arndt, Oct 21 2012

A006342 Coloring a circuit with 4 colors.

Original entry on oeis.org

1, 1, 4, 10, 31, 91, 274, 820, 2461, 7381, 22144, 66430, 199291, 597871, 1793614, 5380840, 16142521, 48427561, 145282684, 435848050, 1307544151, 3922632451, 11767897354, 35303692060, 105911076181, 317733228541, 953199685624, 2859599056870, 8578797170611

Offset: 0

N. J. A. Sloane, Simon Plouffe

Also equal to the number of set partitions of {1,2,...,n+2} with at most 4 parts such that each part does not contain both i,i+1 for 1<=iMike Zabrocki, Sep 08 2020

Also a(n) equals the number of color-complete multipoles with n terminals (that is, having all the states allowed by the Parity Lemma). - Miquel A. Fiol, May 27 2024

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
F. R. Bernhart, Topics in Graph Theory Related to the Five Color Conjecture, Ph.D. Dissertation, Kansas State Univ., 1974.
F. R. Bernhart, Fundamental chromatic numbers, Unpublished. (Annotated scanned copy)
F. R. Bernhart & N. J. A. Sloane, Correspondence, 1977
G. D. Birkhoff, D. C. Lewis, Chromatic polynomials, Trans. Amer. Math. Soc. 60, (1946). 355-451.
Gesualdo Delfino and Jacopo Viti, Potts q-color field theory and scaling random cluster model, arXiv preprint arXiv:1104.4323 [hep-th], 2011.
M. A. Fiol and J. Vilaltella, Some results on the structure of multipoles in the study of snarks, Electron. J. Combin. 22(1) (2015), #P1.45.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (3,1,-3).

A214142, A214167

[3*3^n/8+1/4+3*(-1)^n/8: n in [0..30]]; // Vincenzo Librandi, Aug 20 2011

A006342:=-(-1+2*z)/(z-1)/(3*z-1)/(z+1); # conjectured by Simon Plouffe in his 1992 dissertation
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+3*a[n-2]-1 od: seq(a[n], n=1..26); # Zerinvary Lajos, Apr 28 2008

CoefficientList[Series[(1-2 x)/((1-x^2) (1-3 x)),{x,0,30}],x] (* or *) LinearRecurrence[{3,1,-3},{1,1,4},30] (* Harvey P. Dale, Aug 16 2016 *)

Vec((1 - 2*x) / ((1 - x)*(1 + x)*(1 - 3*x)) + O(x^30)) \\ Colin Barker, Nov 07 2017

G.f.: (1 - 2 x ) / (( 1 - x^2 ) ( 1 - 3 x )).
Binomial transform of A002001 (with interpolated zeros). Partial sums of A054878. E.g.f.: exp(x)(3*cosh(2*x) + 1)/4; a(n) = 3*3^n/8 + 1/4 + 3(-1)^n/8 = Sum_{k=0..n} (3^k + 3(-1)^k)/4. - Paul Barry, Sep 03 2003
a(n) = 2*a(n-1) + 3*a(n-2) - 1, n > 1. - Gary Detlefs, Jun 21 2010
a(n) = a(n-1) + A054878(n-2). - Yuchun Ji, Sep 12 2017
From Colin Barker, Nov 07 2017: (Start)
a(n) = (3^(n+1) + 5) / 8 for n even.
a(n) = (3^(n+1) - 1) / 8 for n odd.
a(n) = 3*a(n-1) + a(n-2) - 3*a(n-3) for n > 2.
(End)
a(n) = 3*a(n-1) + (3*(-1)^n - 1)/2 for n > 0. - Yuchun Ji, Dec 05 2019

A084567 Binomial transform of (1,-1,4,-16,64,-256,1024,...) = (3*0^n-(-4)^n)/4.

Original entry on oeis.org

1, 0, 3, -6, 21, -60, 183, -546, 1641, -4920, 14763, -44286, 132861, -398580, 1195743, -3587226, 10761681, -32285040, 96855123, -290565366, 871696101, -2615088300, 7845264903, -23535794706, 70607384121, -211822152360, 635466457083, -1906399371246

Offset: 0

Paul Barry, May 30 2003

Partial sums of (1,-1,3,-9,27,-81,....) (with g.f. (1+2x)/(1+3x) ).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2,3)

Cf. A054878 (absolute values).

I:=[1,0]; [n le 2 select I[n] else -2*Self(n-1)+3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Oct 07 2013

CoefficientList[Series[(1 + 2 x)/((1 - x) (1 + 3 x)), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 07 2013 *)
LinearRecurrence[{-2,3},{1,0},30] (* Harvey P. Dale, Aug 26 2024 *)

Vec((1+2*x)/((1-x)*(1+3*x))+O(x^66)) \\ Joerg Arndt, Jul 14 2013

G.f.: (1+2*x)/((1-x)*(1+3*x)).

G.f.: 1+ x -x/Q(0), where Q(k) = 1 + 3*x^2 + (3*k+4)*x - x*(3*k+1 + 3*x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 07 2013

Removed incorrect g.f. and e.g.f., Joerg Arndt, Jul 14 2013

A091562 Triangle read by rows, related to Pascal's triangle, starting with 1, 0, 0.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 1, 2, 2, 1, 2, 5, 7, 5, 2, 3, 10, 17, 17, 10, 3, 5, 20, 41, 51, 41, 20, 5, 8, 38, 91, 136, 136, 91, 38, 8, 13, 71, 195, 339, 405, 339, 195, 71, 13, 21, 130, 403, 799, 1107, 1107, 799, 403, 130, 21, 34, 235, 812, 1807, 2845, 3297, 2845, 1807, 812, 235, 34

Offset: 0

Christian G. Bower, Jan 20 2004

			Triangle begins:
  1;
  0,0;
  1,1,1;
  1,2,2,1;
  2,5,7,5,2;
  ...

