A015536 -id:A015536 - cp's OEIS frontend

A057088 Scaled Chebyshev U-polynomials evaluated at i*sqrt(5)/2. Generalized Fibonacci sequence.

1, 5, 30, 175, 1025, 6000, 35125, 205625, 1203750, 7046875, 41253125, 241500000, 1413765625, 8276328125, 48450468750, 283633984375, 1660422265625, 9720281250000, 56903517578125, 333118994140625, 1950112558593750, 11416157763671875, 66831351611328125, 391237546875000000

Offset: 0

Wolfdieter Lang, Aug 11 2000

a(n) gives the length of the word obtained after n steps with the substitution rule 0->11111, 1->111110, starting from 0. The number of 1's and 0's of this word is 5*a(n-1) and 5*a(n-2), resp.
a(n) / a(n-1) converges to (5 + (3 * sqrt(5))) / 2 as n approaches infinity. (5 + (3 * sqrt(5))) / 2 can also be written as phi^2 + (2 * phi), phi^3 + phi, phi + sqrt(5) + 2, (3 * phi) + 1, (3 * phi^2) - 2, phi^4 - 1 and (5 + (3 * (L(n) / F(n)))) / 2, where L(n) is the n-th Lucas number and F(n) is the n-th Fibonacci number as n approaches infinity. - Ross La Haye, Aug 18 2003, on another version
Pisano period lengths: 1, 3, 3, 6, 1, 3, 24, 12, 9, 3, 10, 6, 56, 24, 3, 24,288, 9, 18, 6, ... - R. J. Mathar, Aug 10 2012

Indranil Ghosh, Table of n, a(n) for n = 0..1300
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.
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=5, q=5.
Tanya Khovanova, Recursive Sequences
W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs.(39) and (45),rhs, m=5.
Eric Weisstein's World of Mathematics, Horadam Sequence
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (5,5)

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

I:=[1, 5]; [n le 2 select I[n] else 5*Self(n-1) + 5*Self(n-2): n in [0..30]]; // G. C. Greubel, Jan 16 2018

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=5*a[n-1]+5*a[n-2]od: seq(a[n], n=1..33); # Zerinvary Lajos, Dec 14 2008

LinearRecurrence[{5,5}, {1,5}, 30] (* G. C. Greubel, Jan 16 2018 *)

x='x+O('x^30); Vec(1/(1 - 5*x - 5*x^2)) \\ G. C. Greubel, Jan 16 2018

[lucas_number1(n,5,-5) for n in range(1, 22)] # Zerinvary Lajos, Apr 24 2009

a(n) = 5*(a(n-1) + a(n-2)), a(-1)=0, a(0)=1.
a(n) = S(n, i*sqrt(5))*(-i*sqrt(5))^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
G.f.: 1/(1 - 5*x - 5*x^2).
a(n) = (1/3)*Sum_{k=0..n} binomial(n, k)*Fibonacci(k)*3^k. - Benoit Cloitre, Oct 25 2003
a(n) = ((5 + 3*sqrt(5))/2)^n(1/2 + sqrt(5)/6) + (1/2 - sqrt(5)/6)((5 - 3*sqrt(5))/2)^n. - Paul Barry, Sep 22 2004
(a(n)) appears to be given by the floretion - 0.75'i - 0.5'j + 'k - 0.75i' + 0.5j' + 0.5k' + 1.75'ii' - 1.25'jj' + 1.75'kk' - 'ij' - 0.5'ji' - 0.75'jk' - 0.75'kj' - 1.25e ("jes"). - Creighton Dement, Nov 28 2004
a(n) = Sum_{k=0..n} 4^k*A063967(n,k). - Philippe Deléham, Nov 03 2006
G.f.: G(0)/(2-5*x), where G(k)= 1 + 1/(1 - x*(9*k-5)/(x*(9*k+4) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 17 2013
From Ehren Metcalfe, Nov 18 2017: (Start)
With F(n) = A000045(n), L(n) = A000032(n), beta = (1-sqrt(5))/2:
a(2*n-1) = 5^n*F(4*n)/3 = (5^(n-1/2)*L(4*n) - 2*5^(n-1/2)*beta^(4*n))/3.
a(2*n) = 5^n*L(4*n+2)/3 = (5^(n+1/2)*F(4*n+2) + 2*5^n*beta^(4*n+2))/3.
a(n) = round 5^((n+1)/2)*F(2*(n+1))/3.
a(n) = round 5^(n/2)*L(2*(n+1))/3. (End)

