A015553 -id:A015553 - 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

A087567 a(n) = (1/5)Sum_{k=0..n} binomial(n,k)Fibonacci(k)*5^k.

Original entry on oeis.org

1, 7, 68, 609, 5555, 50456, 458737, 4169823, 37904764, 344559985, 3132110411, 28471412592, 258809985953, 2352626740919, 21385776919540, 194400346514241, 1767132187070947, 16063531893267208, 146020234807218449, 1327348749622606095, 12065825708695393196

Offset: 0

Benoit Cloitre, Oct 25 2003

Index entries for linear recurrences with constant coefficients, signature (7,19).

Cf. A014445, A057088, A015553.

LinearRecurrence[{7,19},{1,7},25] (* Paolo Xausa, Jan 07 2024 *)

[lucas_number1(n,7,-19) for n in range(1, 20)] # Zerinvary Lajos, Apr 29 2009

a(n) = 7*a(n-1) + 19*a(n-2).

G.f.: 1 / (-19*x^2-7*x+1). - Colin Barker, Aug 08 2013

More terms from Colin Barker, Aug 08 2013

A087603 a(n) = (1/8)Sum_{k=0..n} binomial(n,k)Fibonacci(k)*8^k.

Original entry on oeis.org

1, 10, 155, 2100, 29525, 410750, 5731375, 79905000, 1114275625, 15537531250, 216660471875, 3021168937500, 42128015328125, 587444444843750, 8191485291484375, 114224297381250000, 1592774664844140625, 22210083004410156250, 309703436610529296875

Offset: 0

Benoit Cloitre, Oct 25 2003

More generally a(n)=(1/x)*sum(k=0,n,binomial(n,k)*Fibonacci(k)*x^k) satisfies the recurrence formula a(n)=(x+2)*a(n-1)+(x^2-x-1)*a(n-2).

Harvey P. Dale, Table of n, a(n) for n = 0..873
Index entries for linear recurrences with constant coefficients, signature (10, 55).

Cf. A014445, A057088, A015553.

LinearRecurrence[{10,55},{1,10},30] (* Harvey P. Dale, Nov 26 2014 *)

Vec(1/(1-10*x-55*x^2) + O(x^50)) \\ Colin Barker, Mar 30 2016

a(n) = 10*a(n-1)+55*a(n-2).
G.f.: -1/(-1+10*x+55*x^2). - R. J. Mathar, Dec 05 2007
a(n) = ((-(5-4*sqrt(5))^(1+n)+(5+4*sqrt(5))^(1+n)))/(8*sqrt(5)). - Colin Barker, Mar 30 2016

A087579 a(n) = (1/6)Sum_{k=0..n} binomial(n,k)Fibonacci(k)*6^k.

Original entry on oeis.org

1, 8, 93, 976, 10505, 112344, 1203397, 12885152, 137979729, 1477507240, 15821470061, 169419470448, 1814178395353, 19426591805816, 208023907911765, 2227562425662784, 23853192734743457, 255424852222168392, 2735141407084907389, 29288451971122142480

Offset: 0

Benoit Cloitre, Oct 25 2003

Index entries for linear recurrences with constant coefficients, signature (8,29).

Cf. A014445, A057088, A015553.

Join[{b=1},a=0;Table[c=8*b+29*a;a=b;b=c,{n,30}]] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)
LinearRecurrence[{8,29},{1,8},20] (* Harvey P. Dale, Mar 22 2019 *)

a(n) = 8*a(n-1) + 29*a(n-2).

G.f.: 1 / (-29*x^2-8*x+1). - Colin Barker, Aug 08 2013

More terms from Colin Barker, Aug 08 2013

A087584 a(n) = (1/7)Sum_{k=0..n} binomial(n,k)Fibonacci(k)*7^k.

Original entry on oeis.org

1, 9, 122, 1467, 18205, 223992, 2762333, 34044669, 419657674, 5172750495, 63760719089, 785929242096, 9687552661513, 119411072879553, 1471889315038010, 18142857823403763, 223633182327192277, 2756555811704284776, 33977962780753446341, 418820453306656692885

Offset: 0

Benoit Cloitre, Oct 25 2003

Index entries for linear recurrences with constant coefficients, signature (9,41).

Cf. A014445, A057088, A015553.

a(n) = 9*a(n-1) + 41*a(n-2).

G.f.: 1 / (-41*x^2-9*x+1). - Colin Barker, Aug 08 2013

More terms from Colin Barker, Aug 08 2013

A091929 Expansion of (1-6x)/(1-6x-11x^2).

