A099012 -id:A099012 - 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

A234357 Array T(n,k) by antidiagonals: T(n,k) = n^k * Fibonacci(k).

Original entry on oeis.org

1, 2, 2, 3, 8, 3, 4, 18, 24, 5, 5, 32, 81, 80, 8, 6, 50, 192, 405, 256, 13, 7, 72, 375, 1280, 1944, 832, 21, 8, 98, 648, 3125, 8192, 9477, 2688, 34, 9, 128, 1029, 6480, 25000, 53248, 45927, 8704, 55, 10, 162, 1536, 12005, 62208, 203125, 344064, 223074, 28160, 89, 11, 200, 2187

Offset: 0

Ralf Stephan, Dec 24 2013

			Array starts:
1,  2,   3,    5,     8,     13,    21,   34, 55, 89,...    (A000045)
2,  8,  24,   80,   256,    832,  2688, 8704,...   (A063727, A085449)
3, 18,  81,  405,  1944,   9477, 45927,...         (A122069, A099012)
4, 32, 192, 1280,  8192,  53248,...                         (A099133)
5, 50, 375, 3125, 25000, 203125,...
6, 72, 648, 6480, 62208, 606528,...
...
Columns: A000027, A001105, A117642.

PARI
```
T(n,k)=n^k*fibonacci(k)
```

T(n,k)=polcoeff(Ser(1/(1-n*x-n^2*x^2)),k)

G.f. of n-th row: 1/(1 - n*x - n^2*x^2).

Recurrence: T(n,k) = n*T(n,k-1) + n^2*T(n,k-2), starting n, 2*n^2.

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

A122069 a(n) = 3a(n-1) + 9a(n-2) for n > 1, with a(0)=1, a(1)=3.

Original entry on oeis.org

1, 3, 18, 81, 405, 1944, 9477, 45927, 223074, 1082565, 5255361, 25509168, 123825753, 601059771, 2917611090, 14162371209, 68745613437, 333698181192, 1619805064509, 7862698824255, 38166342053346, 185263315578333

Offset: 0

Philippe Deléham, Oct 15 2006

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,9).

Third row of A234357.

Cf. A000045, A085504, A099012.

List([0..25], n-> 3^n*Fibonacci(n+1) ); # G. C. Greubel, Oct 03 2019

[3^n*Fibonacci(n+1): n in [0..25]]; // G. C. Greubel, Oct 03 2019

with(combinat); seq(3^n*fibonacci(n+1), n=0..25); # G. C. Greubel, Oct 03 2019

Table[3^n*Fibonacci[n+1], {n,0,25}] (* G. C. Greubel, Oct 03 2019 *)
LinearRecurrence[{3,9},{1,3},30] (* Harvey P. Dale, Apr 28 2020 *)

vector(26, n, 3^(n-1)*fibonacci(n) ) \\ G. C. Greubel, Oct 03 2019

[lucas_number1(n,3,-9) for n in range(1, 23)] # Zerinvary Lajos, Apr 22 2009

a(n) = 3^n*Fibonacci(n+1) = 3^n*A000045(n+1).
a(n) = Sum_{k=0..n} 2^k*A016095(n,k).
G.f.: 1/(1-3*x-9*x^2).
Limit_{n->oo} a(n+1)/a(n) = 3*(1+sqrt(5))/2.
a(n) = A099012(n+1). - R. J. Mathar, Aug 02 2008
a(n) = A085504(n) for n >= 2. - Georg Fischer, Nov 03 2018

Corrected by T. D. Noe, Nov 07 2006

A099013 a(n) = Sum_{k=0..n} 3^(k-1)*Fibonacci(k).

Original entry on oeis.org

0, 1, 4, 22, 103, 508, 2452, 11929, 57856, 280930, 1363495, 6618856, 32128024, 155953777, 757013548, 3674624638, 17836995847, 86582609284, 420280790476, 2040085854985, 9902784679240, 48069126732586, 233332442310919

Offset: 0

Paul Barry, Sep 22 2004

Partial sums of A099012. Binomial transform of A063092 (with leading 0).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,6,-9).

Cf. A000045, A014335.