A057089 Scaled Chebyshev U-polynomials evaluated at i*sqrt(6)/2. Generalized Fibonacci sequence.

Original entry on oeis.org

1, 6, 42, 288, 1980, 13608, 93528, 642816, 4418064, 30365280, 208700064, 1434392064, 9858552768, 67757668992, 465697330560, 3200729997312, 21998563967232, 151195763787264, 1039165966526976, 7142170381885440

Offset: 0

Wolfdieter Lang, Aug 11 2000

a(n) gives the length of the word obtained after n steps with the substitution rule 0->1^6, 1->(1^6)0, starting from 0. The number of 1's and 0's of this word is 6*a(n-1) and 6*a(n-2), resp.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=6, q=6.
Tanya Khovanova, Recursive Sequences
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs.(39) and (45),rhs, m=6.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (6,6).

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, A083858, A085939, A090017, A091914, A099012, A135030, A135032, A180222, A180226, A180250.

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

Join[{a=0,b=1},Table[c=6*b+6*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
LinearRecurrence[{6,6},{1,6},40] (* Harvey P. Dale, Nov 05 2011 *)

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

[lucas_number1(n,6,-6) for n in range(1, 21)] # Zerinvary Lajos, Apr 24 2009

a(n) = 6*a(n-1) + 6*a(n-2); a(0)=1, a(1)=6.
a(n) = S(n, i*sqrt(6))*(-i*sqrt(6))^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
G.f.: 1/(1-6*x-6*x^2).
a(n) = Sum_{k=0..n} 5^k*A063967(n,k). - Philippe Deléham, Nov 03 2006

A015537 Expansion of x/(1 - 5x - 4x^2).

Original entry on oeis.org

0, 1, 5, 29, 165, 941, 5365, 30589, 174405, 994381, 5669525, 32325149, 184303845, 1050819821, 5991314485, 34159851709, 194764516485, 1110461989261, 6331368012245, 36098688018269, 205818912140325, 1173489312774701, 6690722212434805

Offset: 0

Olivier Gérard

First differences give A122690(n) = {1, 4, 24, 136, 776, 4424, 25224, ...}. Partial sums of a(n) are {0, 1, 6, 35, 200, ...} = (A123270(n) - 1)/8. - Alexander Adamchuk, Nov 03 2006
For n >= 2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 5's along the main diagonal, and 2's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 19 2011
Pisano period lengths:  1, 1, 8, 1, 4, 8, 48, 1, 24, 4, 40, 8, 42, 48, 8, 2, 72, 24, 360, 4, ... - R. J. Mathar, Aug 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Lucyna Trojnar-Spelina and 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 (5,4).

Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015443, A015447, A030195, A053404, A057087, A083858, A085939, A090017, A091914, A099012, A122690, A123270, A180222, A180226.

a:=[0,1];; for n in [3..30] do a[n]:=5*a[n-1]+4*a[n-2]; od; a; # G. C. Greubel, Dec 26 2019

[n le 2 select n-1 else 5*Self(n-1)+4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 12 2012

seq( simplify((2/I)^(n-1)*ChebyshevU(n-1, 5*I/4)), n=0..20); # G. C. Greubel, Dec 26 2019