Original entry on oeis.org

1, 0, 11, 66, 517, 3828, 28655, 214038, 1599433, 11951016, 89299859, 667260330, 4985860429, 37255026204, 278374621943, 2080053019902, 15542438960785, 116135216983632, 867778130470427, 6484156169642514, 48450496453029781

Offset: 0

Paul Barry, Feb 16 2004

Let the generating matrix of the Golay G_24 code be [I|A]. Then a(n)=(A^n)_1,1.

S. Roman, Introduction to Coding and Information Theory, Springer-Verlag, 1996, p. 224

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

Cf. A091927, A015553.

CoefficientList[Series[(1-6x)/(1-6x-11x^2),{x,0,30}],x] (* or *) LinearRecurrence[{6,11},{1,0},30] (* Harvey P. Dale, Apr 25 2018 *)

a(n) = (1/2 - 3*sqrt(5)/20)*(3 + 2*sqrt(5))^n + (3 - 2*sqrt(5))^n*(1/2 + 3*sqrt(5)/20).

A225799 a(n) = Sum_{k=0..n} binomial(n,k) * 10^(n-k) * Fibonacci(n+k).

Original entry on oeis.org

0, 11, 143, 3058, 55341, 1052755, 19717984, 371084087, 6973353387, 131101759514, 2464418392865, 46327530894271, 870879506447808, 16371134451297043, 307750614069672631, 5785211638097121890, 108752568228856901349, 2044371455527726003547, 38430858858805840293152

Offset: 0

John Molokach, Jul 27 2013

This sequence is part of a family of Fibonacci-like sequences, where:
Sum_{k=0..n} binomial(n,k)*m^(n-k)*Fibonacci(n+k) produces a sequence whose terms are divisible by (m+1); m>=1.
A recurrence relation for a(n) (m not equal to zero) is:
a(n) = (m+3)*a(n-1) + (m^2+m-1)*a(n-2); a(0)=0, a(1)=m+1.
Notable values of m include:
  m = 1: Fibonacci(3n),
  m = 0: Fibonacci(2n) (using recurrence relation only - the sum above is undefined for m=0),
  m = -1: the zero sequence,
  m = -2: (-1)*Fibonacci(n), or A152163(n+2).
For any value of m, the sequence gives a(n*k) divisible by a(n); n>=1, k>=1, m not equal to -1 (zero is not divisible by zero).
Equivalent sequences are given by: Sum_{k=0..n} binomial(n,k) * (m+1)^k * Fibonacci(k).
When these sequences are divided by m+1, we obtain the family of sequences A057088, A015553, A087567, A087579, A087584, A087603, and so on.
Another interesting value of m, m = -3, gives a(2n-1)= -2 * 5^(n-1); a(2n)=0.

Index entries for linear recurrences with constant coefficients, signature (13,109).

Cf. A000045, A027941, A152163, A014445, A057088, A015553, A087567, A087579, A087584, A087603.

Table[Sum[Binomial[n, k]*10^(n - k)*Fibonacci[n + k], {k, 0, n}], {n, 0, 25}]
FullSimplify[Table[((13 + 11 Sqrt[5])^n - (13 - 11 Sqrt[5])^n)/(2^n Sqrt[5]), {n, 0, 25}]]
LinearRecurrence[{13,109},{0,11},30] (* Harvey P. Dale, Jul 31 2018 *)

a(n) = ((13 + 11*sqrt(5))^n - (13 - 11*sqrt(5))^n)/(2^n*sqrt(5)).
a(n) = 13*a(n-1) + 109*a(n-2); a(0)=0, a(1)=11.
G.f.: 11*x*/(1 - 13*x - 109*x^2). - Corrected by Georg Fischer, May 10 2019

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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A087567 a(n) = (1/5)*Sum_{k=0..n} binomial(n,k)*Fibonacci(k)*5^k.

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A087603 a(n) = (1/8)*Sum_{k=0..n} binomial(n,k)*Fibonacci(k)*8^k.

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A087579 a(n) = (1/6)*Sum_{k=0..n} binomial(n,k)*Fibonacci(k)*6^k.

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A087584 a(n) = (1/7)*Sum_{k=0..n} binomial(n,k)*Fibonacci(k)*7^k.

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A091929 Expansion of (1-6x)/(1-6x-11x^2).

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A225799 a(n) = Sum_{k=0..n} binomial(n,k) * 10^(n-k) * Fibonacci(n+k).

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Formula

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

A087567 a(n) = (1/5)Sum_{k=0..n} binomial(n,k)Fibonacci(k)*5^k.

A087603 a(n) = (1/8)Sum_{k=0..n} binomial(n,k)Fibonacci(k)*8^k.

A087579 a(n) = (1/6)Sum_{k=0..n} binomial(n,k)Fibonacci(k)*6^k.

A087584 a(n) = (1/7)Sum_{k=0..n} binomial(n,k)Fibonacci(k)*7^k.