A015524 -id:A015524 - cp's OEIS frontend

A015551 Expansion of x/(1 - 6x - 5x^2).

0, 1, 6, 41, 276, 1861, 12546, 84581, 570216, 3844201, 25916286, 174718721, 1177893756, 7940956141, 53535205626, 360916014461, 2433172114896, 16403612761681, 110587537144566, 745543286675801, 5026197405777636

Offset: 0

Olivier Gérard

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)=(B^n)1,2 for instance. - _Paul Barry, Feb 13 2004

Pisano period lengths: 1, 2, 4, 4, 1, 4, 42, 8, 12, 2, 10, 4, 12, 42, 4, 16, 96, 12, 360, 4, ... - R. J. Mathar, Aug 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Lucyna Trojnar-Spelina, Iwona Włoch, On Generalized Pell and Pell-Lucas Numbers, Iranian Journal of Science and Technology, Transactions A: Science (2019), 1-7.
Index entries for linear recurrences with constant coefficients, signature (6,5).

Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015440, A015441, A015443, A015444, A015445, A015447, A015548, A030195, A053404, A057087, A057088, A057089, A083858, A085939, A090017, A091914, A099012, A135030, A135032, A180222, A180226, A180250.

I:=[0,1]; [n le 2 select I[n] else 6*Self(n-1)+5*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 14 2011

Join[{a=0,b=1},Table[c=6*b+5*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
CoefficientList[Series[x/(1-6x-5x^2),{x,0,20}],x] (* or *) LinearRecurrence[ {6,5},{0,1},30] (* Harvey P. Dale, Oct 30 2017 *)

a(n)=([0,1; 5,6]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

[lucas_number1(n,6,-5) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009

a(n) = 6*a(n-1) + 5*a(n-2).

a(n) = sqrt(14)*(3+sqrt(14))^n/28 - sqrt(14)*(3-sqrt(14))^n/28. - Paul Barry, Feb 13 2004

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

A015541 Expansion of x/(1 - 5x - 7x^2).

Original entry on oeis.org

0, 1, 5, 32, 195, 1199, 7360, 45193, 277485, 1703776, 10461275, 64232807, 394392960, 2421594449, 14868722965, 91294775968, 560554940595, 3441838134751, 21133075257920, 129758243232857, 796722742969725, 4891921417478624, 30036666288181195

Offset: 0

Olivier Gérard

Pisano period lengths: 1, 3, 8, 6, 8, 24, 6, 6, 24, 24, 5, 24, 12, 6, 8, 12, 16, 24, 120, 24, ... - R. J. Mathar, Aug 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,7).

Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015443, A015447, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A180222, A180226.

[n le 2 select n-1 else 5*Self(n-1) + 7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 13 2012

Join[{a=0,b=1},Table[c=5*b+7*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
LinearRecurrence[{5, 7}, {0, 1}, 30] (* Vincenzo Librandi, Nov 13 2012 *)

x='x+O('x^30); concat([0], Vec(x/(1-5*x-7*x^2))) \\ G. C. Greubel, Jan 24 2018

[lucas_number1(n,5,-7) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009

a(n) = 5*a(n-1) + 7*a(n-2).

A015544 Lucas sequence U(5,-8): a(n+1) = 5a(n) + 8a(n-1), a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 5, 33, 205, 1289, 8085, 50737, 318365, 1997721, 12535525, 78659393, 493581165, 3097180969, 19434554165, 121950218577, 765227526205, 4801739379641, 30130517107845, 189066500576353, 1186376639744525, 7444415203333449, 46713089134623445

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,8).

Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015441, A015443, A015447, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A180222, A180226, A015555 (binomial transform).

[n le 2 select n-1 else 5*Self(n-1) + 8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 13 2012

a[n_]:=(MatrixPower[{{1,2},{1,-6}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{5, 8}, {0, 1}, 30] (* Vincenzo Librandi, Nov 13 2012 *)

A015544(n)=imag((2+quadgen(57))^n) \\ M. F. Hasler, Mar 06 2009

x='x+O('x^30); concat([0], Vec(x/(1 - 5*x - 8*x^2))) \\ G. C. Greubel, Jan 01 2018

[lucas_number1(n,5,-8) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009

a(n) = 5*a(n-1) + 8*a(n-2).

G.f.: x/(1 - 5*x - 8*x^2). - M. F. Hasler, Mar 06 2009

More precise definition by M. F. Hasler, Mar 06 2009

A179602 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 2x)/(1 - 3x - 7*x^2).