LinearRecurrence[{5,4}, {0,1}, 30] (* Vincenzo Librandi, Nov 12 2012 *)
Table[2^(n-1)*Fibonacci[n, 5/2], {n, 0, 30}] (* G. C. Greubel, Dec 26 2019 *)

x='x+O('x^30); concat([0], Vec(x/(1-5*x-4*x^2))) \\ G. C. Greubel, Jan 01 2018

[lucas_number1(n,5,-4) for n in range(0, 22)] # Zerinvary Lajos, Apr 24 2009

a(n) = 5*a(n-1) + 4*a(n-2).
a(n) = Sum_{k=0..floor((n-1)/2)} C(n-k-1, k)*4^k*5^(n-2*k-1). - Paul Barry, Apr 23 2005
a(n) = Sum_{k=0..(n-1)} A122690(k). - Alexander Adamchuk, Nov 03 2006
a(n) = 2^(n-1)*Fibonacci(n, 5/2) = (2/i)^(n-1)*ChebyshevU(n-1, 5*i/4). - G. C. Greubel, Dec 26 2019

A135030 Generalized Fibonacci numbers: a(n) = 6a(n-1) + 2a(n-2).

Original entry on oeis.org

0, 1, 6, 38, 240, 1516, 9576, 60488, 382080, 2413456, 15244896, 96296288, 608267520, 3842197696, 24269721216, 153302722688, 968355778560, 6116740116736, 38637152257536, 244056393778688, 1541612667187200

Offset: 0

Rolf Pleisch, Feb 10 2008, Feb 14 2008

For n>0, a(n) equals the number of words of length n-1 over {0,1,...,7} in which 0 and 1 avoid runs of odd lengths. - Milan Janjic, Jan 08 2017

Joshua Zucker and Robert Israel, Table of n, a(n) for n = 0..1000 (n=0..51 from Joshua Zucker).
Index entries for linear recurrences with constant coefficients, signature (6, 2).

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, A083858, A085939, A090017, A091914, A099012, A180222, A180226, A180250.

[n le 2 select n-1 else 6*Self(n-1) + 2*Self(n-2): n in [1..35]]; // Vincenzo Librandi, Sep 18 2016

A:= gfun:-rectoproc({a(0) = 0, a(1) = 1, a(n) = 2*(3*a(n-1) + a(n-2))},a(n),remember):
seq(A(n),n=1..30); # Robert Israel, Sep 16 2014

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

a(n)=([0,1; 2,6]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

[lucas_number1(n,6,-2) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009

a(0) = 0; a(1) = 1; a(n) = 2*(3*a(n-1) + a(n-2)).
a(n) = 1/(2*sqrt(11))*( (3 + sqrt(11))^n - (3 - sqrt(11))^n ).
G.f.: x/(1 - 6*x - 2*x^2). - Harvey P. Dale, Jun 20 2011
a(n+1) = Sum_{k=0..n} A099097(n,k)*2^k. - Philippe Deléham, Sep 16 2014
E.g.f.: (1/sqrt(11))*exp(3*x)*sinh(sqrt(11)*x). - G. C. Greubel, Sep 17 2016

More terms from Joshua Zucker, Feb 23 2008

A180250 a(n) = 5a(n-1) + 10a(n-2), with a(1)=0 and a(2)=1.

Original entry on oeis.org

0, 1, 5, 35, 225, 1475, 9625, 62875, 410625, 2681875, 17515625, 114396875, 747140625, 4879671875, 31869765625, 208145546875, 1359425390625, 8878582421875, 57987166015625, 378721654296875, 2473479931640625, 16154616201171875, 105507880322265625

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 16 2011

Nathaniel Johnston, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (5,10).

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, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A180222, A180226.

[n le 2 select n-1 else 5*Self(n-1) +10*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 16 2018

Join[{a=0,b=1},Table[c=5*b+10*a;a=b;b=c,{n,100}]]
LinearRecurrence[{5,10}, {0,1}, 30] (* G. C. Greubel, Jan 16 2018 *)

a(n)=([0,1;10,5]^(n-1))[1,2] \\ Charles R Greathouse IV, Oct 03 2016

my(x='x+O('x^30)); concat([0], Vec(x^2/(1-5*x-10*x^2))) \\ G. C. Greubel, Jan 16 2018

A180250= BinaryRecurrenceSequence(5,10,0,1)
[A180250(n-1) for n in range(1,41)] # G. C. Greubel, Jul 21 2023

a(n) = ((5+sqrt(65))^(n-1) - (5-sqrt(65))^(n-1))/(2^(n-1)*sqrt(65)). - Rolf Pleisch, May 14 2011
G.f.: x^2/(1-5*x-10*x^2).
a(n) = (i*sqrt(10))^(n-1) * ChebyshevU(n-1, -i*sqrt(5/8)). - G. C. Greubel, Jul 21 2023

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

Original entry on oeis.org

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

A180035 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1+x)/(1-5x-3x^2).

