A375101 a(0) = 1; a(n+1) = 10*a(n) + A010888(a(n)), where A010888 = digital root.

1, 11, 112, 1124, 11248, 112487, 1124875, 11248751, 112487512, 1124875124, 11248751248, 112487512487, 1124875124875, 11248751248751, 112487512487512, 1124875124875124, 11248751248751248, 112487512487512487, 1124875124875124875, 11248751248751248751, 112487512487512487512

Offset

0,2

Comments

The digital roots of the terms are (1, 2, 4, 8, 7, 5) with cyclic repetitions.

Initial values 2, 4, 5, 7 or 8 yield the same repeating pattern; for initial values 3 or 6 the repeating pattern is (3, 6), and for a(0) = 9 it is 9.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200

Eric Angelini, Fun with roots, personal blog CinquanteSignes.blogspot.com (and post to the SeqFan list), Jul 28 2024.

Index entries for linear recurrences with constant coefficients, signature (11,-10,-1,11,-10).

Formula

a(6*n) = a(6*n-6)*10^6 + 124875.

G.f.: (1 + x^2 + 3*x^3 + 4*x^4)/((1 - x)*(1 + x)*(1 - 10*x)*(1 - x + x^2)). - Andrew Howroyd, Nov 13 2025

Example

The digital root of a(0) = 1 is 1, thus a(1) = 11.
Then, the digital root of a(1) = 11 is 2, thus a(2) = 112, etc.

Mathematica

NestList[10*# + Mod[#-1, 9] + 1 &, 1, 20] (* Paolo Xausa, Jan 19 2026 *)

Prog

(PARI) A375101_upto(N, a=1)=vector(N, i, a+=if(i>1, 9*a+(a-1)%9+1))

Crossrefs

Cf. A010888.

Sequence in context: A094704 A019523 A359143 * A361350 A361501 A372074

Adjacent sequences: A375098 A375099 A375100 * A375102 A375103 A375104

Keyword

nonn,base,easy

Author

M. F. Hasler, Jul 30 2024

Status

approved