Row sums: A054878, column 0: A000045(n-1), column 1: A001629.
Cf. A090171, A090172, A090173, A090174, A091533, A205575 (same recurrence).
Cf. A090172.

T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k) + T(n-2, k-1) + T(n-2, k-2) for n >= 2, k >= 0, with initial conditions specified by first two rows.

G.f.: A(x, y) = (1-x-x*y)/(1-x-x*y-x^2-x^2*y-x^2*y^2).

A109502 Array read by antidiagonals: T(m,n) is the number of closed walks of length n on the complete graph on m nodes, m >= 1, n >= 0.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 1, 0, 3, 2, 1, 0, 1, 0, 4, 6, 6, 0, 0, 1, 0, 5, 12, 21, 10, 1, 0, 1, 0, 6, 20, 52, 60, 22, 0, 0, 1, 0, 7, 30, 105, 204, 183, 42, 1, 0, 1, 0, 8, 42, 186, 520, 820, 546, 86, 0, 0, 1, 0, 9, 56, 301, 1110, 2605, 3276, 1641, 170, 1, 0

Offset: 1

Mitch Harris, Jun 30 2005

			Array begins:
  m\n| 0  1  2  3   4    5     6      7       8        9        10
  ---+------------------------------------------------------------
   1 | 1  0  0  0   0    0     0      0       0        0         0
   2 | 1  0  1  0   1    0     1      0       1        0         1
   3 | 1  0  2  2   6   10    22     42      86      170       342
   4 | 1  0  3  6  21   60   183    546    1641     4920     14763
   5 | 1  0  4 12  52  204   820   3276   13108    52428    209716
   6 | 1  0  5 20 105  520  2605  13020   65105   325520   1627605
   7 | 1  0  6 30 186 1110  6666  39990  239946  1439670   8638026
   8 | 1  0  7 42 301 2100 14707 102942  720601  5044200  35309407
   9 | 1  0  8 56 456 3640 29128 233016 1864136 14913080 119304648
  10 | 1  0  9 72 657 5904 53145 478296 4304673 38742048 348678441

M. Dukes and C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.

Rows are A078008, A054878, A109499, A109500, A109501.

Cf. A062160.

T := proc(m, n); ((m-1)^n + (m-1)*(-1)^n)/m end:
seq(print(seq(T(m, n), n = 0..10)), m = 1..10); # Peter Bala, May 30 2024

T(m,n) = ((m-1)^n + (m-1)(-1)^n)/m.
G.f.: T(m, n) = [z^n](1 - (m-2)z)/(1 - (m-2)z - (m-1)z^2).
From Peter Bala, May 29 2024: (Start)
Binomial transform of the m-th row: Sum_{k = 0..n} binomial(n, k)*T(m, k) = m^(n-1) for n >= 1.
Let R(m, x) denote the g.f. of the m-th row of the square array. Then R(m_1, x) o R(m_2, x) = R(m_1*m_2, x), where o denotes the black diamond product of power series as defined by Dukes and White. Cf. A062160.
T(m_1*m_2, n) = Sum_{k = 0..n} Sum_{i = k..n} binomial(n, k)*binomial(n-k, i-k)*T(m_1, i)*T(m_2, n-k). (End)

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

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)

A092810 Binomial transform of a Jacobsthal trisection.

Original entry on oeis.org

1, 6, 54, 486, 4374, 39366, 354294, 3188646, 28697814, 258280326, 2324522934, 20920706406, 188286357654, 1694577218886, 15251194969974, 137260754729766, 1235346792567894, 11118121133111046, 100063090197999414, 900567811781994726, 8105110306037952534

Offset: 0

Paul Barry, Mar 10 2004

Binomial transform of A082311.

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

Cf. A001045.

[1] cat [2*3^(2*n-1): n in [1..20]]; // Vincenzo Librandi, Jun 20 2015

Table[EulerPhi[9^n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Nov 10 2009 *)

Vec((1-3*x)/(1-9*x) + O(x^30)) \\ Michel Marcus, Jun 18 2015

G.f.: (1-3*x)/(1-9*x).
E.g.f.: 2*exp(9*x)/3 + 1/3.
a(n) = 2*9^n/3 + 0^n/3.
a(n) = A054878(2n+1) - A054878(2n-1) + 0^n/3 = A015518(2n+1) - A015518(2n-1) + 0^n/3.
a(n) = 2*3^(2*n-1), for n>0. - Gionata Neri, Jun 18 2015

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

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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A218034 Number of ways to seat 4 types of people in n labeled seats around a circle such that no two adjacent people are of the same type.

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A006342 Coloring a circuit with 4 colors.

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Magma

Maple

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A084567 Binomial transform of (1,-1,4,-16,64,-256,1024,...) = (3*0^n-(-4)^n)/4.

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A091562 Triangle read by rows, related to Pascal's triangle, starting with 1, 0, 0.

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A109502 Array read by antidiagonals: T(m,n) is the number of closed walks of length n on the complete graph on m nodes, m >= 1, n >= 0.

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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).