A015536 -id:A015536 - cp's OEIS frontend

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

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

A106569 a(n) = 5a(n-1) + 3a(n-2), where a(0) = 0, a(1) = 3.

Original entry on oeis.org

0, 3, 15, 84, 465, 2577, 14280, 79131, 438495, 2429868, 13464825, 74613729, 413463120, 2291156787, 12696173295, 70354336836, 389860204065, 2160364030833, 11971400766360, 66338095924299, 367604681920575

Offset: 0

Roger L. Bagula, May 30 2005

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,3).

Cf. A015536.

[n le 2 select 3*(n-1) else 5*Self(n-1) + 3*Self(n-2): n in [1..41]]; // G. C. Greubel, Sep 07 2021

a[0]:=0: a[1]:=3: for n from 2 to 23 do a[n]:= 5*a[n-1]+3*a[n-2] od: seq(a[n], n=0..23);

LinearRecurrence[{5,3}, {0,3}, 40] (* G. C. Greubel, Sep 07 2021 *)

[3*lucas_number1(n,5,-3) for n in (0..40)] # G. C. Greubel, Sep 07 2021

a(n) = 5*a(n-1) + 3*a(n-2), a(0) = 0, a(1) = 3.
a(n) = 3*A015536(n).
G.f.: 3*x/(1 - 5*x - 3*x^2). - G. C. Greubel, Sep 07 2021

Edited by N. J. A. Sloane, Apr 30 2006

New (corrected) name by G. C. Greubel, Sep 07 2021

A271451 Triangle read by rows of coefficients of polynomials Q_n(x) = 2^(-n)((x + sqrt(x(x + 6) - 3) + 1)^n - (x - sqrt(x(x + 6) - 3) + 1)^n)/sqrt(x(x + 6) - 3).

Original entry on oeis.org

1, 1, 1, 0, 3, 1, -1, 3, 5, 1, -1, -1, 10, 7, 1, 0, -6, 7, 21, 9, 1, 1, -6, -10, 31, 36, 11, 1, 1, 1, -29, 7, 79, 55, 13, 1, 0, 9, -24, -63, 81, 159, 78, 15, 1, -1, 9, 15, -123, -54, 264, 279, 105, 17, 1, -1, -1, 57, -69, -321, 132, 624, 447, 136, 19, 1, 0, -12, 50, 126, -459, -507, 741, 1245, 671, 171, 21, 1, 1, -12, -20, 302, -81, -1419, -258, 2163, 2227, 959, 210, 23, 1

Offset: 1

Ilya Gutkovskiy, Apr 08 2016

The polynomials Q_n(x) have generating function G(x,t) = t/(1 - (x + 1)*t - (x - 1)*t^2) = t + (x + 1)*t^2 + x*(x + 3)*t^3 + (x^3 + 5*x^2 + 3*x - 1)*t^4 + ...
Q_n(x) can be defined by the recurrence relation Q_n(x) = (x + 1)*Q_(n-1)(x) + (x - 1)*Q_(n-2)(x), Q_0(x)=0, Q_1(x)=1.
Discriminants of Q_n(x) gives the sequence: 0, 1, 1, 9, 320, 35600, 10948608, 8664190976, 16836271800320, 77757312009240576, 833309554769920000000, 20346889104219547132493824,...
Q_n(0)  = A128834(n).
Q_n(1)  = A000079(n-1), n>0.
Q_n(2)  = A006190(n).
Q_n(3)  = A090017(n).
Q_n(4)  = A015536(n).
Q_n(5)  = A135032(n).
Q_n(6)  = A015562(n).
Q_n(7)  = A190560(n).
Q_n(8)  = A015583(n).
Q_n(9)  = A190957(n).
Q_n(10) = A015603(n).

			Triangle begins:
   1;
   1,  1;
   0,  3,  1;
  -1,  3,  5,  1;
  -1, -1, 10,  7,  1;
...
The first few polynomials are:
Q_0(x) = 0;
Q_1(x) = 1;
Q_2(x) = x + 1;
Q_3(x) = x^2 + 3*x;
Q_4(x) = x^3 + 5*x^2 + 3*x - 1;
Q_5(x) = x^4 + 7*x^3 + 10*x^2 - x - 1,
...

G. C. Greubel, Table of n, a(n) for the first 101 rows, flattened
Ilya Gutkovskiy, Polynomials Q_n(x)
Eric Weisstein's World of Mathematics, Fibonacci Polynomial

Cf. A049310.

Flatten[Table[CoefficientList[((x + Sqrt[x (x + 6) - 3] + 1)^n - (x - Sqrt[x (x + 6) - 3] + 1)^n)/2^n/Sqrt[x (x + 6) - 3], x], {n, 0, 13}]]

A111365 a(n) = 5a(n-1) + 3a(n-2) where a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 8, 43, 239, 1324, 7337, 40657, 225296, 1248451, 6918143, 38336068, 212434769, 1177182049, 6523214552, 36147618907, 200307738191, 1109981547676, 6150830952953, 34084099407793, 188872989897824, 1046617247712499

Offset: 0

Parthasarathy Nambi, Nov 07 2005

nonn

			a(2) = 5*a(1) + 3*a(0) = 5*1 + 3*1 = 8 which is the third term in the sequence.

Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", Wiley, 2001

Index entries for linear recurrences with constant coefficients, signature (5, 3).

Cf. A000045, A072264.

Transpose[NestList[Flatten[{Rest[#],ListCorrelate[{3,5},#]}]&, {1,1},40]][[1]]  (* Harvey P. Dale, Mar 23 2011 *)

a(n)=A015536(n+1)-4*A015536(n). G.f.: (1-4x)/(1-5x-3x^2). [From R. J. Mathar, Jul 08 2009]

More terms from Robert G. Wilson v, Nov 10 2005

cp's OEIS Frontend

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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A106569 a(n) = 5*a(n-1) + 3*a(n-2), where a(0) = 0, a(1) = 3.

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A271451 Triangle read by rows of coefficients of polynomials Q_n(x) = 2^(-n)*((x + sqrt(x*(x + 6) - 3) + 1)^n - (x - sqrt(x*(x + 6) - 3) + 1)^n)/sqrt(x*(x + 6) - 3).

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A111365 a(n) = 5*a(n-1) + 3*a(n-2) where a(0) = a(1) = 1.

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

A106569 a(n) = 5a(n-1) + 3a(n-2), where a(0) = 0, a(1) = 3.

A271451 Triangle read by rows of coefficients of polynomials Q_n(x) = 2^(-n)((x + sqrt(x(x + 6) - 3) + 1)^n - (x - sqrt(x(x + 6) - 3) + 1)^n)/sqrt(x(x + 6) - 3).

A111365 a(n) = 5a(n-1) + 3a(n-2) where a(0) = a(1) = 1.