A123851 A cubic recurrence: a(0) = 1, a(n) = n*a(n-1)^3 for n >= 1.
1, 1, 2, 24, 55296, 845378412871680, 3624972460853492659595005581182702601633792000
Offset: 0
Examples
a(3) = 3*a(2)^3 = 3*(2*a(1)^3)^3 = 3*(2*(1*a(0)^3)^3)^3 = 3*(2*(1*1^3)^3)^3 = 3*(2*1)^3 = 3*8 = 24. G.f. = 1 + x + 2*x^2 + 24*x^3 + 55296*x^4 + 845378412871680*x^5 + ...
References
- Steven R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003, p. 446.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..8
- Sung-Hyuk Cha, On the k-ary Tree Combinatorics.
- Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, arXiv:0610499 [math.CA], 2006.
- Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl. 332 (2007), 292-314.
- Eric Weisstein's World of Mathematics, Somos's Quadratic Recurrence Constant.
- Aimin Xu, Asymptotic expansion related to the Generalized Somos Recurrence constant, International Journal of Number Theory 15(10) (2019), 2043-2055.
Programs
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GAP
List([0..8], n-> Product([0..n-1], k-> (n-k)^(3^k)) ); # G. C. Greubel, Aug 10 2019
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Magma
[n eq 0 select 1 else (&*[(n-k)^(3^k): k in [0..n-1]]):n in [0..8]]; // G. C. Greubel, Aug 10 2019
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Mathematica
a[n_]:= If[n==0, 1, n*a[n-1]^3]; Table[a[n], {n,0,7}] nxt[{n_,a_}]:={n+1, (n+1)a^3}; NestList[nxt,{0,1},7][[All,2]] (* Harvey P. Dale, May 25 2019 *) -
PARI
{a(n) = if( n<1, n==0, prod(k=0, n-1, (n - k)^3^k))}; /* Michael Somos, Aug 07 2016 */ -
Sage
[1]+[prod((n-k)^(3^k) for k in (0..n-1)) for n in (1..8)] # G. C. Greubel, Aug 10 2019
Formula
a(n) ~ c^(3^n)*n^(-1/2)/(1 + 3/(4*n) - 15/(32*n^2) + 113/(128*n^3) + ...) where c = 1.1563626843322... is the cubic recurrence constant A123852.
Extensions
Corrected by Harvey P. Dale, May 25 2019
Comments