A015551 -id:A015551 - cp's OEIS frontend

A090018 a(n) = 6a(n-1) + 3a(n-2) for n > 2, a(0)=1, a(1)=6.

1, 6, 39, 252, 1629, 10530, 68067, 439992, 2844153, 18384894, 118841823, 768205620, 4965759189, 32099171994, 207492309531, 1341251373168, 8669985167601, 56043665125110, 362271946253463, 2341762672896108, 15137391876137037, 97849639275510546, 632510011281474387

Offset: 0

Paul Barry, Nov 19 2003

From Johannes W. Meijer, Aug 09 2010: (Start)

a(n) represents the number of n-move routes of a fairy chess piece starting in a given corner or side square on a 3 X 3 chessboard. This fairy chess piece behaves like a white queen on the eight side and corner squares but on the central square the queen explodes with fury and turns into a red queen, see A180032. The central square leads to A180028. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (6,3).

Cf. A141041, A180028, A180032.

Sequences with g.f. of the form 1/(1 - 6*x - k*x^2): A106392 (k=-10), A027471 (k=-9), A006516 (k=-8), A081179 (k=-7), A030192 (k=-6), A003463 (k=-5), A084326 (k=-4), A138395 (k=-3), A154244 (k=-2), A001109 (k=-1), A000400 (k=0), A005668 (k=1), A135030 (k=2), this sequence (k=3), A135032 (k=4), A015551 (k=5), A057089 (k=6), A015552 (k=7), A189800 (k=8), A189801 (k=9), A190005 (k=10), A015553 (k=11).

[n le 2 select 6^(n-1) else 6*Self(n-1)+3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 15 2011

a:= n-> (<<0|1>, <3|6>>^n. <<1,6>>)[1,1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 17 2011

Join[{a=1,b=6},Table[c=6*b+3*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
LinearRecurrence[{6,3}, {1,6}, 41] (* G. C. Greubel, Oct 10 2022 *)

my(x='x+O('x^30)); Vec(1/(1-6*x-3*x^2)) \\ G. C. Greubel, Jan 24 2018

[lucas_number1(n,6,-3) for n in range(1, 31)] # Zerinvary Lajos, Apr 24 2009

a(n) = (3+2*sqrt(3))^n*(sqrt(3)/4+1/2) + (1/2-sqrt(3)/4)*(3-2*sqrt(3))^n.
a(n) = (-i*sqrt(3))^n * ChebyshevU(n, isqrt(3)), i^2=-1.
From Johannes W. Meijer, Aug 09 2010: (Start)
G.f.: 1/(1 - 6*x - 3*x^2).
Limit_{k->oo} a(n+k)/a(k) = A141041(n) + A090018(n-1)*sqrt(12) for n >= 1.
Limit_{n->oo} A141041(n)/A090018(n-1) = sqrt(12). (End)
a(n) = Sum_{k=0..n} A099089(n,k)*3^k. - Philippe Deléham, Nov 21 2011
E.g.f.: exp(3*x)*(2*cosh(2*sqrt(3)*x) + sqrt(3)*sinh(2*sqrt(3)*x))/2. - Stefano Spezia, Apr 23 2025

Typo in Mathematica program corrected by Vincenzo Librandi, Nov 15 2011

A015553 Expansion of x/(1 - 6x - 11x^2).

Original entry on oeis.org

0, 1, 6, 47, 348, 2605, 19458, 145403, 1086456, 8118169, 60660030, 453260039, 3386820564, 25306783813, 189095729082, 1412948996435, 10557746998512, 78888920951857, 589468742694774, 4404590586639071, 32911699689476940

Offset: 0

Olivier Gérard

Let the generator matrix for the binary Golay G_24 code be [I|B]. Then a(n)=(A^n)1,2 for instance. Third binomial transform of (0,1,0,20,0,400,0,8000,...). - _Paul Barry, Feb 13 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,11).

Cf. A015551.

[n le 2 select n-1 else 6*Self(n-1) + 11*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 13 2012

a[n_]:=(MatrixPower[{{1,4},{1,-7}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{6, 11}, {0, 1}, 30] (* Vincenzo Librandi, Nov 13 2012 *)

x='x+O('x^30); concat([0], Vec(x/(1 - 6*x - 11*x^2))) \\ G. C. Greubel, Dec 30 2017

[lucas_number1(n,6,-11) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009

a(n) = 6*a(n-1) + 11*a(n-2).
a(n) = (1/4)*Sum_{k=0..n} binomial(n,k)*Fibonacci(k)*4^k. - Benoit Cloitre, Oct 25 2003
a(n) = sqrt(5)(3 + 2*sqrt(5))^n/20 - sqrt(5)(3 - 2*sqrt(5))^n/20. - Paul Barry, Feb 13 2004
E.g.f.: (exp(x*(3 + 2*sqrt(5))) - exp(x*(3 - 2*sqrt(5))))/(4*sqrt(5)). - Iain Fox, Dec 31 2017

A091928 a(0)=1, a(1)=5; a(n) = 6a(n-1) + 5a(n-2) for n > 1.

