A081291 -id:A081291 - cp's OEIS frontend

A236859 The length of the initial ascent 123... in the n-th Catalan numeral, A239903(n).

0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2

Offset: 0

Antti Karttunen, Apr 18 2014

nonn

			A239903(1) = 1, thus a(1) = 1.
A239903(2) = 10, thus a(2) = 1.
A239903(4) = 12, thus a(4) = 2.
A239903(39) = 1232, thus a(39) = 3.
A239903(58784) = 1234567899, thus a(58784) = 9.
Note that although the range of validity of A239903 is inherently limited by the decimal representation employed, it doesn't matter here: We have a(58785) = 10, as the corresponding 58785th Catalan String is [1,2,3,4,5,6,7,8,9,10], even though A239903 cannot represent that unambiguously.

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

Cf. A239903, A236855, A081291, A126307.

(define (A236859 n) (if (zero? n) n (- (A126307 (A081291 n)) 1)))

a(0) = 0, and for n>=1, a(n) = A126307(A081291(n))-1.

Each n occurs for the first time (as a record) at the position (C_{n+1})-1, so we have a(A001453(n+1)) = n for all n.

A082361 Permutation of natural numbers induced by the contraction of Catalan bijection signature-permutation A082351.

Original entry on oeis.org

0, 1, 4, 2, 3, 10, 11, 12, 5, 6, 13, 7, 8, 9, 28, 29, 30, 31, 32, 33, 34, 35, 14, 15, 36, 16, 17, 18, 37, 38, 39, 19, 20, 40, 21, 22, 23, 41, 24, 25, 26, 27, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 42, 43, 106, 44, 45, 46

Offset: 0

Antti Karttunen, Apr 17 2003

nonn

A. Karttunen, Gatomorphisms (with the complete Scheme source)
Index entries for signature-permutations induced by Catalan automorphisms

Inverse of A082362. Cf. also A082363-A082364.

a(n) = A082853(A082351(A081291(n))).

A082362 Permutation of natural numbers induced by the contraction of Catalan bijection signature-permutation A082352.

Original entry on oeis.org

0, 1, 3, 4, 2, 8, 9, 11, 12, 13, 5, 6, 7, 10, 22, 23, 25, 26, 27, 31, 32, 34, 35, 36, 38, 39, 40, 41, 14, 15, 16, 17, 18, 19, 20, 21, 24, 28, 29, 30, 33, 37, 64, 65, 67, 68, 69, 73, 74, 76, 77, 78, 80, 81, 82, 83, 92, 93, 95, 96, 97, 101, 102, 104, 105, 106, 108, 109, 110, 111

Offset: 0

Antti Karttunen, Apr 17 2003

nonn

A. Karttunen, Gatomorphisms (with the complete Scheme source)
Index entries for signature-permutations induced by Catalan automorphisms

Inverse of A082361. Cf. also A082363-A082364.

a(n) = A082853(A082352(A081291(n))).

A085193 Repeating part of A085192.

Original entry on oeis.org

2, 6, 2, 4, 18, 2, 6, 2, 4, 10, 2, 4, 8, 58, 2, 6, 2, 4, 18, 2, 6, 2, 4, 10, 2, 4, 8, 26, 2, 6, 2, 4, 10, 2, 4, 8, 18, 2, 4, 8, 16, 202, 2, 6, 2, 4, 18, 2, 6, 2, 4, 10, 2, 4, 8, 58, 2, 6, 2, 4, 18, 2, 6, 2, 4, 10, 2, 4, 8, 26, 2, 6, 2, 4, 10, 2, 4, 8, 18, 2, 4, 8, 16, 74, 2, 6, 2, 4, 18, 2, 6, 2, 4, 10

Offset: 0

Antti Karttunen, Jun 14 2003

nonn

Antti Karttunen, Notes concerning A080237-tree and related sequences.

Same sequence divided by 2: A085194. Cf. A085190.

a(n) = A085192(A081291(n+1)-1).

a(n) = 4*A085193(A085182(n+1)-1) + 2 - (2^A007001(n+1)) if A007001(n+2)==1, otherwise 2^A007001(n+1).

A220656 The positions of those permutations in A030298 where the first element is not fixed.

Original entry on oeis.org

3, 6, 7, 8, 9, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100

Offset: 1

Antti Karttunen, Dec 17 2012

nonn

Correspondingly, gives the positions of those terms in A030299 whose first digit is not 1, as long as the decimal encoding system employed is valid.

Antti Karttunen, Table of n, a(n) for n = 1..5039

Complement: A220696. Cf. A081291.

(define (A220656 n) (+ (A003422 (+ 1 (A084558 n))) (A000142 (A084558 n)) (A212598 n)))

a(n) = A003422(1+A084558(n)) + A000142(A084558(n)) + A212598(n).

a(n) = A220655(n)+1.

A244314 Nonnegative integers m such that the Semigreedy Catalan representation A244159(m) contains at least one zero.

Original entry on oeis.org

0, 2, 5, 6, 7, 14, 15, 16, 17, 18, 19, 20, 21, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157

Offset: 0

Antti Karttunen, Jun 25 2014

nonn

Starting offset is zero because A244159(0) = 0 is a borderline case (either one zero, or no zeros if leading zeros are discarded).
From a(1)=2 onward the positions of zeros in A244233.
After zero consists of successive subsequences containing terms from A000108(k) to (A000108(k)+A014138(k-2)-1) computed from k >= 2 onward, as: [2], [5,6,7], [14 .. 21], [42 .. 63], [132 .. 195], [429 .. 624], [1430 .. 2054], [4862 .. 6916], etc.

