A179907 Numerators in the approximation of sqrt(2) satisfying the recurrence: a(n)= [a(n-1)*a(n-2)+2]/[a(n-1)+a(n-2)] with a(1)=a(2)=1.

1, 1, 3, 7, 41, 577, 47321, 54608393, 5168247530883, 564459384575477049359, 5834531641231893991002972081099601, 6586712278991805873205386201784303378672881239591942401, 76860762406936659889820302898486134128259057213749687093311841304353879849599982549213361

Offset

1,3

Links

Table of n, a(n) for n=1..13.

Peter Bala, Notes on A179823 and A179907.

Formula

From Peter Bala, Jul 05 2026: (Start)

a(n) = A001333(Fibonacci(n)) = (1/2) * ( (1 + sqrt(2))^Fibonacci(n) + (1 - sqrt(2))^Fibonacci(n) ).

a(n) ~ 1/2 * exp(c*phi^n), where c = 1/sqrt(5) * log(1 + sqrt(2)) and phi = (1 + sqrt(5))/2 is the golden ratio. (End)

Maple

with(combinat):
seq( simplify( (1/2) * ( (1 + sqrt(2))^fibonacci(n) + (1 - sqrt(2))^fibonacci(n) ) ), n = 1..13); # Peter Bala, Jul 05 2026

Crossrefs

Cf. A000045, A001333, A002203, A179823

Sequence in context: A381361 A006383 A181148 * A080581 A086397 A229941

Adjacent sequences: A179904 A179905 A179906 * A179908 A179909 A179910

Keyword

frac,nonn

Author

Mark Dols, Aug 01 2010

Extensions

a(11)-a(13) from Peter Bala, Jul 05 2026

Status

approved