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 1-4 of 4 results.

A062073 Decimal expansion of Fibonacci factorial constant.

Original entry on oeis.org

1, 2, 2, 6, 7, 4, 2, 0, 1, 0, 7, 2, 0, 3, 5, 3, 2, 4, 4, 4, 1, 7, 6, 3, 0, 2, 3, 0, 4, 5, 5, 3, 6, 1, 6, 5, 5, 8, 7, 1, 4, 0, 9, 6, 9, 0, 4, 4, 0, 2, 5, 0, 4, 1, 9, 6, 4, 3, 2, 9, 7, 3, 0, 1, 2, 1, 4, 0, 2, 2, 1, 3, 8, 3, 1, 5, 3, 1, 2, 1, 6, 8, 4, 5, 2, 6, 2, 1, 5, 6, 2, 4, 9, 4, 7, 9, 7, 7, 4, 1, 2, 5, 9, 1, 3
Offset: 1

Views

Author

Jason Earls, Jun 27 2001

Keywords

Comments

The Fibonacci factorial constant is associated with the Fibonacci factorial A003266.
Two closely related constants are A194159 and A194160. [Johannes W. Meijer, Aug 21 2011]

Examples

			1.226742010720353244417630230455361655871409690440250419643297301214...

References

  • Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.5.
  • R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison Wesley, 1990, pp. 478, 571.

Links

Crossrefs

Cf. A003266, A003267, A003268, A056569, A062072, A062381, A135407, A181926.
Cf. A218490, A253924, A256831, A259314, A259405.

Programs

  • Mathematica
    RealDigits[N[QPochhammer[-1/GoldenRatio^2], 105]][[1]] (* Alonso del Arte, Dec 20 2010 *)
RealDigits[N[Re[(-1)^(1/24) * GoldenRatio^(1/12) / 2^(1/3) * EllipticThetaPrime[1,0,-I/GoldenRatio]^(1/3)], 120]][[1]] (* Vaclav Kotesovec, Jul 19 2015, after Eric W. Weisstein *)
  • PARI
    \p 1300 a=-1/(1/2+sqrt(5)/2)^2; prod(n=1,17000,(1-a^n))
  • PARI
    { default(realprecision, 5080); p=-1/(1/2 + sqrt(5)/2)^2; x=prodinf(k=1, 1-p^k); for (n=1, 5000, d=floor(x); x=(x-d)*10; write("b062073.txt", n, " ", d)) } \\ Harry J. Smith, Jul 31 2009

Formula

C = (1-a)*(1-a^2)*(1-a^3)... 1.2267420... where a = -1/phi^2 and where phi is the Golden ratio = 1/2 + sqrt(5)/2.
C = QPochhammer[ -1/GoldenRatio^2]. [Eric W. Weisstein, Dec 01 2009]
C = A194159 * A194160. [Johannes W. Meijer, Aug 21 2011]
C = exp( Sum_{k>=1} 1/(k*(1-(-(3+sqrt(5))/2)^k)) ). - Vaclav Kotesovec, Jun 08 2013
C = Sum_{k = -inf .. inf} (-1)^((k-1)*k/2) / phi^((3*k-1)*k), where phi = (1 + sqrt(5))/2. - Vladimir Reshetnikov, Sep 20 2016

A126772 Padovan factorials: a(n) is the product of the first n terms of the Padovan sequence. Similar to the Fibonacci factorial.

Original entry on oeis.org

1, 1, 1, 2, 4, 12, 48, 240, 1680, 15120, 181440, 2903040, 60963840, 1706987520, 63158538240, 3094768373760, 201159944294400, 17299755209318400, 1972172093862297600, 297797986173206937600, 59559597234641387520000
Offset: 1

Views

Author

John Lien, Feb 17 2007

Keywords

Links

Crossrefs

Cf. A000931, A003266, A060006, A253924.

