A190723 Numbers m for which A055778(m) > A055778(m-1).

1, 2, 4, 8, 9, 11, 15, 19, 20, 22, 26, 27, 29, 33, 37, 38, 40, 44, 48, 49, 51, 55, 56, 58, 62, 66, 67, 69, 73, 74, 76, 80, 84, 85, 87, 91, 95, 96, 98, 102, 103, 105, 109, 113, 114, 116, 120, 124, 125, 127, 131, 132, 134, 138, 142, 143, 145, 149, 150, 152, 156

Offset: 1

Author

Carmine Suriano, May 17 2011

Keywords

• nonn

Comments

The sequence (a(n+1)-1) = 1,3,7,8,10,... is the union of two generalized Beatty sequences, namely (floor(n*phi)+2*n) = A003231, and the sequence (4*floor(n*phi)+3*n+1), the latter with offset 0. For a proof see my paper "Points of increase...". - Michel Dekking, Apr 01 2020

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michel Dekking, Points of increase of the sum of digits function of the base phi expansion, arXiv:2003.14125 [math.CO], 2020.

Crossrefs

Cf. A055778, A190720, A190721, A003231. The morphism in the Formula is a change of alphabet of the morphism generating A284749.

Programs

Mathematica

Block[{nn = 160, k}, k = 2 Ceiling[Log[GoldenRatio, nn]]; Position[Differences@ Array[Total@ First@ RealDigits[#, GoldenRatio, k] &, nn, 0], ?(# > 0 &)][[All, 1]]] (* _Michael De Vlieger, Apr 02 2020, after T. D. Noe at A055778 *)

Formula

a(n) = 1 + Sum_{k=1..n-1} x(k), where x is the unique fixed point of the morphism 1->12, 2->4, 4->1244 on the alphabet {1,2,4}. - Michel Dekking, Apr 01 2020