A396384 - OEIS

A396384

Decimal expansion of the constant D = Sum_{n>=0} 1/(2*n! + 1).

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

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OFFSET

0,1

COMMENTS

D arises in three independent contexts:

(1) Game theory: expected number of green-slot outcomes in a hierarchical roulette where wheel n has 2*(n-1)! red/black slots and 1 green slot.

(2) Cryptography: expected number of honeypot triggers in a cascading cipher where level n has 2*(n-1)! real keys and 1 decoy key.

(3) Biology: expected number of premature stop-codon reading events. In all three cases D < 1 is the stability threshold.

LINKS

Table of n, a(n) for n=0..104.

FORMULA

Equals Sum_{n>=1} n/(2*n!+n).

Equals e/2 - Sum_{n>=0} 1/(2*n!*(2*n!+1)).

EXAMPLE

D = 1/3 + 1/3 + 1/5 + 1/13 + 1/49 + ... = 0.96895437342956...

MATHEMATICA

N[Sum[1/(2*(n-1)! + 1), {n, 1, Infinity}], 50]

PROG

(Python)

from mpmath import mp, nsum, factorial, inf

mp.dps = 50

print(nsum(lambda n: 1/(2*factorial(n-1)+1), [1, inf]))

(PARI) suminf(k = 1, 1/(2*(k-1)!+1)) \\ Amiram Eldar, May 24 2026

CROSSREFS

Cf. A001113 (decimal expansion of e), A000142 (factorials), A372392.

Sequence in context: A161484 A103985 A153071 * A336085 A363539 A086279

Adjacent sequences: A396381 A396382 A396383 * A396385 A396386 A396387

KEYWORD

nonn,cons

AUTHOR

Danylo Ivanov, May 24 2026

STATUS

approved