cp's OEIS Frontend

Number of edges in an n-dimensional hypercube.

Number of 132-avoiding permutations of [n+2] containing exactly one 123 pattern. - Emeric Deutsch, Jul 13 2001

Number of ways to place n-1 nonattacking kings on a 2 X 2(n-1) chessboard for n >= 2. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 22 2001

Arithmetic derivative of 2^n: a(n) = A003415(A000079(n)). - Reinhard Zumkeller, Feb 26 2002

(-1) times the determinant of matrix A_{i,j} = -|i-j|, 0 <= i,j <= n.

a(n) is the number of ones in binary numbers 1 to 111...1 (n bits). a(n) = A000337(n) - A000337(n-1) for n = 2,3,... . - Emeric Deutsch, May 24 2003

The number of 2 X n 0-1 matrices containing n+1 1's and having no zero row or column. The number of spanning trees of the complete bipartite graph K(2,n). This is the case m = 2 of K(m,n). See A072590. - W. Edwin Clark, May 27 2003

Binomial transform of 0,1,2,3,4,5,... (A001477). Without the initial 0, binomial transform of odd numbers.

With an additional leading zero, [0,0,1,4,...] this is the binomial transform of the integers repeated A004526. Its formula is then (2^n*(n-1) + 0^n)/4. - Paul Barry, May 20 2003

Number of zeros in all different (n+1)-bit integers. - Ralf Stephan, Aug 02 2003

From Lekraj Beedassy, Jun 03 2004: (Start)

Final element of a summation table (as opposed to a difference table) whose first row consists of integers 0 through n (or first n+1 nonnegative integers A001477); illustrating the case n=5:

0 1 2 3 4 5

1 3 5 7 9

4 8 12 16

12 20 28

32 48

80

and the final element is a(5)=80. (End)

This sequence and A001871 arise in counting ordered trees of height at most k where only the rightmost branch at the root actually achieves this height and the count is by the number of edges, with k = 3 for this sequence and k = 4 for A001871.

Let R be a binary relation on the power set P(A) of a set A having n = |A| elements such that for all elements x,y of P(A), xRy if x is a proper subset of y and there are no z in P(A) such that x is a proper subset of z and z is a proper subset of y. Then a(n) = |R|. - Ross La Haye, Sep 21 2004

Number of 2 X n binary matrices avoiding simultaneously the right-angled numbered polyomino patterns (ranpp) (00;1) and (10;1). 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, j1 < j2 and these elements are in same relative order as those in the triple (x,y,z). - Sergey Kitaev, Nov 11 2004

Number of subsequences 00 in all binary words of length n+1. Example: a(2)=4 because in 000,001,010,011,100,101,110,111 the sequence 00 occurs 4 times. - Emeric Deutsch, Apr 04 2005

If you expand the n-factor expression (a+1)*(b+1)*(c+1)*...*(z+1), there are a(n) variables in the result. For example, the 3-factor expression (a+1)*(b+1)*(c+1) expands to abc+ab+ac+bc+a+b+c+1 with a(3) = 12 variables. - David W. Wilson, May 08 2005

An inverse Chebyshev transform of n^2, where g(x)->(1/sqrt(1-4*x^2))*g(x*c(x^2)), c(x) the g.f. of A000108. - Paul Barry, May 13 2005

Sequences A018215 and A058962 interleaved. - Graeme McRae, Jul 12 2006

The number of never-decreasing positive integer sequences of length n with a maximum value of 2*n. - Ben Paul Thurston, Nov 13 2006

Total size of all the subsets of an n-element set. For example, a 2-element set has 1 subset of size 0, 2 subsets of size 1 and 1 of size 2. - Ross La Haye, Dec 30 2006

Convolution of the natural numbers [A000027] and A045623 beginning [0,1,2,5,...]. - Ross La Haye, Feb 03 2007

If M is the matrix (given by rows) [2,1;0,2] then the sequence gives the (1,2) entry in M^n. - Antonio M. Oller-Marcén, May 21 2007

If X_1,X_2,...,X_n is a partition of a 2n-set X into 2-blocks then, for n > 0, a(n) is equal to the number of (n+1)-subsets of X intersecting each X_i (i=1,2,...,n). - Milan Janjic, Jul 21 2007

Number of n-permutations of 3 objects u,v,w, with repetition allowed, containing exactly one u. