A230412 a(n) = the number of ways to express n as a sum d1*(k1!-1) + d2*(k2!-1) + ... + dj*(kj!-1), where all k's are distinct and greater than one and each di is in range [1,ki]; the characteristic function of A219650.

1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1

Offset

0

Links

Antti Karttunen, Table of n, a(n) for n = 0..10079

Formula

a(0)=1 and for n>=1, a(n) = [(A219650(A230413(n-1)) + A230405(A230413(n-1))) = n], where [] stands for Iverson bracket.

Example

a(0)=1 as the only solution is an empty sum.
1 can be represented as 1*(2!-1), and this is the only solution, thus a(1) = 1.
2 can be represented (also uniquely) as 2*(2!-1) thus a(2) = 1.
3 and 4 cannot be represented as such a sum, thus a(3) = a(4) = 0.
5 can be represented (uniquely) as 1*(3!-1) thus a(5) = 1.
6 can be represented (uniquely) as 1*(3!-1) + 1*(2!-1), thus a(6) = 1.
7 can be represented (uniquely) as 1*(3!-1) + 2*(2!-1), thus a(7) = 1.
17 can be represented (uniquely) as 3*(3!-1) + 2*(2!-1), thus a(17) = 1.

Prog

(Scheme, with memoizing definec-macro from Antti Karttunen's IntSeq-library)
(definec (A230412 n) (if (zero? n) 1 (let ((k (A230413 (- n 1)))) (if (= (+ (A219650 k) (A230405 k)) n) 1 0))))

Crossrefs

The characteristic function of A219650. A219658 gives the positions of zeros.

Together with A230413 (the partial sums) can be used to compute A230414, A230423 and A230424.

This sequence relates to the factorial base representation (A007623) in a similar way as A079559 relates to the binary system.

Sequence in context: A014024 A014039 A016410 * A115361 A115358 A325898

Adjacent sequences: A230409 A230410 A230411 * A230413 A230414 A230415

Keyword

nonn

Author

Antti Karttunen, Nov 02 2013

Status

approved