A236855 a(n) is the sum of digits in A239903(n).

0, 1, 1, 2, 3, 1, 2, 2, 3, 4, 3, 4, 5, 6, 1, 2, 2, 3, 4, 2, 3, 3, 4, 5, 4, 5, 6, 7, 3, 4, 4, 5, 6, 5, 6, 7, 8, 6, 7, 8, 9, 10, 1, 2, 2, 3, 4, 2, 3, 3, 4, 5, 4, 5, 6, 7, 2, 3, 3, 4, 5, 3, 4, 4, 5, 6, 5, 6, 7, 8, 4, 5, 5, 6, 7, 6, 7, 8, 9, 7, 8, 9, 10, 11, 3, 4

Offset

0,4

Links

Antti Karttunen, Table of n, a(n) for n = 0..16796

Formula

a(n) = A034968(x) - A060130(x), where x = A071155(A081291(n)).

For up to n = A000108(11)-2 = 58784, a(n) = A007953(A239903(n)).

Catalan numbers, A000108, give the positions of ones, and the n-th triangular number occurs for the first time at the position immediately before that, i.e., a(A001453(n)) = A000217(n-1).

For each n, a(n) >= A000217(A236859(n)).

Example

As the 0th Catalan String is empty, indicated by A239903(0)=0, a(0)=0.
As the 18th Catalan String is [1,0,1,2] (A239903(18)=1012), a(18) = 1+0+1+2 = 4.
Note that although the range of validity of A239903 is inherently limited by the decimal representation employed, it doesn't matter here: We have a(58785) = 55, as the corresponding 58785th Catalan String is [1,2,3,4,5,6,7,8,9,10], even though A239903 cannot represent that unambiguously.

Mathematica

A236855list[m_] := With[{r = 2*Range[2, m]-1}, Reverse[Map[Total[r-#] &, Select[Subsets[Range[2, 2*m-1], {m-1}], Min[r-#] >= 0 &]]]];
A236855list[6] (* Generates C(6) terms *) (* Paolo Xausa, Feb 19 2024 *)

Prog

(Scheme)
(define (A236855 n) (apply + (A239903raw n)))
(define (A239903raw n) (if (zero? n) (list) (let loop ((n n) (row (- (A081288 n) 1)) (col (- (A081288 n) 2)) (srow (- (A081288 n) 2)) (catstring (list 0))) (cond ((or (zero? row) (negative? col)) (reverse! (cdr catstring))) ((> (A009766tr row col) n) (loop n srow (- col 1) (- srow 1) (cons 0 catstring))) (else (loop (- n (A009766tr row col)) (+ row 1) col srow (cons (+ 1 (car catstring)) (cdr catstring))))))))
;; Alternative definition:
(define (A236855 n) (let ((x (A071155 (A081291 n)))) (- (A034968 x) (A060130 x))))

Crossrefs

Cf. A239903, A014420, A237449, A236859, A009766, A071155, A081291, A034968, A060130.

Sequence in context: A079087 A324920 A358638 * A244232 A227781 A366294

Adjacent sequences: A236852 A236853 A236854 * A236856 A236857 A236858

Keyword

nonn,base

Author

Antti Karttunen, Apr 18 2014

Status

approved