Original entry on oeis.org

1, 5, 35, 235, 1585, 10685, 72035, 485635, 3273985, 22072085, 148802435, 1003175035, 6763062385, 45594249485, 307380808835, 2072256100435, 13970440646785, 94183924382885, 634955749531235, 4280654119101835

Offset: 0

Paul Barry, Feb 13 2004

Let the generator matrix for the ternary Golay G_12 code be [I|B], where the elements of B are taken from the set {0,1,2}. Then a(n)=sum of first row of B^n.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,5).

Cf. A015551.

[n le 2 select 5^(n-1) else 6*Self(n-1) +5*Self(n-2): n in [1..41]]; // G. C. Greubel, Oct 27 2024

LinearRecurrence[{6,5},{1,5},30] (* Harvey P. Dale, Apr 09 2022 *)

A091928= BinaryRecurrenceSequence(6,5,1,5)
[A091928(n) for n in range(41)] # G. C. Greubel, Oct 27 2024

G.f.: (1-x)/(1-6*x-5*x^2).
a(n) = (1/2 +1/sqrt(14))*(3 +sqrt(14))^n + (1/2 -1/sqrt(14))*(3 -sqrt(14))^n.
From Philippe Deléham, Sep 22 2006: (Start)
a(n) = Sum_{k=0..n} 5^k*A122542(n,k).
Lim_{n->infinity} a(n+1)/a(n) = 3 + sqrt(14) = 6.741657386773... . (End)

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 05 2007

A189800 a(n) = 6a(n-1) + 8a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 6, 44, 312, 2224, 15840, 112832, 803712, 5724928, 40779264, 290475008, 2069084160, 14738305024, 104982503424, 747801460736, 5326668791808, 37942424436736, 270267896954880, 1925146777223168, 13713023838978048, 97679317251653632, 695780094221746176

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,8).

Sequences of the form a(n) = c*a(n-1) + d*a(n-2), with a(0)=0, a(1)=1:
c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10
.A000045,A001045,A006130,A006131,A015440,A015441,A015442,A015443,A015445,A015446
.A000129,A002605,A015518,A063727,A002532,A083099,A015519,A003683,A002534,A083102
.A006190,A007482,A030195,A015521,A015523,A083858,A015524,A015525,A099012,A015528
.A001076,A090017,A015530,A057087,A015531,A085939,A015532,A180222,A015533,A180226
.A052918,A015535,A015536,A015537,A057088,A015540,A015541,A015544,A015545,A180250
.A005668,A135030,A090018,A135032,A015551,A057089,A015552,A189800,A189801,A190005
.A054413,A015555,A015559,A015561,A015562,A015564,A057090,A015565,A015566,A015568
.A041025,A190331,A015574,A190510,A015575,A190560,A015576,A057091,A015577,A190953
.A099371,A015579,A181353,A015580,A015581,A153191,A015583,A015584,A057092,A015585
A041041,A191014,A015588,A190954,A190955,A190956,A015589,A190957,A015591,A057093

I:=[0,1]; [n le 2 select I[n] else 6*Self(n-1)+8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 14 2011

LinearRecurrence[{6, 8}, {0, 1}, 50]
CoefficientList[Series[-(x/(-1+6 x+8 x^2)),{x,0,50}],x] (* Harvey P. Dale, Jul 26 2011 *)

a(n)=([0,1; 8,6]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

G.f.: x/(1 - 2*x*(3+4*x)). - Harvey P. Dale, Jul 26 2011

A091927 Expansion of (1-6x)/(1-6x-5x^2).

Original entry on oeis.org

1, 0, 5, 30, 205, 1380, 9305, 62730, 422905, 2851080, 19221005, 129581430, 873593605, 5889468780, 39704780705, 267676028130, 1804580072305, 12165860574480, 82018063808405, 552937685722830, 3727716433379005

Offset: 0

Paul Barry, Feb 13 2004

Let the generating matrix of the Golay G_12 code be [I|B]. Then a(n)=(B^n)_1,1.

Index entries for linear recurrences with constant coefficients, signature (6,5).

Cf. A015551.

CoefficientList[Series[(1-6x)/(1-6x-5x^2),{x,0,30}],x] (* or *) LinearRecurrence[ {6,5},{1,0},30](* Harvey P. Dale, Aug 11 2019 *)

a(n)=5(sqrt(14)(3+sqrt(14))^(n-1)/28-sqrt(14)(3-sqrt(14))^(n-1)/28)

cp's OEIS Frontend

A090018 a(n) = 6*a(n-1) + 3*a(n-2) for n > 2, a(0)=1, a(1)=6.

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A015553 Expansion of x/(1 - 6*x - 11*x^2).

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A091928 a(0)=1, a(1)=5; a(n) = 6*a(n-1) + 5*a(n-2) for n > 1.

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A189800 a(n) = 6*a(n-1) + 8*a(n-2), with a(0)=0, a(1)=1.

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Magma

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A091927 Expansion of (1-6x)/(1-6x-5x^2).

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A090018 a(n) = 6a(n-1) + 3a(n-2) for n > 2, a(0)=1, a(1)=6.

A015553 Expansion of x/(1 - 6x - 11x^2).

A091928 a(0)=1, a(1)=5; a(n) = 6a(n-1) + 5a(n-2) for n > 1.

A189800 a(n) = 6a(n-1) + 8a(n-2), with a(0)=0, a(1)=1.