A244232 - OEIS

%I #18 Jul 05 2014 14:10:29

%S 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,

%T 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,

%U 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

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

%C 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.

%C 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).

%H Antti Karttunen, <a href="/A244232/b244232.txt">Table of n, a(n) for n = 0..16796</a>

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

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

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

%e 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.

%e For n=128, we see that is likewise represented as A197433(31) + A197433(31) = 64 + 64. Thus a(128) = 2*A000120(31) = 10.

%o (Scheme, two alternative implementations)

%o ;; One based on recurrence:

%o (definec (A244232 n) (if (not (zero? (A176137 n))) (A000120 (A244230 n)) (+ (A000120 (-1+ (A244230 n))) (A244232 (- n (A197433 (-1+ (A244230 n))))))))

%o ;; Another using function A244159raw given in A244159:

%o (define (A244232 n) (apply + (vector->list (A244159raw n))))

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

%K nonn

%O 0,4

%A _Antti Karttunen_, Jun 25 2014