cp's OEIS Frontend

Glaisher calls this zeta(n) or zeta_1(n). - N. J. A. Sloane, Nov 24 2018

Coefficients in expansion of Sum_{n >= 1} x^n/(1+x^n)^2 = Sum_{n >= 1} (-1)^(n-1)*n*x^n/(1-x^n).

Unsigned sequence is A113184. - Peter Bala, Dec 14 2020

Number of nodes in rooted tree of height n in which every node (including the root) has valency 3.

Pascal diamond numbers: reflect Pascal's n-th triangle vertically and sum all elements. E.g., a(3)=1+(1+1)+(1+2+1)+(1+1)+1. - Paul Barry, Jun 23 2003

Number of 2 X n binary matrices avoiding simultaneously the right-angled numbered polyomino patterns (ranpp) (00;1), (10;0) and (11;0). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1 < i2 and j1 < j2 and these elements are in the same relative order as those in the triple (x,y,z). - Sergey Kitaev, Nov 11 2004

Binomial and inverse binomial transform are in A001047 (shifted) and A122553. - R. J. Mathar, Sep 02 2008

a(n) = (Sum_{k=0..n-1} a(n)) + (2*n + 1); e.g., a(3) = 22 = (1 + 4 + 10) + 7. - Gary W. Adamson, Jan 21 2009

Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if either 0) x is a proper subset of y or y is a proper subset of x and x and y are disjoint, or 1) x equals y. Then a(n) = |R|. - Ross La Haye, Mar 19 2009

Equals the Jacobsthal sequence A001045 convolved with (1, 3, 4, 4, 4, 4, 4, ...). - Gary W. Adamson, May 24 2009

Equals the eigensequence of a triangle with the odd integers as the left border and the rest 1's. - Gary W. Adamson, Jul 24 2010

An elephant sequence, see A175655. For the central square four A[5] vectors, with decimal values 58, 154, 178 and 184, lead to this sequence. For the corner squares these vectors lead to the companion sequence A097813. - Johannes W. Meijer, Aug 15 2010

a(n+2) is the integer with bit string "10" * "1"^n * "10".

a(n) = A027383(2n). - Jason Kimberley, Nov 03 2011

a(n) = A153893(n)-1 = A083416(2n+1). - Philippe Deléham, Apr 14 2013

a(n) = A082560(n+1,A000079(n)) = A232642(n+1,A128588(n+1)). - Reinhard Zumkeller, May 14 2015

a(n) is the sum of the entries in the n-th and (n+1)-st rows of Pascal's triangle minus 2. - Stuart E Anderson, Aug 27 2017

Also the number of independent vertex sets and vertex covers in the complete tripartite graph K_{n,n,n}. - Eric W. Weisstein, Sep 21 2017

Apparently, a(n) is the least k such that the binary expansion of A000045(k) ends with exactly n+1 ones. - Rémy Sigrist, Sep 25 2021

a(n) is the number of root ancestral configurations for a pair consisting of a matching gene tree and species tree with the modified lodgepole shape and n+1 cherry nodes. - Noah A Rosenberg, Jan 16 2025

Apart from leading term (which should really be 3/2), same as A055010.

Binomial transform of A040001. Inverse binomial transform of A053156.

a(n) = A105728(n+1,2). - Reinhard Zumkeller, Apr 18 2005

Row sums of triangle A133567. - Gary W. Adamson, Sep 16 2007

Row sums of triangle A135226. - Gary W. Adamson, Nov 23 2007

a(n) = number of partitions Pi of [n+1] (in standard increasing form) such that the permutation Flatten[Pi] avoids the patterns 2-1-3 and 3-1-2. Example: a(3)=11 counts all 15 partitions of [4] except 13/24, 13/2/4 which contain a 2-1-3 and 14/23, 14/2/3 which contain a 3-1-2. Here "standard increasing form" means the entries are increasing in each block and the blocks are arranged in increasing order of their first entries. - David Callan, Jul 22 2008

An elephant sequence, see A175654. For the corner squares four A[5] vectors, with decimal values 42, 138, 162, 168, lead to this sequence. For the central square these vectors lead to the companion sequence A003945. - Johannes W. Meijer, Aug 15 2010

The binary representation of a(n) has n+1 digits, where all digits are 1's except digit n-1. For example: a(4) = 23 = 10111 (2). - Omar E. Pol, Dec 02 2012

Row sums of triangle A209561. - Reinhard Zumkeller, Dec 26 2012

If a Stern's sequence based enumeration system of positive irreducible fractions is considered (for example, A007305/A047679, A162909/A162910, A071766/A229742, A245325/A245326, ...), and if it is organized by blocks or levels (n) with 2^n terms (n >= 0), and the fractions, term by term, are summed at each level n, then the resulting sequence of integers is a(n) + 1/2, apart from leading term (which should be 1/2). - Yosu Yurramendi, May 23 2015

For n >= 2, A083329(n) in binary representation is a string [101..1], also 10 followed with (n-1) 1's. For n >= 3, A036563(n) in binary representation is a string [1..101], also (n-2) 1's followed with 01. Thus A083329(n) is a reflection of the binary representation of A036563(n+1). Example: A083329(5) = 101111 in binary, A036563(6) = 111101 in binary. - Ctibor O. Zizka, Nov 06 2018

For n > 0, a(n) is the minimum number of turns in (n+1)-dimensional Euclidean space needed to visit all 2^(n+1) vertices of the (n+1)-cube (e.g., {0,1}^(n+1)) and return to the starting point, moving along straight-line segments between turns (turns may occur elsewhere in R^(n+1)). - Marco Ripà, Aug 14 2025

The k-th row contains 2k+1 numbers.

