A059563 -id:A059563 - cp's OEIS frontend

A059564 Beatty sequence for (e^2 + 1)/(e^2 - e + 1).

1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 36, 38, 39, 41, 42, 44, 45, 47, 48, 50, 51, 53, 54, 56, 57, 59, 60, 62, 63, 65, 66, 68, 69, 71, 72, 73, 75, 76, 78, 79, 81, 82, 84, 85, 87, 88, 90, 91, 93, 94, 96, 97, 99, 100, 102, 103

Offset: 1

Mitch Harris, Jan 22 2001

Harry J. Smith, Table of n, a(n) for n = 1..2000
Aviezri S. Fraenkel, Jonathan Levitt, Michael Shimshoni, Characterization of the set of values f(n)=[n alpha], n=1,2,..., Discrete Math. 2 (1972), no.4, 335-345.
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Beatty complement is A059563.

A059564:=n->floor(n*(exp(1)^2+1)/(exp(1)^2-exp(1)+1)): seq(A059564(n), n=1..100); # Wesley Ivan Hurt, Jan 03 2016

Table[Floor[n (E^2 + 1)/(E^2 - E + 1)], {n, 80}] (* Wesley Ivan Hurt, Jan 03 2016 *)

{ default(realprecision, 100); e=exp(1); b=(e^2 + 1)/(e^2 - e + 1); for (n = 1, 2000, write("b059564.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

a(n) = floor(n*(e^2+1)/(e^2-e+1)). - Michel Marcus, Jan 05 2015

A109231 a(n) = floor(n*cosh(1)).

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 20, 21, 23, 24, 26, 27, 29, 30, 32, 33, 35, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 54, 55, 57, 58, 60, 61, 63, 64, 66, 67, 69, 70, 72, 74, 75, 77, 78, 80, 81, 83, 84, 86, 87, 89, 91, 92, 94, 95, 97, 98, 100

Offset: 1

Reinhard Zumkeller, Jun 23 2005

nonn

Beatty sequence for cosh(1) = (e+1/e)/2 = 1.54308...= A073743; complement of A109232.

Eric Weisstein's World of Mathematics, Hyperbolic Cosine
Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences related to Beatty sequences

Cf. A073743, A109232, A109233, A109234, A109237, A059563.

A127451 Beatty sequence for 1/(1 - e^Pi + Pi^e), complement of A127450.

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 21, 25, 28, 31, 34, 37, 40, 43, 47, 50, 53, 56, 59, 62, 65, 69, 72, 75, 78, 81, 84, 87, 91, 94, 97, 100, 103, 106, 109, 113, 116, 119, 122, 125, 128, 131, 135, 138, 141, 144, 147, 150, 153, 157, 160, 163, 166, 169, 172, 175, 178, 182, 185, 188

Offset: 1

Robert G. Wilson v, Jan 14 2007

Differs from A022844 first at a(57). - L. Edson Jeffery, Dec 01 2013

1/(1 - e^Pi + Pi^e) = 3.140061643.., so a(n)<=A022844(n). - R. J. Mathar, May 30 2025

Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Cf. A022844, A059563, A063504.

Table[Floor[n/(1 - Exp[Pi] + Pi^E)], {n, 60}]

a(n) = floor(n/(1 - e^Pi + Pi^e))

A380408 a(n) = Sum_{k>=0} floor(n/(2k)!).

Original entry on oeis.org

0, 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21, 22, 24, 25, 27, 28, 30, 31, 33, 34, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 74, 75, 77, 78, 80, 81, 83, 84, 86, 87, 89, 90, 92, 93, 95, 96, 98, 99, 101, 102, 104, 105, 107

Offset: 0

Akiva Weinberger, Jan 23 2025

nonn

Partial sum of A060832 except for the first term in the sum.
Congruent to A034968(n) mod 2. Therefore, the parity of a(n) is the parity of the n-th permutation of k elements (k>=n) in lexicographic order.
For even n, a(n) equals A059563(n/2) whenever cosh(1)*n - a(n) < 1. The first time this fails is n=70, as a(70)=107 but A059563(35)=108. For small n, such failures appear to be very rare; however, the asymptotic density of these failures approaches 1.

Cf. A060832, A034968, A013936, A013939, A038663.

a(n) = round(sumpos(k=0, n\(2*k)!)); \\ Michel Marcus, Jan 24 2025

a(n) = cosh(1)*n - f(n) where f(n) = Sum_{k>=0} fract(n/(2k)!). Here, fract() is the fractional part. The error term f(n) is unbounded above, and the greatest lower bound is 0 (even excluding n=0). The first values for which f(n) > s for s=1,2,3 are f(13)=1.06005, f(407) = 2.03382, and f(22319) = 3.01669. The error is almost periodic: for large m, f(n) is approximately f(n+(2m)!). If n is odd, f(n) > 1/2. f(n) alternately rises and descends, that is, f(2*n)f(2*n+2) for all n.

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A059564 Beatty sequence for (e^2 + 1)/(e^2 - e + 1).

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A109231 a(n) = floor(n*cosh(1)).

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A127451 Beatty sequence for 1/(1 - e^Pi + Pi^e), complement of A127450.

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A380408 a(n) = Sum_{k>=0} floor(n/(2k)!).

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