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A114551
Continued fraction expansion of the constant (A114550) equal to Sum_{n>=0} 1/A112373(n) such that the partial quotients satisfy a(2n) = A112373(n) for n > 0 and a(2n+1) = A112373(n+1)/A112373(n) for n >= 0.
6
2, 1, 1, 2, 2, 6, 12, 78, 936, 73086, 68408496, 4999703411742, 342022190843338960032, 1710009514450915230711940280907486, 584861200495456320274313200204390612579749188443599552
OFFSET
0,1
COMMENTS
A112373 is defined by the recurrence: let b(n) = A112373(n), then
b(n) = (b(n-1)^3 + b(n-1)^2)/b(n-2) for n >= 2 with b(0)=b(1)=1.
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.
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
Cf. A112373, A114550 (constant), A114552 (bisection).
Sequence in context: A359652 A179974 A246402 * A191620 A214751 A306512
KEYWORD
cofr,nonn
AUTHOR
Paul D. Hanna, Dec 08 2005
STATUS
approved