Columns in the right half consist of convolutions of the Lucas numbers with the natural numbers.

T(n,k) = number of strings s(0),...,s(n) such that s(n)=n-k. s(0) in {0,1,2}, s(1)=1 if s(0) in {1,2}, s(1) in {0,1,2} if s(0)=0 and for 1 <= i <= n, s(i) = s(i-1)+d, with d in {0,2} if s(i)=2i, in {0,1,2} if s(i)=2i-1, in {0,1} if 0 <= s(i) <= 2i-2.

a(n) represents the number of n-move routes of a fairy chess piece starting in the central square (m = 5) on a 3 X 3 chessboard. This fairy chess piece behaves like a bishop on the eight side and corner squares but on the central square the bishop turns into a raging elephant, see A175654.

For the central square the 512 elephants lead to 46 different elephant sequences, see the cross-references for examples.

The sequence above corresponds to 16 A[5] vectors with decimal values 71, 77, 101, 197, 263, 269, 293, 323, 326, 329, 332, 353, 356, 389, 449 and 452. These vectors lead for the side squares to A000079 and for the corner squares to A175654.

Note the indexing: the domain starts from 0, while the range excludes zero.

This sequence can be represented as a binary tree. Each left hand child is produced as A019565(n), and each right hand child as A065642(n), when the parent node contains n >= 2:

1

|

...................2...................

3 4

6......../ \........9 5......../ \........8

/ \ / \ / \ / \

/ \ / \ / \ / \

/ \ / \ / \ / \

15 12 14 27 10 25 7 16

210 45 35 18 105 28 462 81 21 20 154 125 30 49 11 32

etc.

Where will 38 appear in this tree? It is a reasonable assumption that by iterating A087207 starting from 38, as A087207(38) = 129, A087207(129) = 8194, A087207(8194) = 1501199875790187, ..., we will eventually hit a prime A000040(k), most likely with a largish index k. This prime occurs at the penultimate edge at right, as a(A000918(k)) = a((2^k)-2), and thus 38 occurs somewhere below it as a(m) = 38, m > k. All the numbers that share prime factors with 38, namely 76, 152, 304, 608, 722, ..., occur similarly late in this tree, as they form the rightward branch starting from 38. Alternatively, by iterating A285330 (each iteration moves one step towards the root) starting from 38, we might instead first hit some power of 3, or say, one of the terms of A033845 (the rightward branch starting from 6), in which case the first prime encountered would be a(2)=3 and 38 would appear on the left-hand side instead of the right-hand side subtree.

As long as it remains conjecture that A019565 has no cycles, it is certainly also an open question whether this is a permutation of the natural numbers: If A019565 has any cycles, then neither any of the terms in those cycles nor any A065642-trajectories starting from those terms (that is, numbers sharing same prime factors) may occur in this tree.

Sequence exhibits some outrageous swings, for example, a(703) = 224, but a(704) is 1427 decimal digits (4739 binary digits) long, thus it no longer fits into a b-file.

However, the scatter plot of A286543 gives some flavor of the behavior of this sequence even after that point. - Antti Karttunen, Dec 25 2017

Fifth diagonal of the array which contains m-acci numbers in the m-th row.

The base array is constructed from m-acci numbers starting each with 1, 1, and 2 and filling one row of the table (see the examples).

The main and the upper diagonals of the table are the powers of 2, A000079.

The first subdiagonal is essentially A000225, followed by essentially A036563.

The next subdiagonal is this sequence here, followed by A159742, A159743, A159744, A159746, A159747, A159748.

a(n) written in base 2: 1000, 11000, 111000, 1111000, ..., i.e., n times 1 and 3 times 0 (A161770). - Jaroslav Krizek, Jun 18 2009

Also numbers for which n^8/(n+8) is an integer. - Vicente Izquierdo Gomez, Jan 03 2013

From Reinhard Zumkeller, Aug 29 2009: (Start)

A023416(a(n)) = 1;

apart from the initial term the sequence can be seen as a triangle read by rows, see A164874;

A055010 and A086224 are subsequences, see also A000918 and A036563. (End)

Zero and numbers of form 2^m-2^k-1, 2 <= m, 0 <= k <= m-2. - Zak Seidov, Aug 06 2010

Square array read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

The topmost row (row 1) of the array is A000079 (powers of 2), and in general each row 2^k contains the sequence (2^n - k), starting from the term (2^(k+1) - k). This follows from the properties (3) and (4) of A004001 given on page 227 of Kubo & Vakil paper (page 3 in PDF).

Moreover, each row 2^k - 1 (for k >= 2) contains the sequence 2^n - n - (k-2), starting from the term (2^(k+1) - (2k-1)). To see why this holds, consider the definitions of sequences A162598 and A265332, the latter which also illustrates how the frequency counts Q_n for A004001 are recursively constructed (in the Kubo & Vakil paper).

Inverse binomial transform (by columns) of A090888.