A059558 -id:A059558 - cp's OEIS frontend

A192544 Bases b such that all integers m having the commuting property r(m)^2 = r(m^2), where r is cyclic replacement of digits d->(d+1) mod b, are of the form m = (b/2 - 1)*(b^k - 1)/(b - 1) + 1 for k >= 1.

8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232, 236, 240, 244, 248, 252, 256, 260, 264

Offset: 1

Walter Kehowski, Jul 04 2011

The bases b form the arithmetic sequence 8+4*k, k>=0, so b/2 is necessarily even. The bases b=2 and b=4 have b/2 as the only number with the commuting property. No odd base b has the commuting property.

			In base 8, the numbers with the commuting property are 4, 34, 334, 3334, 33334, 333334 etc, given by the formula 3*(8^k - 1)/7 + 1.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Except for initial terms, same as A008586 and A124354.

Cf. A059558, A192544, A117755, A127856, A127857, A127859, A127860, A127861.

a[n_] := 4*(n + 1); Table[a[n], {n, 1, 65}] (* Robert P. P. McKone, Aug 25 2023 *)

From Chai Wah Wu, Dec 29 2021: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 2.
G.f.: x*(8 - 4*x)/(x - 1)^2. (End)

More terms from Chai Wah Wu, Dec 29 2021

Edited by Max Alekseyev, Aug 24 2023

A059557 Beatty sequence for 1 + gamma^2, (gamma is the Euler-Mascheroni constant A001620).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94

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 A059558.

R:=RealField(100); [Floor((1+EulerGamma(R)^2)*n): n in [1..100]]; // G. C. Greubel, Aug 27 2018

Table[Floor[(1 + EulerGamma^2)*n], {n,1,100}] (* G. C. Greubel, Aug 27 2018 *)

{ default(realprecision, 100); b=1 + Euler^2; for (n = 1, 2000, write("b059557.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

a(n) = A042968(n-1), 1<=n<2146. - R. J. Mathar, Oct 05 2008

cp's OEIS Frontend

A192544 Bases b such that all integers m having the commuting property r(m)^2 = r(m^2), where r is cyclic replacement of digits d->(d+1) mod b, are of the form m = (b/2 - 1)*(b^k - 1)/(b - 1) + 1 for k >= 1.

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