Antti Karttunen, Table of n, a(n) for n = 0..9891

Subsequence of A244217.

Cf. A000108, A014138, A014143, A244233, A244159, A244217, A244317, A081291, A081293.

(define (A244314 n) (cond ((zero? n) 0) ((= 1 n) 2) (else (+ -1 n (- (A000108 (+ 1 (A244317 n))) (A014143 (- (A244317 n) 2)))))))

a(0) = 0, a(1) = 2, and for n >= 2, a(n) = n + A000108(1+A244317(n)) - A014143(A244317(n)-2) - 1.

A080336 Partial sums of A007001.

Original entry on oeis.org

0, 1, 3, 4, 6, 9, 10, 12, 13, 15, 18, 19, 21, 24, 28, 29, 31, 32, 34, 37, 38, 40, 41, 43, 46, 47, 49, 52, 56, 57, 59, 60, 62, 65, 66, 68, 71, 75, 76, 78, 81, 85, 90, 91, 93, 94, 96, 99, 100, 102, 103, 105, 108, 109, 111, 114, 118, 119, 121

Offset: 0

Benoit Cloitre, Mar 18 2003

nonn

A. Karttunen, Notes concerning A080237-tree and related sequences.

The repeating subsequences in A085196 increase to this sequence. Difference between the position of (n+1)-th 1 in A007001 (= A085197(n)) and A007001(n+1). Also a(n) = A085196(A081291(n)).

A082327 Permutation of natural numbers induced by the contraction of Catalan bijection signature-permutation A082325.

Original entry on oeis.org

0, 1, 3, 2, 4, 8, 9, 7, 5, 6, 12, 10, 11, 13, 22, 23, 26, 25, 27, 21, 24, 19, 14, 15, 20, 17, 16, 18, 35, 36, 33, 28, 29, 34, 30, 31, 32, 40, 37, 38, 39, 41, 64, 65, 68, 67, 69, 77, 78, 76, 73, 74, 82, 80, 81, 83, 63, 66, 75, 72, 79, 61, 70, 56, 42, 43, 57, 45, 44, 46

Offset: 0

Antti Karttunen, Apr 17 2003

nonn

A. Karttunen, Gatomorphisms (with the complete Scheme source)
Index entries for signature-permutations induced by Catalan automorphisms

Inverse of A082328. a(n) = A072798(A082328(A072798(n))). Cf. also A082329-A082330.

a(n) = A082853(A082325(A081291(n))).

A082328 Permutation of natural numbers induced by the contraction of Catalan bijection signature-permutation A082326.

Original entry on oeis.org

0, 1, 3, 2, 4, 8, 9, 7, 5, 6, 11, 12, 10, 13, 22, 23, 26, 25, 27, 21, 24, 19, 14, 15, 20, 17, 16, 18, 31, 32, 34, 35, 36, 30, 33, 28, 29, 38, 39, 40, 37, 41, 64, 65, 68, 67, 69, 77, 78, 76, 73, 74, 81, 82, 80, 83, 63, 66, 75, 72, 79, 61, 70, 56, 42, 43, 57, 45, 44, 46

Offset: 0

Antti Karttunen, Apr 17 2003

nonn

A. Karttunen, Gatomorphisms (with the complete Scheme source)
Index entries for signature-permutations induced by Catalan automorphisms

Inverse of A082327. a(n) = A072798(A082327(A072798(n))). Cf. also A082329-A082330.

a(n) = A082853(A082326(A081291(n))).

A082329 Permutation of positive natural numbers: a(n) = A082327(n-1)+1.

Original entry on oeis.org

1, 2, 4, 3, 5, 9, 10, 8, 6, 7, 13, 11, 12, 14, 23, 24, 27, 26, 28, 22, 25, 20, 15, 16, 21, 18, 17, 19, 36, 37, 34, 29, 30, 35, 31, 32, 33, 41, 38, 39, 40, 42, 65, 66, 69, 68, 70, 78, 79, 77, 74, 75, 83, 81, 82, 84, 64, 67, 76, 73, 80, 62, 71, 57, 43, 44, 58, 46, 45, 47

Offset: 1

Antti Karttunen, Apr 17 2003

nonn

Index entries for sequences that are permutations of the natural numbers

Inverse of A082330. a(n) = A082854(A082325(A081291(n-1))).

cp's OEIS Frontend

A236859 The length of the initial ascent 123... in the n-th Catalan numeral, A239903(n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Scheme

Formula

A082361 Permutation of natural numbers induced by the contraction of Catalan bijection signature-permutation A082351.

Views

Author

Keywords

Links

Crossrefs

Formula

A082362 Permutation of natural numbers induced by the contraction of Catalan bijection signature-permutation A082352.

Views

Author

Keywords

Links

Crossrefs

Formula

A085193 Repeating part of A085192.

Views

Author

Keywords

Links

Crossrefs

Formula

A220656 The positions of those permutations in A030298 where the first element is not fixed.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A244314 Nonnegative integers m such that the Semigreedy Catalan representation A244159(m) contains at least one zero.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A080336 Partial sums of A007001.

Views

Author

Keywords

Links

Crossrefs

A082327 Permutation of natural numbers induced by the contraction of Catalan bijection signature-permutation A082325.

Views

Author

Keywords

Links

Crossrefs

Formula

A082328 Permutation of natural numbers induced by the contraction of Catalan bijection signature-permutation A082326.

Views

Author

Keywords

Links

Crossrefs

Formula

A082329 Permutation of positive natural numbers: a(n) = A082327(n-1)+1.

Views

Author

Keywords

Links

Crossrefs