A080677 a(n) = n + 1 - A004001(n).

1, 2, 2, 3, 3, 3, 4, 5, 5, 5, 5, 6, 6, 7, 8, 9, 9, 9, 9, 9, 10, 10, 10, 11, 11, 12, 13, 13, 14, 15, 16, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 20, 20, 21, 22, 22, 22, 23, 23, 24, 25, 25, 26, 27, 28, 28, 29, 30, 31, 32, 33, 33, 33, 33, 33, 33, 33, 34, 34, 34, 34, 34, 35, 35, 35

Offset

1,2

Comments

From Antti Karttunen, Jan 10 2016: (Start)

This is the sequence b(n) mentioned on page 229 (page 5 of PDF) in Kubo & Vakil paper, but using starting offset 1 instead of 2.

The recursive sum formula for A004001, a(n) = a(a(n-1)) + a(n-a(n-1)) can be written also as a(n) = a(a(n-1)) + a(A080677(n-1)).

This is the least monotonic left inverse for sequence A087686. Proof: Taking the first differences of this sequence yields the characteristic function for the complement of A188163, because A188163 gives the positions where A004001 increases, and this sequence increases by one whenever A004001 does not increase (and vice versa). Sequence A188163 is also 1 followed by A088359 (see comment in former), whose complement A087686 is, thus A087686 is also the complement of A188163, apart from the initial one. Note also how A087686 is closed with respect to A004001 (see A266188).

(End)

References

J. Arkin, D. C. Arney, L. S. Dewald and W. E. Ebel, Jr., Families of recursive sequences, J. Rec. Math., 22 (No. 22, 1990), 85-94.

Links

Antti Karttunen, Table of n, a(n) for n = 1..8192

T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.

Index entries for Hofstadter-type sequences

Formula

a(n) = n + 1 - A004001(n).

Other identities. For all n >= 1:

a(A087686(n)) = n. [See comments.] - Antti Karttunen, Jan 10 2016

Prog

(Scheme) (define (A080677 n) (- (+ 1 n) (A004001 n))) ;; Scheme code for A004001 given in that entry. - Antti Karttunen, Jan 10 2016

Crossrefs

Cf. A004001, A087686, A088359, A162598, A188163, A265332, A266188, A267111.

Sequence in context: A101646 A166079 A269381 * A344497 A316628 A153112

Adjacent sequences: A080674 A080675 A080676 * A080678 A080679 A080680

Keyword

nonn

Author

N. J. A. Sloane, Mar 03 2003

Status

approved