Programs

  • Maple
    From R. J. Mathar, Sep 14 2010: (Start)
A000931 := proc(n) option remember; if n = 0 then 1; elif n <=2 then 0; else procname(n-2)+procname(n-3) ; end if; end proc:
A126772 := proc(n) mul( A000931(i),i=5..n+4) ; end proc: seq(A126772(n),n=1..40) ; (End)
  • Mathematica
    Rest[FoldList[Times,1,LinearRecurrence[{0,1,1},{1,1,1},30]]] (* Harvey P. Dale, Apr 29 2013 *)

Formula

a(n) ~ c * d^(n/2) * r^(n^2/2), where r = 1.324717957244746... (see A060006) is the root of the equation r^3 = r + 1, d = 0.393641282401116385386658448446561... is the root of the equation 1 + 7*d + 184*d^2 - 529*d^3 = 0, c = 1.25373683131537208838997864311903035079685338006712312402418098138010834953... (see A253924). - Vaclav Kotesovec, Jan 26 2015

Extensions

More terms from R. J. Mathar, Sep 14 2010

A256831 Decimal expansion of Pell factorial constant.

Original entry on oeis.org

1, 1, 4, 1, 9, 8, 2, 5, 6, 9, 6, 6, 7, 7, 9, 1, 2, 0, 6, 0, 2, 8, 0, 4, 3, 3, 3, 8, 3, 6, 7, 8, 6, 0, 1, 5, 0, 8, 6, 4, 7, 3, 0, 4, 8, 2, 4, 0, 8, 5, 4, 0, 7, 9, 1, 5, 5, 6, 2, 5, 4, 3, 5, 2, 4, 4, 9, 8, 4, 3, 7, 8, 5, 4, 8, 0, 6, 2, 0, 8, 6, 0, 7, 8, 2, 5, 0, 6, 3, 7, 0, 6, 0, 9, 2, 5, 3, 3, 4, 7, 8, 1, 6, 3, 6
Offset: 1

Views

Author

Vaclav Kotesovec, Apr 10 2015

Keywords

Examples

			1.141982569667791206028043338367860150864730482408540791556...

Crossrefs

Cf. A000129, A099929, A256832.
Cf. A062073, A218490, A253924.

Programs

  • Mathematica
    RealDigits[N[QPochhammer[2*Sqrt[2]-3], 105]][[1]]

Formula

Equals limit n->infinity A256832(n) / ((1+sqrt(2))^(n*(n+1)/2) / 2^(3*n/2)).

A259314 Decimal expansion of partition factorial constant.

Original entry on oeis.org

9, 1, 1, 0, 1, 6, 7, 3, 1, 3, 3, 2, 2, 4, 9, 9, 5, 1, 8, 6, 1, 5, 4, 7, 4, 6, 9, 5, 9, 4, 6, 8, 3, 4, 5, 2, 7, 8, 0, 7, 3, 8, 6, 0, 9, 7, 8, 0, 0, 8, 0, 9, 3, 0, 2, 8, 1, 3, 2, 1, 4, 9, 0, 2, 2, 7, 5, 9, 1, 4, 9, 1, 2, 4, 0, 4, 5, 5, 5, 7, 5, 1, 1, 6, 5, 0, 2, 5, 3, 7, 0, 7, 0, 2, 7, 5, 3, 9, 2, 1, 0, 4, 4, 7, 5, 0
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 24 2015

Keywords

Examples

			0.91101673133224995186154746959468345278073860978008093028132149022759...

Links

Crossrefs

Cf. A000041, A058694, A062073, A218490, A253924, A256831, A259405.

Programs

  • Mathematica
    (* The iteration cycle: *) Do[Print[Product[N[PartitionsP[k]/((E^(Sqrt[2/3]*Sqrt[k-1/24]*Pi) * (1 - Sqrt[3/2]/(Sqrt[k-1/24]*Pi))) / (4*Sqrt[3]*(k-1/24))), 150], {k, 1, n}]], {n, 500, 50000, 500}]

Formula

Equals limit n->infinity Product_{k=1..n} p(k) / (exp(Pi*sqrt(2/3*(k-1/24))) / (4*sqrt(3)*(k-1/24)) * (1 - sqrt(3/(2*(k-1/24)))/Pi)), where p(k) is the partition function A000041.
Showing 1-4 of 4 results.

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