Original entry on oeis.org

1, 6, 33, 183, 1014, 5619, 31137, 172542, 956121, 5298231, 29359518, 162692283, 901539969, 4995776694, 27683503377, 153404846967, 850074744966, 4710588265731, 26103165563553, 144647592614958, 801547459765449

Offset: 0

Johannes W. Meijer, Aug 09 2010

The a(n) represent the number of n-move routes of a fairy chess piece starting in the corner and side squares (m = 1, 3, 7, 9; 2, 4, 6, 8) 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 sequence above corresponds to 56 red queen vectors, i.e. A[5] vector, with decimal values between 7 and 448. The central squares lead for these vectors to A180036.
For n>=1, a(n) equals the numbers of words of length n-1 on alphabet {0,1,2,3,5} containing no subwords 00, 11 and 22. - Milan Janjic, Jan 31 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..200
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, example 17.
Index entries for linear recurrences with constant coefficients, signature (5, 3).

I:=[1,6]; [n le 2 select I[n] else 5*Self(n-1)+3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 15 2011

with(LinearAlgebra): nmax:=20; m:=1; A[5]:= [0,0,0,0,0,0,1,1,1]: A:=Matrix([[0,1,1,1,1,0,1,0,1], [1,0,1,1,1,1,0,1,0], [1,1,0,0,1,1,1,0,1], [1,1,0,0,1,1,1,1,0], A[5], [0,1,1,1,1,0,0,1,1], [1,0,1,1,1,0,0,1,1], [0,1,0,1,1,1,1,0,1], [1,0,1,0,1,1,1,1,0]]): 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);

LinearRecurrence[{5,3},{1,6},50] (* Vincenzo Librandi, Nov 15 2011 *)

G.f.: (1+x)/(1-5*x-3*x^2).
a(n) = 5*a(n-1) + 3*a(n-2) with a(0) = 1 and a(1) = 6.
a(n) = ((7+A)*A^(-n-1)+(7+B)*B^(-n-1))/37 with A = (-5+sqrt(37))/6 and B = (-5-sqrt(37))/6.
a(n) = Sum_{k, 0<=k<=n} A202396(n,k)*2^k. - Philippe Deléham, Dec 21 2011
a(n) = A015536(n+1)+A015536(n). - R. J. Mathar, Aug 04 2019

A083861 Square array T(n,k) of second binomial transforms of generalized Fibonacci numbers, read by ascending antidiagonals, with n, k >= 0.

Original entry on oeis.org

0, 0, 1, 0, 1, 5, 0, 1, 5, 19, 0, 1, 5, 20, 65, 0, 1, 5, 21, 75, 211, 0, 1, 5, 22, 85, 275, 665, 0, 1, 5, 23, 95, 341, 1000, 2059, 0, 1, 5, 24, 105, 409, 1365, 3625, 6305, 0, 1, 5, 25, 115, 479, 1760, 5461, 13125, 19171, 0, 1, 5, 26, 125, 551, 2185, 7573, 21845, 47500, 58025

Offset: 0

Paul Barry, May 06 2003

Row n >= 0 of the array gives the solution to the recurrence b(k) = 5*b(k-1) + (n - 6)*b(k-2) for k >= 2 with b(0) = 0 and b(1) = 1. The rows are the binomial transforms of the rows of array A083857. The rows are the second binomial transforms of the generalized Fibonacci numbers in array A083856.

