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

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

Offset: 1

Author

John Lien, Feb 17 2007

Keywords

• nonn
• easy

Links

• Ian Stewart, Tales of a Neglected Number
• Ian Stewart, Tales of a Neglected Number, Mathematical Recreations, Scientific American, Vol. 274, No. 6 (1996), pp. 102-103.
• Eric Weisstein's World of Mathematics, Padovan Sequence
• E. Wilson, The Scales of Mt. Meru

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