A244232 Sum of "digit values" in Semigreedy Catalan Representation of n, A244159.

0, 1, 1, 2, 3, 1, 2, 2, 3, 4, 4, 5, 6, 4, 1, 2, 2, 3, 4, 2, 3, 3, 4, 5, 5, 6, 7, 5, 6, 6, 7, 8, 8, 9, 10, 8, 5, 6, 6, 7, 8, 6, 1, 2, 2, 3, 4, 2, 3, 3, 4, 5, 5, 6, 7, 5, 2, 3, 3, 4, 5, 3, 4, 4, 5, 6, 6, 7, 8, 6, 7, 7, 8, 9, 9, 10, 11, 9, 6, 7, 7, 8, 9, 7, 8, 8, 9, 10, 10, 11, 12, 10, 11, 11, 12, 13, 13, 14, 15, 13, 10, 11, 11, 12, 13, 11, 6, 7, 7, 8, 9, 7, 8, 8, 9, 10, 10, 11, 12, 10, 7, 8, 8, 9, 10, 8, 9, 9, 10, 11, 11, 12, 1

Offset

0,4

Comments

Note that a(33604) = A000217(10) = 55 because the sum is computed from the underlying list (vector) of numbers, and thus is not subject to any corruption by decimal representation as A244159 itself is.

Equivalent description: partition n "greedily" as terms of A197433, i.e. n = A197433(i) + A197433(j) + ... + A197433(k), always using the largest term of A197433 that still "fits in" (i.e. is <= n remaining). Then a(n) = A000120(i) + A000120(j) + ... + A000120(k).

Links

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

Formula

If A176137(n) = 1, a(n) = A000120(A244230(n)), otherwise a(n) = A000120(A244230(n)-1) + a(n-A197433(A244230(n)-1)).

For all n, a(A000108(n)) = 1. [And moreover, Catalan numbers, A000108, give all such k that a(k) = 1].

For all n, a(A014138(n)) = n and a(A014143(n)) = A000217(n+1).

Example

For n=18, using the alternative description, we see that it is partitioned  into the terms of A197433 as a greedy sum A197433(11) + A197433(1) = 17 + 1. Thus a(18) = A000120(11) + A000120(1) = 3+1 = 4.
For n=128, we see that is likewise represented as A197433(31) + A197433(31) = 64 + 64. Thus a(128) = 2*A000120(31) = 10.

Prog

(Scheme, two alternative implementations)
;; One based on recurrence:
(definec (A244232 n) (if (not (zero? (A176137 n))) (A000120 (A244230 n)) (+ (A000120 (-1+ (A244230 n))) (A244232 (- n (A197433 (-1+ (A244230 n))))))))
;; Another using function A244159raw given in A244159:
(define (A244232 n) (apply + (vector->list (A244159raw n))))

Crossrefs

Cf. A000108, A000120, A176137, A197433, A244230, A244159, A244231, A244233, A244234, A014420, A236855.

Sequence in context: A324920 A358638 A236855 * A227781 A366294 A351441

Adjacent sequences: A244229 A244230 A244231 * A244233 A244234 A244235

Keyword

nonn

Author

Antti Karttunen, Jun 25 2014

Status

approved