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.

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A198834 Number of sequences of n coin flips that win on the last flip, if the sequence of flips ends with (0,1,1) or (1,1,1).

Original entry on oeis.org

0, 0, 2, 2, 4, 6, 10, 16, 26, 42, 68, 110, 178, 288, 466, 754, 1220, 1974, 3194, 5168, 8362, 13530, 21892, 35422, 57314, 92736, 150050, 242786, 392836, 635622, 1028458, 1664080, 2692538, 4356618, 7049156, 11405774, 18454930, 29860704, 48315634, 78176338
Offset: 1

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Author

Paul Weisenhorn, Oct 30 2011

Keywords

Comments

If the sequence ends with (011) Abel wins; if it ends with (111) Kain wins.
Kain(n)=0 for n <> 3; Kain(3)=1.
Abel(n) = A128588(n-2) for n > 2.
a(n) = A006355(n-1) for n > 2.
Win probability for Abel: Sum_{n>=1} Abel(n)/2^n = 7/8.
Win probability for Kain: Kain(3)/8 = 1/8.
Mean length of the game: Sum_{n>=1} n*a(n)/2^n = 7.
Appears to be essentially the same as A163733, A118658, A055389. - R. J. Mathar, Oct 31 2011

Examples

			For n=6 the a(6)=6 solutions are (0,0,0,0,1,1), (1,0,0,0,1,1); (0,1,0,0,1,1), (1,1,0,0,1,1), (0,0,1,0,1,1), (1,0,1,0,1,1) all for Abel.

References

  • A. Engel, Wahrscheinlichkeit und Statistik, Band 2, Klett, 1978, pages 25-26.

Links

Crossrefs

Cf. A006355, A055389, A118658, A128588, A163733.

Programs

  • Maple
    a(1):=0: a(2):=0: a(3):=2:
ml:=0.75: pot:=8:
for n from 4 to 100 do
  pot:=2*pot:
  a(n):=a(n-1)+a(n-2):
  ml:=ml+n*a(n)/pot:
end do:
printf("%12.8f",ml);
seq(a(n),n=1..100);
  • Mathematica
    Join[{0, 0}, Table[2*Fibonacci[n], {n, 70}]] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2012 *)
Join[{0},LinearRecurrence[{1,1},{0,2},50]] (* Vincenzo Librandi, Feb 19 2012 *)

Formula

a(n) = a(n-1) + a(n-2) for n > 3.
G.f.: 2*x^3/(1 - x - x^2).
a(n) = 2*A000045(n-2). - R. J. Mathar, Jan 11 2017
E.g.f.: 2 - 2*x + 2*exp(x/2)*(3*sqrt(5)*sinh(sqrt(5)*x/2) - 5*cosh(sqrt(5)*x/2))/5. - Stefano Spezia, Feb 19 2023

A258575 Numbers n such that Lucas(n)-Fibonacci(n) is squarefree.

Original entry on oeis.org

0, 2, 3, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 48, 50, 53, 54, 56, 59, 60, 62, 63, 65, 66, 68, 69, 71, 72, 74, 75, 77, 78, 80, 81, 83, 84, 86, 87, 89, 90, 93, 95, 96, 98, 99, 102, 104, 105
Offset: 1

Views

Author

Vincenzo Librandi, Jun 04 2015

Keywords

Comments

These numbers belong to the sequence A007494 (see Chai Wah Wu argumentation in A258574).
Also numbers n such that 2*Fibonacci(n-1) is squarefree. [Bruno Berselli, Jun 05 2015]
A258575(n) = A258574(n+1)-1. - Chai Wah Wu, Jun 09 2015

Crossrefs

Cf. A000032, A000045, A005117, A007494, A037918, A118658, A258574.

Programs

  • Magma
    [0] cat [n: n in [2..150] | IsSquarefree(Lucas(n)-Fibonacci(n))];
  • Mathematica
    Select[Range[0, 200], SquareFreeQ[LucasL[#] - Fibonacci[#]] &]
  • Sage
    [n for n in (0..110) if is_squarefree(2*fibonacci(n-1))] # Bruno Berselli, Jun 05 2015

Extensions

Name corrected by Bruno Berselli, Jun 05 2015

A282465 a(n) = 11*Fibonacci(n+3) + Fibonacci(n-8) with n>=0.

Original entry on oeis.org

1, 46, 47, 93, 140, 233, 373, 606, 979, 1585, 2564, 4149, 6713, 10862, 17575, 28437, 46012, 74449, 120461, 194910, 315371, 510281, 825652, 1335933, 2161585, 3497518, 5659103, 9156621, 14815724, 23972345, 38788069, 62760414, 101548483, 164308897, 265857380, 430166277
Offset: 0

Views

Author

Bruno Berselli, Feb 20 2017

Keywords

Comments

Similar sequences with the formula h*Fibonacci(n+k) + Fibonacci(n+k-h):
h= 1, k=-1: A000045;
h= 2, k= 1: A013655;
h= 3, k=-2: A118658 = 2*A212804;
h= 4, k= 2: A022379 = 3*A000204;
h= 5, k= 1: A022113;
h= 6, k= 2: A022125;
h= 7, k= 3: A097657;
h= 8, k= 2: A022355 = 21*A000045;
h= 9, k= 3: 10, 32, 42, 74, 116, 190, 306, 496, 802, ... = 2*A022140;
h=10, k= 3: 33, 22, 55, 77, 132, 209, 341, 550, 891, ... = 11*A013655;
h=11, k= 3: this sequence.

Links

Crossrefs

Cf. A000045, A013655, A022113, A022125, A022355, A022379, A022379, A097657, A118658.
Cf. sequences with g.f. (1 + r*x)/(1 - x - x^2) for r = 2..31, respectively: A000204, A000285, A022095 - A022110, A022391 - A022402.

Programs

  • Magma
    [11*Fibonacci(n+3)+Fibonacci(n-8): n in [0..40]];
  • Mathematica
    Table[11 Fibonacci[n + 3] + Fibonacci[n - 8], {n, 0, 40}]
LinearRecurrence[{1,1},{1,46},36] (* or *) CoefficientList[Series[(1 + 45*x)/(1 - x - x^2) , {x,0,35}],x] (* Indranil Ghosh, Feb 22 2017 *)
  • PARI
    a(n) = 11*fibonacci(n+3) + fibonacci(n-8) \\ Indranil Ghosh, Feb 23 2017

Formula

G.f.: (1 + 45*x)/(1 - x - x^2).
a(n) = a(n-1) + a(n-2).
a(n) = a(i)*Fibonacci(n-i+1) + a(i-1)*Fibonacci(n-i). Examples:
for i= 3, a(3)=93, a(2)= 47: a(n) = 93*Fibonacci(n-2) + 47*Fibonacci(n-3);
for i=-1, a(-1)=45, a(-2)=-44: a(n) = 45*Fibonacci(n+2) - 44*Fibonacci(n+1).
Other formulae:
a(n) = 44*Fibonacci(n) + Fibonacci(n+2),
a(n) = 45*Fibonacci(n) + Fibonacci(n+1),
a(n) = 46*Fibonacci(n) + Fibonacci(n-1),
a(n) = 47*Fibonacci(n) - Fibonacci(n-2).
a(n) = ((91 + sqrt(5))*((1 + sqrt(5))/2)^n - (91 - sqrt(5))*((1 - sqrt(5))/2)^n)/sqrt(20).
Previous Showing 11-13 of 13 results.

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