A093609 Upper Beatty sequence for e^G, G = Euler's gamma constant.

2, 5, 8, 11, 13, 16, 19, 22, 25, 27, 30, 33, 36, 38, 41, 44, 47, 50, 52, 55, 58, 61, 63, 66, 69, 72, 75, 77, 80, 83, 86, 88, 91, 94, 97, 100, 102, 105, 108, 111, 114, 116, 119, 122, 125, 127, 130, 133, 136, 139, 141, 144, 147, 150, 152, 155, 158, 161, 164, 166, 169

Offset

1,1

Comments

Determine the continued fraction convergents to e^(-G) = .561459484...; the first few being 1/1, 1/2, 4/7, 5/9, 9/16, 32/57...(check: 32/57 = .561403508...). Pick a convergent, a/b say 5/9. Then through (a+b) = n = 14, 5 of those integers are in the upper Beatty pair set: 2, 5, 8, 11, 13; while 9 terms are in the lower Beatty pair set, being 1, 3, 4, 6, 7, 9, 10, 12, 14. Since the upper Beatty pair set is derived from (k+1) and the lower from (k+1)/k, the ratio of upper to lower converges to k = 1.789107241...= e^G.

Links

Table of n, a(n) for n=1..61.

Formula

a(n) = floor (n*(k+1)) where k = 1.781072417...= e^G, G = Euler's gamma constant, .577215664901...

Example

a(7) = 19 since floor(n*2.7810724...) = 19.

Mathematica

Table[ Floor[ n*(E^EulerGamma + 1)], {n, 65}]

Crossrefs

Beatty complement is A093610.

Sequence in context: A329924 A330112 A206911 * A249118 A292643 A140101

Adjacent sequences: A093606 A093607 A093608 * A093610 A093611 A093612

Keyword

nonn

Author

Gary W. Adamson, Apr 04 2004

Extensions

More terms from Robert G. Wilson v, Apr 05 2004

Status

approved