			Array T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
  0, 1, 5, 19,  65, 211,  665,  2059,  6305,  19171, ...
  0, 1, 5, 20,  75, 275, 1000,  3625, 13125,  47500, ...
  0, 1, 5, 21,  85, 341, 1365,  5461, 21845,  87381, ...
  0, 1, 5, 22,  95, 409, 1760,  7573, 32585, 140206, ...
  0, 1, 5, 23, 105, 479, 2185,  9967, 45465, 207391, ...
  0, 1, 5, 24, 115, 551, 2640, 12649, 60605, 290376, ...
  0, 1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, ...
  ...

G. C. Greubel, Antidiagonals n = 0..100, flattened
OEIS, Transformations of integer sequences.

Rows include A001047 (n=0), A093131 (n=1), A002450 (n=2), A004254 (n=5), A000351 (n=6), A052918 (n=7), A015535 (n=8), A015536 (n=9), A015537 (n=10).

Cf. A083856 (second inverse binomial transform), A083856 (first inverse binomial transform), A082297 (main diagonal).

T:= func< n,k | Round( (((5+Sqrt(4*n+1))/2)^k - ((5-Sqrt(4*n+1))/2)^k)/Sqrt(4*n + 1) ) >;
[T(n-k,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 27 2019

seq(seq(round( (((5+sqrt(4*(n-k)+1))/2)^k - ((5-sqrt(4*(n-k)+1))/2)^k)/sqrt(4*(n-k)+1) ), k=0..n), n=0..10); # G. C. Greubel, Dec 27 2019

T[n_, k_]:= Round[(((5 +Sqrt[4*n+1])/2)^k - ((5 -Sqrt[4*n+1])/2)^k)/Sqrt[4*n+1]]; Table[T[n-k, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 27 2019 *)

T(n, k) = round( (((5+sqrt(4*n+1))/2)^k - ((5-sqrt(4*n+1))/2)^k)/sqrt(4*n + 1) );
for(n=0,10, for(k=0,n, print1(T(n-k,k), ", "))) \\ G. C. Greubel, Dec 27 2019

[[round( (((5+sqrt(4*(n-k)+1))/2)^k - ((5-sqrt(4*(n-k)+1))/2)^k)/sqrt(4*(n-k)+1) ) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 27 2019

T(n, k) = (((5 + sqrt(4*n + 1))/2)^k - ((5 - sqrt(4*n + 1))/2)^k)/sqrt(4*n + 1).
O.g.f. for row n >= 0: -x/(-1 + 5*x + (n-6)*x^2) . - R. J. Mathar, Dec 02 2007
From Petros Hadjicostas, Dec 25 2019: (Start)
T(n,k) = 5*T(n,k-1) + (n - 6)*T(n,k-2) for k >= 2 with T(n,0) = 0 and T(n,1) = 1 for all n >= 0.
T(n,k) = Sum_{i = 0..k} binomial(k,i) * A083857(n,i).
T(n,k) = Sum_{i = 0..k} Sum_{j = 0..i} binomial(k,i) * binomial(i,j) * A083856(n,j). (End)

Name and various sections edited by Petros Hadjicostas, Dec 25 2019

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).

cp's OEIS Frontend

A057088 Scaled Chebyshev U-polynomials evaluated at i*sqrt(5)/2. Generalized Fibonacci sequence.

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A057089 Scaled Chebyshev U-polynomials evaluated at i*sqrt(6)/2. Generalized Fibonacci sequence.

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A015537 Expansion of x/(1 - 5*x - 4*x^2).

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A135030 Generalized Fibonacci numbers: a(n) = 6*a(n-1) + 2*a(n-2).

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A180250 a(n) = 5*a(n-1) + 10*a(n-2), with a(1)=0 and a(2)=1.

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

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A015537 Expansion of x/(1 - 5x - 4x^2).

A135030 Generalized Fibonacci numbers: a(n) = 6a(n-1) + 2a(n-2).

A180250 a(n) = 5a(n-1) + 10a(n-2), with a(1)=0 and a(2)=1.

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

A180035 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1+x)/(1-5x-3x^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).