A307086 Decimal expansion of 4*(5 - sqrt(5)*log(phi))/25, where phi is the golden ratio (A001622).

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

Offset: 0

Author

Ilya Gutkovskiy, Mar 23 2019

Keywords

• nonn
• cons

Comments

Decimal expansion of the alternating sum of the reciprocals of the central binomial coefficients (A000984).

Examples

			1/1 - 1/2 + 1/6 - 1/20 + 1/70 - 1/252 + ... = 0.62783642361439838444422670681975782983017172698388...
		

Links

• Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, INTEGERS 6 (2006) #A27
• Eric Weisstein's World of Mathematics, Central Binomial Coefficient
• Index entries for transcendental numbers

Crossrefs

Cf. A000984, A001622, A073016, A086465, A091682, A145433, A145434.

Programs

Mathematica

RealDigits[4 (5 - Sqrt[5] Log[GoldenRatio])/25, 10, 120][[1]]

PARI

4*(5 - sqrt(5)*log((sqrt(5)+1)/2))/25 \\ Charles R Greathouse IV, May 15 2019

Formula

Equals Sum_{k>=0} (-1)^k/binomial(2*k,k).

Equals Sum_{k>=0} (-1)^k*(k!)^2/(2*k)!.