OFFSET
0,3
COMMENTS
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..381
S. Janson, A divergent generating function that can be summed and analysed analytically, Discrete Mathematics and Theoretical Computer Science; 2010, Vol. 12, No. 2, 1-22.
FORMULA
a(n) = (Pi/3) * (BesselJ(1/3+n,2/3) * BesselY(1/3,2/3) - BesselJ(1/3,2/3) * BesselY(1/3+n,2/3)).
a(n) = Sum_{k = 0..floor((n-1)/2)} (-1)^k*3^(n-2*k-1)*(n-2*k-1)!*binomial(n-k-1,k)*binomial(n-k-2/3,k+1/3), cf. A058798. - Peter Bala, Aug 01 2013
a(n) ~ n! * BesselJ(1/3, 2/3) * 3^(n-2/3) * n^(-2/3). - Vaclav Kotesovec, Jul 31 2014
a(n) = 3^n*Gamma(n+1/3)*hypergeometric([1/2-n/2, 1-n/2], [4/3, 2/3-n, 1-n], -4/9)/Gamma(1/3) for n >= 2. - Peter Luschny, Sep 10 2014
MATHEMATICA
f[n_Integer] = Module[{a}, a[n] /. RSolve[{a[n] == (3*n - 2)*a[n - 1] - a[n - 2], a[0] == 0, a[1] == 1}, a[n], n][[1]] // FullSimplify] Rationalize[N[Table[f[n], {n, 0, 25}], 100], 0]
RecurrenceTable[{a[0]==0, a[1]==1, a[n]==(3n-2)*a[n-1]-a[n-2]}, a, {n, 20}] (* Vaclav Kotesovec, Jul 31 2014 *)
nxt[{n_, a_, b_}]:={n+1, b, b(3n+1)-a}; NestList[nxt, {1, 0, 1}, 20][[;; , 2]] (* Harvey P. Dale, Jun 03 2023 *)
PROG
(Sage)
def A121353(n):
if n < 2: return n
return 3^n*gamma(n+1/3)*hypergeometric([1/2-n/2, 1-n/2], [4/3, 2/3 -n, 1-n], -4/9)/gamma(1/3)
[round(A121353(n).n(100)) for n in (0..19)] # Peter Luschny, Sep 10 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Roger L. Bagula and Bob Hanlon (hanlonr(AT)cox.net), Sep 05 2006
STATUS
approved