cp's OEIS Frontend

Differences of the Meta-Fibonacci sequence for s=0. - Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca)

Fixed point of morphism 0-->0, 1-->110 - Joerg Arndt, Jun 07 2007

A006697(k) gives number of distinct subwords of length k, conjectured to be equal to A094913(k)+1. - M. F. Hasler, Dec 19 2007

Characteristic function for the range of A005187: a(A055938(n))=0; a(A005187(n))=1; if a(m)=1 then either a(m-1)=1 or a(m+1)=1. - Reinhard Zumkeller, Mar 18 2009

The number of zeros between successive pairs of ones in this sequence is A007814. - Franklin T. Adams-Watters, Oct 05 2011

Length of n-th run = abs(A088705) + 1. - Reinhard Zumkeller, Dec 11 2011

Number of ways to write n as a sum a_1 + ... + a_k where the a_i are positive integers and a_i = 2 * a_{i-1}, cf. A000929.

Number of divisors of n of the form 2^k - 1 (A000225) for k >= 1. - Jeffrey Shallit, Jan 23 2017

The decapitation of such a partition (delete the greatest part) is term-wise less than its negated first-differences.

The decapitation of such a partition (delete the greatest part) is term-wise greater than or equal to its negated first-differences.

The decapitation of such a partition (delete the greatest part) is term-wise greater than its negated first-differences.

The decapitation of such a partition (delete the greatest part) is term-wise greater than its negated first-differences.

Define a unit column of a binary matrix to be a column with only one 1. a(n) = number of 3 X n binary matrices with 1 unit column up to row and column permutations (if offset is 1). - Vladeta Jovovic, Sep 09 2000

Generally, number of 3 X n binary matrices with k=0,1,2,... unit columns, up to row and column permutations, is the coefficient of x^k in 1/6*(Z(S_n; 5 + 3*x,5 + 3*x^2, ...) + 3*Z(S_n; 3 + x,5 + 3*x^2,3 + x^3,5 + 3*x^4, ...) + 2*Z(S_n; 2,2,5 + 3*x^3,2,2,5 + 3*x^6, ...)), where Z(S_n; x_1,x_2,...,x_n) is the cycle index of symmetric group S_n of degree n.

First differences of a(n) give number of minimal 3-covers of an unlabeled n-set that cover 4 points of that set uniquely (if offset is 4).

Transform of tetrahedral numbers, binomial(n+3,3), under the Riordan array (1/(1-x^2), x). - Paul Barry, Apr 16 2005

Equals triangle A152205 as an infinite lower triangular matrix * [1, 2, 3, ...]. - Gary W. Adamson, Feb 14 2010

With a leading zero, number of all possible octahedra of any size, formed when intersecting a regular tetrahedron by planes parallel to its sides and dividing its edges into n equal parts. - V.J. Pohjola, Sep 13 2012

With 2 leading zeros and offset 1, the sequence becomes 0,0,1,4,11,... for n=1,2,3,... Call this b(n). Consider the partitions of n into two parts (p,q) with p <= q. Then b(n) is the total volume of the family of rectangular prisms with dimensions p, |q - p| and |q - p|. - Wesley Ivan Hurt, Apr 14 2018

Conjecture: For n > 2, a(n-3) is the absolute value of the coefficient of the term [x^(n-2)] in the characteristic polynomial of an n X n square matrix M(n) defined as the n-th principal submatrix of the array A010751 whose general element is given by M[i,j] = floor((j - i + 1)/2). - Stefano Spezia, Jan 12 2022

Consider the following drawing of the complete graph on n vertices K_n: Vertices 1, 2, ..., n are on a straight line. Any pair of nonconsecutive vertices (i, j) with i < j is connected by a semicircle that goes above the line if i is odd, and below if i is even. With four leading zeros and offset 1, a(n) gives the number of edge crossings of the aforementioned drawing of K_n. - Carlo Francisco E. Adajar, Mar 17 2022

The decapitation of such a partition (delete the greatest part) is term-wise greater than or equal to its negated first-differences.

