login
A265754
Reduced frequency counts for A004001: a(n) = A265332(n+1) - A036987(n).
6
1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 2, 3, 4, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 5, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5
OFFSET
1,3
COMMENTS
Can be generated recursively by first setting R_1 = (1), after which each R_n is obtained by replacing in R_{n-1} each term k with terms 1 .. k, followed by final n. This sequence is then obtained by concatenating all levels R_1, R_2, ..., R_inf together. See page 230 in Kubo-Vakil paper (page 6 in PDF).
Deleting all 1's and decrementing the remaining terms by one gives the sequence back.
Comment from N. J. A. Sloane, Nov 05 2017: (Start)
The following simple Pascal-like triangle produces the same sequence. Construct a triangle T(n,k) of strings (with 0 <= k <= n), where T(0,0) = {1}, T(n,n) = {n+1}, and otherwise T(n,k) is the concatenation of T(n-1,k-1) and T(n-1,k). The first few rows of the triangle (where the strings T(n,k) are shown without spaces for legibility) are:
1
1,2
1,12,3
1,112,123,4
1,1112,112123,1234,5
1,11112,1112112123,1121231234,12345,6
...
Now read the strings across the rows to get the sequence. T(n,k) has length binomial(n,k). (End)
LINKS
T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.
FORMULA
a(n) = A265332(n+1) - A036987(n).
As a recurrence: If A036987(n) = 1 [when n is of the form 2^k -1], a(n) = A070939(n), else if a(n+1) = 1, a(n) = a(2^A000523(n) - A266349(n)), otherwise a(n) = a(n+1)-1.
Other identities. For all n >= 1:
a(n) = A266640(A054429(n)).
a(A000225(n)) = n.
EXAMPLE
Illustration of the sequence as a tree:
1
/ \
1 2
/ /|\
1 1 2 3_________
/ / /| | \ \ \
1 1 1 2 1 2 3__ 4________
/ / / /| | / \ |\ \ \ \ \ \ \
1 1 1 1 2 1 1 2 1 2 3 1 2 3 4 5
etc.
Compare with the illustration in A265332.
PROG
(Scheme, two variants)
(define (A265754 n) (- (A265332 (+ 1 n)) (A036987 n)))
(definec (A265754 n) (cond ((= 1 (A036987 n)) (A070939 n)) ((> (A265754 (+ 1 n)) 1) (- (A265754 (+ 1 n)) 1)) (else (A265754 (- (A000079 (A000523 n)) (A266349 n))))))
CROSSREFS
Cf. A000225 (positions of records, where n appears first time).
Cf. A266640 (obtained from the mirror image of the same tree).
See A293959 for another version.
Sequence in context: A309736 A368010 A237453 * A089309 A126387 A038374
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jan 10 2016
STATUS
approved