A385508 Decimal expansion of Sum_{k>=1} Fibonacci(k)/Catalan(k).

2, 3, 5, 1, 9, 6, 8, 9, 1, 5, 4, 4, 7, 3, 8, 0, 7, 2, 5, 8, 3, 2, 2, 4, 9, 3, 5, 4, 0, 3, 5, 5, 6, 1, 8, 8, 1, 1, 2, 0, 3, 6, 7, 5, 9, 6, 8, 5, 3, 2, 7, 7, 0, 8, 7, 2, 4, 6, 2, 0, 9, 6, 4, 6, 4, 1, 6, 1, 8, 9, 1, 5, 4, 2, 5, 5, 7, 5, 8, 3, 0, 3, 0, 3, 3, 6, 2, 6, 9, 5, 3, 8, 0, 0, 6, 9, 7, 9, 8, 9, 2, 5, 0, 0, 6

Offset

1,1

Links

Table of n, a(n) for n=1..105.

Formula

Equals 2*(4015 + 6*sqrt(110*(1541*sqrt(5) - 3351)) * arctanh(sqrt((2*sqrt(5) - 3)/11)) + 6*sqrt(110*(3351 + 1541*sqrt(5))) * arctan(sqrt((3 + 2*sqrt(5))/11))) / 6655.

Equals Sum_{n>=1} Fibonacci(n)*n!*(n + 1)!/(2*n)!. - Artur Jasinski, Jul 01 2025

Example

2.351968915447380725832249354035561881120367596853277087246209646416189...

Mathematica

RealDigits[2*(4015 + 6*Sqrt[110*(1541*Sqrt[5] - 3351)] * ArcTanh[Sqrt[(2*Sqrt[5] - 3)/11]] + 6*Sqrt[110*(3351 + 1541*Sqrt[5])] * ArcTan[Sqrt[(3 + 2*Sqrt[5])/11]])/6655, 10, 120][[1]] (* or *)
RealDigits[Sum[Fibonacci[k]/CatalanNumber[k], {k, 1, Infinity}], 10, 105][[1]]

Crossrefs

Cf. A000045, A000108, A385450.

Sequence in context: A132597 A030335 A256611 * A276350 A262215 A030790

Adjacent sequences: A385505 A385506 A385507 * A385509 A385510 A385511

Keyword

nonn,cons

Author

Vaclav Kotesovec, Jul 01 2025

Status

approved