A217760 Cumulative counting sequence: (adjective-before-noun) pairs with first term 0; see Comments.

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

Offset: 1

Author

Clark Kimberling, Mar 24 2013

Keywords

• nonn
• easy

Comments

Write 0 followed by segments defined inductively as follows: each segment

tells how many times each previously written integer occurs, in the order

of first occurrence. This is Method A (adjective-before-noun pairs); for

Method B (noun-before-adjective), see A055168. The sequence A217760 was

originally A055186 (Apr 27 2000); the present A055186 has a shorter

definition and differs from A217760 beginning at the 62nd term.

Examples

			Start with 0, followed by the adjective-noun pair 1,0; followed by
adjective-noun pairs 2,0 then 1,1; etc. Writing the pairs vertically,
the initial segments are
0..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 4 1
...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
The order of appearance is 0,1,2,3,4,5,6,9,7,11,8,... conjectured at A055170 to include all the nonnegative integers.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..9666
• Clark Kimberling, Unsolved Problems and Rewards, Problem 4

Crossrefs

Cf. A055170, A055186.

Programs

Mathematica

s = {0}; Do[s = Flatten[{s, {Count[s, #], #} & /@ DeleteDuplicates[s]}], {14}]; s  (* A217760 *)
s = {0}; Do[s = Flatten[{s, {Count[s, #], #} & /@ (a = DeleteDuplicates[s])}], {24}]; a;  (* A055170 *) (* Peter J. C. Moses, Mar 21 2013 *)