A055168 Cumulative counting sequence: method B (noun,adjective)-pairs with first term 0.

0, 0, 1, 0, 2, 1, 1, 0, 3, 1, 3, 2, 1, 0, 4, 1, 5, 2, 2, 3, 2, 0, 5, 1, 6, 2, 5, 3, 3, 4, 1, 5, 1, 0, 6, 1, 9, 2, 6, 3, 5, 4, 2, 5, 4, 6, 1, 0, 7, 1, 11, 2, 8, 3, 6, 4, 4, 5, 6, 6, 4, 9, 1, 0, 8, 1, 13, 2, 9, 3, 7, 4, 7, 5, 7, 6, 7, 9, 2, 7, 1, 11, 1, 8, 1, 0, 9, 1, 17

Offset

1,5

Comments

Write 0 followed by segments defined inductively as follows: each segment tells how many times each previously written integer occurs, stating first the integer being counted and then its frequency. This is Method B (noun-before-adjective); for Method A (adjective-before-noun), see A217760. - Clark Kimberling, Mar 25 2013

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Example

Start with 0, then 0,1; then 2,0 and 1,1; etc.
Writing pairs vertically, the initial segments are
0..0..0 1..0 1 2..0 1 2 3..0 1 2 3 4 5..0 1 2 3 4 5 6..0  1 2 3 4 5 6 9
...1..2 1..3 3 1..4 5 2 2..5 6 5 3 1 1..6 9 6 5 2 4 1..7 11 8 6 4 6 3 1
The 5th segment tells that 0 has been written 4 times, 1 5 times, 2 2 times, and 3 2 times. The nouns are 1 2 3; the adjectives, 5 2 2.  - Clark Kimberling, Mar 25 2013

Mathematica

s = {0}; Do[s = Flatten[{s, {#, Count[s, #]} & /@ DeleteDuplicates[s]}], {24}]; s (* Peter J. C. Moses, Mar 21 2013 *)

Crossrefs

Cf. A217760, A055170.

Cf. A055171, A055174, A055177, A055180, A055183.

See A001155 and A022470 for well-known counting sequences using methods A and B.

Sequence in context: A340031 A029352 A340011 * A085144 A156578 A171846

Adjacent sequences: A055165 A055166 A055167 * A055169 A055170 A055171

Keyword

nonn

Author

Clark Kimberling, Apr 27 2000

Extensions

Conjecture removed by Clark Kimberling, Oct 24 2009

Status

approved