OFFSET
0,1
COMMENTS
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..20
Taras Goy and Mark Shattuck, Determinant Formulas of Some Hessenberg Matrices with Jacobsthal Entries Jacobsthal Entries, Applications and Appl. Math. (2021) Vol. 16, Issue 1, Art. 10.
A. Hone, Curious continued fractions, nonlinear recurrences and transcendental numbers, arXiv:1507.00063 [math.NT], 2015 and J. Int. Seq. 18 (2015) # 15.8.4.
FORMULA
a(2n) = a(2n-1)*a(2n-2) for n>=2, a(2n+1) = a(2n)*a(2n-1) + a(2n-1) for n>=1, with a(0)=2, a(1)=a(2)=1. - Jeffrey Shallit
EXAMPLE
2.584401724019776724812076147153331342112382090467969...
= Sum_{n>=0} 1/A112373(n) = 1/1 + 1/1 + 1/2 + 1/12 + 1/936 + 1/68408496 + ...
= [2;1,1,2,2,6,12,78,936,73086,68408496,...] (continued fraction).
The recurrence of partial quotients is demonstrated by:
(odd-index) a(7) = 78 = a(6)*a(5) + a(5) = 12*6 + 6;
(even-index) a(8) = 936 = a(7)*a(6) = 78*12.
MATHEMATICA
a[0] = 2; a[1] = a[2] = 1;
a[n_] := a[n] = a[n-1] a[n-2] + Mod[n, 2] a[n-2];
a /@ Range[0, 14] (* Jean-François Alcover, Oct 01 2019 *)
PROG
(PARI) a(n)=if(n<0, 0, if(n<3, [2, 1, 1][n+1], a(n-1)*a(n-2)+(n%2)*a(n-2)))
CROSSREFS
KEYWORD
cofr,nonn
AUTHOR
Paul D. Hanna, Dec 08 2005
STATUS
approved