The number of binary strings sharing the same autocorrelations.

Appears to be row sums of A155092, beginning from a(2). - Mats Granvik, Jan 20 2009

The number of binary words of length n (beginning with 0) which do not start with an even palindrome (i.e. which are not of the form ss*t where s is a (nonempty) word, s* is its reverse, and t is any (possibly empty) word). - Mamuka Jibladze, Sep 30 2014

From Gus Wiseman, Mar 08 2021: (Start)

This sequence counts each of the following essentially equivalent things:

1. Sets of distinct positive integers with maximum n in which all adjacent elements have quotients > 1/2. For example, the a(1) = 1 through a(6) = 10 sets are:

{1} {2} {3} {4} {5} {6}

{2,3} {3,4} {3,5} {4,6}

{2,3,4} {4,5} {5,6}

{2,3,5} {3,4,6}

{3,4,5} {3,5,6}

{2,3,4,5} {4,5,6}

{2,3,4,6}

{2,3,5,6}

{3,4,5,6}

{2,3,4,5,6}

2. For n > 1, sets of distinct positive integers with maximum n - 1 whose first-differences are term-wise less than their decapitation (remove the maximum). For example, the set q = {2,4,5} has first-differences (2,1), which are not less than (2,4), so q is not counted under a(5). On the other hand, r = {2,3,5,6} has first-differences {1,2,1}, which are less than {2,3,5}, so r is counted under a(6).

3. Compositions of n where each part after the first is less than the sum of all preceding parts. For example, the a(1) = 1 through a(6) = 10 compositions are:

(1) (2) (3) (4) (5) (6)

(21) (31) (41) (51)

(211) (32) (42)

(311) (411)

(212) (321)

(2111) (312)

(3111)

(2121)

(2112)

(21111)

(End)

Apparently a(n) = number of partitions (p_1, p_2, ..., p_k) of n+1, with p_1 >= p_2 >= ... >= p_k, such that for each i, p_i > p_{i+1}+...+p_k. - John McKay (mac(AT)mathstat.concordia.ca), Mar 06 2009

Comment from John McKay confirmed in paper by Bessenrodt, Olsson, and Sellers. Such partitions are called "strongly decreasing" partitions in the paper, see the function s(n) therein.

Also the number of unlabeled binary rooted trees with 2*n + 3 nodes in which the two branches directly under any given non-leaf node are either equal or at least one of them is a leaf. - Gus Wiseman, Oct 08 2018

From Gus Wiseman, Apr 06 2021: (Start)

This sequence counts both of the following essentially equivalent things:

1. Sets of distinct positive integers with maximum n + 1 in which all adjacent elements have quotients < 1/2. For example, the a(0) = 1 through a(8) = 7 subsets are:

{1} {2} {3} {4} {5} {6} {7} {8} {9}

{1,3} {1,4} {1,5} {1,6} {1,7} {1,8} {1,9}

{2,5} {2,6} {2,7} {2,8} {2,9}

{3,7} {3,8} {3,9}

{1,3,7} {1,3,8} {4,9}

{1,3,9}

{1,4,9}

2. Sets of distinct positive integers with maximum n + 1 whose first differences are term-wise greater than their decapitation (remove the maximum). For example, the set q = {1,4,9} has first differences (3,5), which are greater than (1,4), so q is counted under a(8). On the other hand, r = {1,5,9} has first differences (4,4), which are not greater than (1,5), so r is not counted under a(8).

Also the number of partitions of n + 1 into powers of 2 covering an initial interval of powers of 2. For example, the a(0) = 1 through a(8) = 7 partitions are:

1 11 21 211 221 2211 421 4211 4221

111 1111 2111 21111 2221 22211 22221

11111 111111 22111 221111 42111

211111 2111111 222111

1111111 11111111 2211111

21111111

111111111

(End)