A125758 Numbers congruent to 4 or 7 (mod 9).

4, 7, 13, 16, 22, 25, 31, 34, 40, 43, 49, 52, 58, 61, 67, 70, 76, 79, 85, 88, 94, 97, 103, 106, 112, 115, 121, 124, 130, 133, 139, 142, 148, 151, 157, 160, 166, 169, 175, 178, 184, 187, 193, 196, 202, 205, 211, 214, 220, 223, 229, 232, 238, 241, 247, 250, 256, 259, 265, 268

Offset: 1

Author

N. J. A. Sloane and David Applegate, Feb 02 2007

Keywords

• nonn
• easy

Comments

For a given integer m, write its binary representation in reverse order, as in A125626, A125754, etc.; let a 0 mean "halving" and a 1 mean "k -> 3k+1". Then m specifies an operation on real numbers given by k -> f_m(k). Suppose the equation f_m(k) = k has a positive integer solution for some m. Then we conjecture that the values of k are precisely the terms of this sequence.

25 is a term because we have 25 -> 76 -> 38 -> 19 -> 58 -> 29 -> 88 -> 44 -> 22 -> 11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 25.

In other words, we conjecture that this sequence coincides with A125757 sorted and with duplicates removed.

Links

• David Lovler, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A125626, A125754, A125755, A125756, A125757, A125710, A125711.

Programs

Mathematica

Select[Range[300],MemberQ[{4,7},Mod[#,9]]&]  (* Harvey P. Dale, Mar 12 2011 *)

Formula

From R. J. Mathar, Apr 03 2009: (Start)

a(n) = a(n-1) + a(n-2) - a(n-3) = a(n-2) + 9.

a(n) + a(n+1) = A017185(n).

G.f.: x*(4+3*x+2*x^2)/((1+x)*(x-1)^2). (End)

E.g.f.: 2 + ((9*x - 5/2)*exp(x) - (3/2)*exp(-x))/2. - David Lovler, Aug 21 2022