Original entry on oeis.org

1, 5, 22, 101, 457, 2078, 9433, 42845, 194566, 883613, 4012801, 18223694, 82760689, 375847925, 1706868598, 7751541269, 35202703993, 159868900862, 726025630537, 3297159197645, 14973657006694, 68001085403597, 308818855257649

Offset: 0

Johannes W. Meijer, Jul 28 2010

The a(n) represent the number of n-move routes of a fairy chess piece starting in a given side square (m = 2, 4, 6 or 8) on a 3 X 3 chessboard. This fairy chess piece behaves like a king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.

The sequence above corresponds to 24 red king vectors, i.e., A[5] vectors, with decimal values 95, 119, 125, 215, 221, 245, 287, 311, 317, 347, 350, 371, 374, 377, 380, 407, 413, 437, 467, 470, 473, 476, 497 and 500. These vectors lead for the corner squares to A015524 and for the central square to A179603.

Index entries for linear recurrences with constant coefficients, signature (3, 7).

Cf. A126473 (side squares).

with(LinearAlgebra): nmax:=22; m:=2; A[1]:= [0,1,0,1,1,0,0,0,0]: A[2]:= [1,0,1,1,1,1,0,0,0]: A[3]:= [0,1,0,0,1,1,0,0,0]: A[4]:=[1,1,0,0,1,0,1,1,0]: A[5]:= [1,1,1,0,1,0,0,1,1]: A[6]:= [0,1,1,0,1,0,0,1,1]: A[7]:= [0,0,0,1,1,0,0,1,0]: A[8]:= [0,0,0,1,1,1,1,0,1]: A[9]:= [0,0,0,0,1,1,0,1,0]: A:=Matrix([A[1],A[2],A[3],A[4],A[5],A[6],A[7],A[8],A[9]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

CoefficientList[Series[(1+2x)/(1-3x-7x^2),{x,0,40}],x] (* or *) LinearRecurrence[ {3,7},{1,5},40] (* Harvey P. Dale, Mar 28 2013 *)

G.f.: (1+2*x)/(1 - 3*x - 7*x^2).
a(n) = 3*a(n-1) + 7*a(n-2) with a(0) = 1 and a(1) = 5.
a(n) = ((37+4*37^(1/2))*A^(-n-1) + (37-4*37^(1/2))*B^(-n-1))/259 with A = (-3+sqrt(37))/14 and B = (-3-sqrt(37))/14.

A179603 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 3x)/(1 - 3x - 7*x^2).

Original entry on oeis.org

1, 6, 25, 117, 526, 2397, 10873, 49398, 224305, 1018701, 4626238, 21009621, 95412529, 433304934, 1967802505, 8936542053, 40584243694, 184308525453, 837015282217, 3801205524822, 17262723549985, 78396609323709

Offset: 0

Johannes W. Meijer, Jul 28 2010

The a(n) represent the number of n-move routes of a fairy chess piece starting in the central square (m = 5) on a 3 X 3 chessboard. This fairy chess piece behaves like a king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.

The sequence above corresponds to 24 red king vectors, i.e., A[5] vectors, with decimal values 95, 119, 125, 215, 221, 245, 287, 311, 317, 347, 350, 371, 374, 377, 380, 407, 413, 437, 467, 470, 473, 476, 497 and 500. These vectors lead for the corner squares to A015524 and for the side squares to A179602.

Index entries for linear recurrences with constant coefficients, signature (3, 7).

Cf. A179597 (central square).

with(LinearAlgebra): nmax:=23; m:=5; A[1]:= [0,1,0,1,1,0,0,0,0]: A[2]:= [1,0,1,1,1,1,0,0,0]: A[3]:= [0,1,0,0,1,1,0,0,0]: A[4]:= [1,1,0,0,1,0,1,1,0]: A[5]:= [1,1,1,0,1,0,0,1,1]: A[6]:= [0,1,1,0,1,0,0,1,1]: A[7]:= [0,0,0,1,1,0,0,1,0]: A[8]:= [0,0,0,1,1,1,1,0,1]: A[9]:= [0,0,0,0,1,1,0,1,0]: A:=Matrix([A[1],A[2],A[3],A[4],A[5],A[6],A[7],A[8],A[9]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

G.f.: (1+3*x)/(1 - 3*x - 7*x^2).
a(n) = 3*a(n-1) + 7*a(n-2) with a(0) = 1 and a(1) = 6.
a(n) = ((9+5*A)*A^(-n-1) + (9+5*B)*B^(-n-1))/37 with A = (-3+sqrt(37))/14 and B = (-3-sqrt(37))/14.