I:=[0, 1, 4]; [n le 3 select I[n] else 4*Self(n-1)+6*Self(n-2)-9*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 25 2012

Join[{a=0,b=1},Table[c=3*b+9*a+1;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 18 2011 *)
Table[Sum[3^(k-1) Fibonacci[k],{k,0,n}],{n,0,30}] (* or *) LinearRecurrence[{4,6,-9},{0,1,4},30] (* Harvey P. Dale, Dec 09 2011 *)
CoefficientList[Series[x/((1-x)(1-3x-9x^2)),{x,0,40}],x] (* Vincenzo Librandi, Jun 25 2012 *)

x='x+O('x^30); concat([0], Vec(x/((1-x)*(1 - 3*x - 9*x^2)))) \\ G. C. Greubel, Dec 31 2017

G.f.: x/((1-x)*(1 - 3*x - 9*x^2)).
a(n) = 4*a(n-1) + 6*a(n-2) - 9*a(n-3).
a(n) = 3^(n-1)*Sum_{k=0..n} Fibonacci(n-k)*3^(-k).
a(n) = (3/2 + 3*sqrt(5)/2)^n*(1/22 + 7*sqrt(5)/110) + (1/22 - 7*sqrt(5)/110)*(3/2 - 3*sqrt(5)/2)^n - 1/11.
a(n) = (3^n*A000285(n) - 1)/11, the case m = 3 of Sum_{k=0..n} m^(k-1)*F(k) = (m^n*(m*F(n) + F(n+1)) - 1)/(m^2 + m - 1), F=A000045. - Ehren Metcalfe, Apr 29 2018

Sign in second formula corrected by Harvey P. Dale, Dec 09 2011

A099133 4^(n-1)*Fibonacci(n).

Original entry on oeis.org

0, 1, 4, 32, 192, 1280, 8192, 53248, 344064, 2228224, 14417920, 93323264, 603979776, 3909091328, 25300041728, 163745628160, 1059783180288, 6859062771712, 44392781971456, 287316132233216, 1859549040476160, 12035254277636096, 77893801758162944

Offset: 0

Paul Barry, Sep 29 2004

Binomial transform of A099134.
Second binomial transform of x/(1-20x^2), or (0,1,0,20,0,400,0,8000,....).
In general k^(n-1)*Fibonacci(n) has g.f. x/(1-kx-k^2x^2).
The ratio a(n+1)/a(n) converges to 4 times the golden ratio as n approaches infinity. In general, the ratio a(n+1)/a(n) of the sequence which is the solution to the linear recurrence relation a(n) = m*a(n-1)+m^2*a(n-2) with a(0)=0 and a(1) = 1 converges to m times the golden ratio as n approaches infinity where m is a positive integer. - Felix P. Muga II, Mar 10 2014

			G.f. = x + 4*x^2 + 32*x^3 + 192*x^4 + 1280*x^5 + 8192*x^6 + 53248*x^7 + ...

F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivré Formula, March 2014; Preprint on ResearchGate.

Index entries for linear recurrences with constant coefficients, signature (4,16).

Cf. A000045, A099012, A085449. Fourth row of A234357.

Join[{a=0,b=1},Table[c=4*b+16*a;a=b;b=c,{n,40}]] (* Vladimir Joseph Stephan Orlovsky, Mar 29 2011*)
Table[4^(n-1) Fibonacci[n],{n,0,20}] (* Harvey P. Dale, Aug 22 2012 *)
LinearRecurrence[{4,16},{0,1},30] (* Harvey P. Dale, Aug 22 2012 *)

a(n) = 4^(n-1)*fibonacci(n); \\ Michel Marcus, Jan 10 2014

G.f.: x/(1-4*x-16*x^2).
a(n) = 4*a(n-1) + 16*a(n-2).
a(n) = (2+2*sqrt(5))^n/(4*sqrt(5))-(2-sqrt(5))^n/(4*sqrt(5)).
a(-n) = -(-1)^n * a(n) / 16^n for all n in Z. - Michael Somos, Mar 18 2014

A085504 Horadam sequence (0,1,9,3).

