A061196 If n = Sum_{i} c_i * 10^i then let b(n) = Sum_{i} c_i * (i+1). Order the integers by b(n) and then n.

0, 1, 2, 10, 3, 11, 100, 4, 12, 20, 101, 1000, 5, 13, 21, 102, 110, 1001, 10000, 6, 14, 22, 30, 103, 111, 200, 1002, 1010, 10001, 100000, 7, 15, 23, 31, 104, 112, 120, 201, 1003, 1011, 1100, 10002, 10010, 100001, 1000000, 8, 16, 24, 32, 40, 105, 113, 121, 202

Offset: 0

Author

Henry Bottomley, Apr 20 2001

Keywords

• nonn
• base

Comments

This is in effect a listing of single-digit (nonnegative) solutions to b + 2c + 3d + 4e + ... = k.

The sequence can be considered as an irregular triangle listing partitions in which no part occurs more than 9 times. The row lengths are given by A261776. For example, in row 5 the value 102, corresponds to the partition 1+1+3 (= 2*1 + 0*2 + 1*3). - Andrew Howroyd, Apr 25 2023

Examples

			From _Andrew Howroyd_, Apr 25 2023: (Start)
The sequence as a triangle T(n,k) begins:
  0 | 0;
  1 | 1;
  2 | 2, 10;
  3 | 3, 11, 100;
  4 | 4, 12,  20, 101, 1000;
  5 | 5, 13,  21, 102,  110, 1001, 10000;
  6 | 6, 14,  22,  30,  103,  111,   200, 1002, 1010, 10001, 100000;
  ...
(End)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..96
• Rasa Smidtaite and Minvydas Ragulskis, Commentary: Multidimensional discrete chaotic maps, Front. Phys. (2022) Vol. 10, 1094240.

Crossrefs

Cf. A000041, A028897, A261776.

Programs

Mathematica

With[{k = 7}, {{0}}~Join~Values@ PositionIndex[Array[Total@ MapIndexed[#1*First[#2] &, Reverse@ IntegerDigits[#]] &, 10^k]][[1 ;; k]]] // Flatten (* Michael De Vlieger, Dec 22 2022, solution only suitable for generating the data field *)

PARI

F(p)={my(v=vector(if(#p, p[#p], 1))); for(i=1, #p, v[p[i]]++); v}
row(n)={my(R=[F(p) | p<-partitions(n)]); vecsort([fromdigits(Vecrev(u)) | u<-R, vecmax(u)<=9])}
{ for(n=0, 7, print(row(n))) } \\ Andrew Howroyd, Apr 25 2023

Formula

For n < 10, a(A000070(n)) = n+1 and a(A026905(n)) = 10^(n-1).