A389114 Decimal expansion of (9 + 5*sqrt(3)*Pi)/27.

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

Offset

1,2

Comments

This constant is c(2), where

c(k) = Sum_{n>=0} 1/binomial(2n+k,n) = HypergeometricPFQ((1,1,k+1), ((k+1)/2, (k+2)/2), 1/4).

c(0) = 2*(18 + sqrt(3)*Pi)/27, as in A091682.

c(1) = 2*(9 + 2*sqrt(3)*Pi)/27, as in A248179.

c(2) = (9 + 5*sqrt(3)*Pi)/27, as in this sequence.

c(3) = (45 - 2*sqrt(3)*Pi)/27.

For k>=4, does c(k) have a representation of the form shown here for k = 0..3?

Links

Table of n, a(n) for n=1..86.

Index entries for transcendental numbers.

Example

1.340999646796787694774487920912308740157814582...

Mathematica

s[k_] := Sum[1/Binomial[2 n + k, n], {n, 0, Infinity}];
Column[Table[{s[k], N[s[k], 20]}, {k, 0, 10}]]

Prog

(PARI) (9+5*sqrt(3)*Pi)/27 \\ Charles R Greathouse IV, May 19 2026

Crossrefs

Cf. A000984, A091682, A248179.

Sequence in context: A308642 A158674 A077628 * A213280 A056862 A113035

Adjacent sequences: A389111 A389112 A389113 * A389115 A389116 A389117

Keyword

nonn,cons

Author

Clark Kimberling, Oct 30 2025

Status

approved