A137280 a(n) = 3a(n-1) + 7a(n-2), with a(1) = 1, a(2) = 10.

Original entry on oeis.org

1, 10, 37, 181, 802, 3673, 16633, 75610, 343261, 1559053, 7079986, 32153329, 146019889, 663132970, 3011538133, 13676545189, 62110402498, 282067023817, 1280973888937, 5817390833530, 26418989723149, 119978705004157, 544869043074514, 2474458064252641, 11237457494279521

Offset: 1

Gary W. Adamson, Mar 14 2008

a(n) == 1 mod 9.

a(n)/a(n-1) tends to 4.54138126... = (3 + sqrt(37))/2.

			a(4) = 181 = 3*a(3) + 7*a(2) = 3*37 + 7*10.
a(4) = 181 = upper left term in [1,3; 3,2]^4.

Andrew Howroyd, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (3,7)

LinearRecurrence[{3, 7}, {1, 10}, 25] (* Paolo Xausa, Jan 15 2025 *)

a(1) = 1, a(2) = 10, a(n) = 3*a(n-1) + 7*a(n-2) for n>2.
a(n) = upper left term in [1,3; 3,2]^n
From R. J. Mathar, Mar 17 2008: (Start)
O.g.f.: x*(1+7*x)/(1-3*x-7*x^2).
a(n) = A015524(n) + 7*A015524(n-1). (End)

a(23) onwards from Andrew Howroyd, Jan 12 2025

A228815 Symmetric triangle, read by rows, related to Fibonacci numbers.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 5, 5, 2, 3, 10, 14, 10, 3, 5, 20, 36, 36, 20, 5, 8, 38, 83, 106, 83, 38, 8, 13, 71, 182, 281, 281, 182, 71, 13, 21, 130, 382, 690, 834, 690, 382, 130, 21, 34, 235, 778, 1606, 2268, 2268, 1606, 778, 235, 34, 55, 420, 1546, 3586, 5780, 6750

Offset: 0

Philippe Deléham, Oct 30 2013

Triangles satisfying the same recurrence: A091533, A091562, A185081, A205575, A209137, A209138.

			Triangle begins :
0
1, 1
1, 2, 1
2, 5, 5, 2
3, 10, 14, 10, 3
5, 20, 36, 36, 20, 5
8, 38, 83, 106, 83, 38, 8
13, 71, 182, 281, 281, 182, 71, 13
21, 130, 382, 690, 834, 690, 382, 130, 21
34, 235, 778, 1606, 2268, 2268, 1606, 778, 235, 34
55, 420, 1546, 3586, 5780, 6750, 5780, 3586, 1546, 420, 55

Cf. A000045 (1st column), A001629 (2nd column), A008998, A152011, A261055 (3rd column).

G.f.: x*(1+y)/(1-x-x*y-x^2-x^2*y-x^2*y^2).
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), T(0,0) = 0, T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n.
Sum_{k = 0..n} T(n,k)*x^k = A000045(n), 2*A015518(n), 3*A015524(n), 4*A200069(n) for x = 0, 1, 2, 3 respectively.
Sum_{k = 0..floor(n/2)} T(n-k,k) = A008998(n+1).

cp's OEIS Frontend

A015551 Expansion of x/(1 - 6*x - 5*x^2).

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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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A015541 Expansion of x/(1 - 5*x - 7*x^2).

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A015544 Lucas sequence U(5,-8): a(n+1) = 5*a(n) + 8*a(n-1), a(0)=0, a(1)=1.

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A179602 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 2*x)/(1 - 3*x - 7*x^2).

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A179603 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 3*x)/(1 - 3*x - 7*x^2).

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A137280 a(n) = 3*a(n-1) + 7*a(n-2), with a(1) = 1, a(2) = 10.

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A015551 Expansion of x/(1 - 6x - 5x^2).

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

A015541 Expansion of x/(1 - 5x - 7x^2).

A015544 Lucas sequence U(5,-8): a(n+1) = 5a(n) + 8a(n-1), a(0)=0, a(1)=1.

A179602 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 2x)/(1 - 3x - 7*x^2).

A179603 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 3x)/(1 - 3x - 7*x^2).

A137280 a(n) = 3a(n-1) + 7a(n-2), with a(1) = 1, a(2) = 10.