A140654 Numerators of convergents to transcendental number defined by Yun Gao and Jining Gao.

3, 201, 52690971, 949196149445558443966545, 5549006906251488336142058083587971135417862217678366680875041864474951923

Offset

1,1

Comments

Gao and Gao prove in Example 7, on p.11, that the series converges to a transcendental number.

Links

Table of n, a(n) for n=1..5.

Yun Gao and Jining Gao, Some sufficient conditions of a given series with rational terms converging to an irrational number, arXiv:0807.1376

Formula

a(n) = numerator of SUM[k=1..n] (3^n)/(2^3^n).

Example

a(1)/denom = (3^1)/(2^3^1) = 3/8.
a(2)/denom = (3/8) + (3^2)/(2^3^2) = (3/8)+(9/512) = 201/512.
a(3)/denom = 201/512 + (3^3)/(2^3^3) = 201/512 + 27/(2^27)= 52690971/134217728.
a(4)/denom = 52690971/134217728 + (3^4)/(2^3^4) = 52690971/134217728 + 81/(2^81) = (3/8) + (9/512) + (27/(2^27)) + (81/(2^81)) = 949196149445558443966545/2417851639229258349412352.
a(5)/denom = (3/8) + (9/512) + (27/(227)) + (81/(281)) + (243/(2243)) =
5549006906251488336142058083587971135417862217678366680875041864474951923/ 14134776518227074636666380005943348126619871175004951664972849610340958208 = 0 + 1/2+ 1/1+ 1/1+ 1/4+ 1/1+ 1/3+ 1/1+ 1/3+ 1/18+ 1/1+ 1/2+ 1/3+ 1/13+ 1/1+ 1/4+ 1/6+ 1/2+ 1/1+ 1/1657008+ 1/3+ 1/3+ 1/2+ 1/4+ 1/1+ 1/4+ 1/1+ 1/1+ 1/1+ 1/1+ 1/1+ 1/7+ 1/1+ 1/1+ 1/40+ 1/3+ 1/8+ 1/3+ 1/7+ 1/2+ 1/2+ 1/9.9500067e+21+ 1/1+ 1/1+ 1/2+ 1/3+ 1/1+ 1/9+ 1/3+ 1/2+ 1/4+ 1/3+ 1/2+ 1/1+ 1/1+ 1/1+ 1/1+ 1/1+ 1/13+ 1/2+ 1/1+ 1/2+ 1/1+ 1/1+ 1/1+ 1/1+ 1/1+ 1/2+ 1/2+ 1/1+ 1/2+ 1/2+ 1/184111+ 1/1+ 1/2+ 1/4+ 1/1+ 1/14+ 1/4+ 1/1+ 1/1+ 1/2+ 1/1+ 1/2+ 1/3+ 1/27+ 1/15+ 1/1+ 1/1+ 1/5+ 1/1+ 1/49+ 1/2+ 1/2, to give the continued fraction for the last example.

Crossrefs

Sequence in context: A342578 A264674 A306630 * A229602 A080297 A080274

Adjacent sequences: A140651 A140652 A140653 * A140655 A140656 A140657

Keyword

easy,nonn

Author

Jonathan Vos Post, Jul 09 2008

Status

approved