cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A258160 a(n) = 8*Lucas(n).

Original entry on oeis.org

16, 8, 24, 32, 56, 88, 144, 232, 376, 608, 984, 1592, 2576, 4168, 6744, 10912, 17656, 28568, 46224, 74792, 121016, 195808, 316824, 512632, 829456, 1342088, 2171544, 3513632, 5685176, 9198808, 14883984, 24082792, 38966776, 63049568, 102016344, 165065912
Offset: 0

Views

Author

Bruno Berselli, May 22 2015

Keywords

Links

Crossrefs

Cf. A022112, A039834, A156279.
Cf. A022091: 8*Fibonacci(n).
Cf. A022352: Fibonacci(n+6) + Fibonacci(n-6).
Cf. sequences with the formula Fibonacci(n+k)-Fibonacci(n-k): A000045 (k=1); A000032 (k=2); A022087 (k=3); A022379 (k=4, without initial 6); A022345 (k=5); this sequence (k=6); A022363 (k=7).

Programs

  • Magma
    [8*Lucas(n): n in [0..40]];
  • Mathematica
    Table[8 LucasL[n], {n, 0, 40}]
CoefficientList[Series[8*(2 - x)/(1 - x - x^2), {x, 0, 50}], x] (* G. C. Greubel, Dec 21 2017 *)
  • PARI
    a(n)=([0,1; 1,1]^n*[16;8])[1,1] \\ Charles R Greathouse IV, Oct 07 2015
  • Sage
    [8*lucas_number2(n, 1, -1) for n in (0..40)]

Formula

G.f.: 8*(2 - x)/(1 - x - x^2).
a(n) = Fibonacci(n+6) - Fibonacci(n-6), where Fibonacci(-6..-1) = -8, 5, -3, 2, -1, 1 (see similar sequences listed in Crossrefs).
a(n) = Lucas(n+4) + Lucas(n) + Lucas(n-4), where Lucas(-4..-1) = 7, -4, 3, -1.
a(n) = a(n-1) + a(n-2) for n>1, a(0)=16, a(1)=8.
a(n) = 2*A156279(n).
a(n+1) = 4*A022112(n).

A226328 a(0)=1, a(1)=-2; a(n+2) = a(n+1) + a(n) + (period 3: repeat 3, 1, -1).

Original entry on oeis.org

1, -2, 2, 1, 2, 6, 9, 14, 26, 41, 66, 110, 177, 286, 466, 753, 1218, 1974, 3193, 5166, 8362, 13529, 21890, 35422, 57313, 92734, 150050, 242785, 392834, 635622, 1028457, 1664078, 2692538, 4356617, 7049154, 11405774, 18454929, 29860702, 48315634, 78176337
Offset: 0

Views

Author

Paul Curtz, Jun 04 2013

Keywords

Comments

a(n+1)/a(n) -> the golden ratio, A001622.
a(3*n)+a(3*n+1)+a(3*n+2) = 1,9,49,217,929,... = b(n), and b(n+1)-b(n) = 8*A015448(n+1).

Examples

			a(2)=-2+1+3=2, a(3)=2-2+1=1, a(4)=1+2-1=2, a(5)=2+1+3=6.
a(0)=F(-3)+F(n)-1=2+0-1=1,  a(1)=-1+1-2=-2, a(2)=1+1-0=2.
a(3)=1+4*0=1, a(4)=-2+4*1=2, a(5)=2+4*1=6, a(6)=1+4*2=9.

Links

Crossrefs

Cf. A156279, A022120, A022353.

Programs

  • Magma
    I:=[1,-2,2,1,2]; [n le 5 select I[n] else Self(n-1)+Self(n-2)+Self(n-3)-Self(n-4)-Self(n-5): n in [1..40]]; // Vincenzo Librandi, Jun 05 2013
  • Mathematica
    CoefficientList[Series[(2 x^4 + 3 x^2 - 3 x + 1) / (x^5 + x^4 - x^3 - x^2 - x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 05 2013 *)
LinearRecurrence[{1, 1, 1, -1, -1}, {1, -2, 2, 1, 2}, 40] (* Hugo Pfoertner, Feb 12 2024 *)
  • PARI
    Vec((2*x^4+3*x^2-3*x+1)/(x^5+x^4-x^3-x^2-x+1)+O(x^99)) \\ Charles R Greathouse IV, Jun 04 2013

Formula

a(n) = F(n-3) + F(n) - A010872(n+1).
a(n+3) = a(n) + 4*F(n).
G.f.: (2*x^4+3*x^2-3*x+1)/( (x-1)*(x^2+x-1)*(1+x+x^2) ). [Charles R Greathouse IV, Jun 04 2013]
a(n) = A057078(n+1) +2*A212804(n) -1. - R. J. Mathar, Jun 26 2013

Extensions

a(23) corrected by Charles R Greathouse IV, Jun 04 2013
More terms from Bruno Berselli, Jun 04 2013

A022410 a(n) = a(n-1) + a(n-2) + 1 for n>1, a(0)=3, a(1)=11.

Original entry on oeis.org

3, 11, 15, 27, 43, 71, 115, 187, 303, 491, 795, 1287, 2083, 3371, 5455, 8827, 14283, 23111, 37395, 60507, 97903, 158411, 256315, 414727, 671043, 1085771, 1756815, 2842587, 4599403, 7441991, 12041395, 19483387, 31524783, 51008171, 82532955, 133541127
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000032, A156279.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((3+5*x-7*x^2)/((x-1)*(x^2+x-1)))); // G. C. Greubel, Feb 28 2018
  • Mathematica
    LinearRecurrence[{2, 0, -1}, {3, 11, 15}, 40] (* Bruno Berselli, Jul 27 2017 *)
  • PARI
    Vec((3+5*x-7*x^2)/((x-1)*(x^2+x-1)) + O(x^50)) \\ Colin Barker, Jul 27 2017
  • Python
    from sympy import lucas
def a(n): return 4 * lucas(n + 1) - 1
print([a(n) for n in range(51)]) # Indranil Ghosh, Jul 27 2017

Formula

From R. J. Mathar, Mar 11 2011: (Start)
a(n+1) - a(n) = A156279(n).
G.f.: (3 + 5*x - 7*x^2) / ((x - 1)*(x^2 + x - 1)).
(End)
a(n) = A156279(n+1) - 1. - Bruno Berselli, Jul 27 2017
From Colin Barker, Jul 27 2017: (Start)
a(n) = 2^(-n)*(-2^n + 2*(1-sqrt(5))^(1+n) + 2*(1+sqrt(5))^(1+n)).
a(n) = 2*a(n-1) - a(n-3) for n>2.
(End)
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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