Original entry on oeis.org

0, 1, 18, 81, 405, 1944, 9477, 45927, 223074, 1082565, 5255361, 25509168, 123825753, 601059771, 2917611090, 14162371209, 68745613437, 333698181192, 1619805064509, 7862698824255, 38166342053346, 185263315578333, 899287025215113, 4365230915850336

Offset: 0

Ross La Haye, Aug 18 2003

Lim_{n->infinity} a(n)/a(n-1) = (3/2)*(1 + sqrt(5)), which can also be written as phi^2 + 2*phi - 1, phi^3 + phi - 1, phi + sqrt(5) + 1, 3*phi, 3*phi^2 - 3, phi^4 - 2 and lim_{n->infinity} (3/2)*(1 + Lucas(n)/Fibonacci(n)).

			a(4) = 405 because a(3) = 81, a(2) = 18, s = 3, r = 9 and (3 * 81) + (9 * 18) = 405.

Eric Weisstein, Horadam Sequence
Eric Weisstein, Fibonacci Number
Eric Weisstein, Pell Number
Eric Weisstein, Lucas Number
Eric Weisstein, Lucas Sequence
Index entries for linear recurrences with constant coefficients, signature (3,9).

Cf. A024318, A000032, A000129, A001076, A085939.

Essentially the same as A122069 and A099012.

Join[{0,1},LinearRecurrence[{3,9},{18,81},30]] (* or *) CoefficientList[ Series[x (1+15x+18x^2)/(1-3x-9x^2),{x,0,30}],x] (* Harvey P. Dale, Nov 24 2012 *)

a(n) = s*a(n-1) + r*a(n-2); for n > 3, where a(0) = 0, a(1) = 1, a(2) = 18, a(4) = 81, s = 3, r = 9.

G.f.: x*(1+15*x+18*x^2)/(1-3*x-9*x^2). [Colin Barker, Jun 20 2012]

First formula corrected and more terms from Harvey P. Dale, Nov 24 2012

A368157 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 + 2x^2.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 3, 10, 16, 16, 5, 20, 46, 56, 44, 8, 40, 108, 184, 188, 120, 13, 76, 244, 496, 692, 608, 328, 21, 142, 520, 1248, 2088, 2480, 1920, 896, 34, 260, 1074, 2936, 5764, 8256, 8592, 5952, 2448, 55, 470, 2156, 6616, 14764, 24760, 31200, 28992

Offset: 1

Clark Kimberling, Jan 20 2024

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

			First eight rows:
   1
   1    2
   2    4    6
   3   10   16    16
   5   20   46    56    44
   8   40  108   184   188   120
  13   76  244   496   692   608   328
  21  142  520  1248  2088  2480  1920  896
Row 4 represents the polynomial p(4,x) = 3 + 10*x + 16*x^2 + 16*x^3, so (T(4,k)) = (3,10,16,16), k=0..3.

Rigoberto Flórez, Robinson A. Higuita, and Antara Mikherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers 18 (2018) 1-28.

Cf. A000045 (column 1); A002605, (p(n,n-1)); A030195 (row sums), (p(n,1)); A182228 (alternating row sums), (p(n,-1)); A015545, (p(n,2)); A099012, (p(n,-2)); A087567, (p(n,3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151, A368152, A368153, A368154, A368155, A368156.

p[1, x_] := 1; p[2, x_] := 1 + 2 x; u[x_] := p[2, x]; v[x_] := 1 + 2x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 1 + 2*x, u = p(2,x), and v = 1 + 2*x^2.

p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 + 4*x + 13*x^2), b = (1/2)*(2*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).

cp's OEIS Frontend

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

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A234357 Array T(n,k) by antidiagonals: T(n,k) = n^k * Fibonacci(k).

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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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A122069 a(n) = 3*a(n-1) + 9*a(n-2) for n > 1, with a(0)=1, a(1)=3.

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A099013 a(n) = Sum_{k=0..n} 3^(k-1)*Fibonacci(k).

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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.

A122069 a(n) = 3a(n-1) + 9a(n-2) for n > 1, with a(0)=1, a(1)=3.

A368157 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 + 2x^2.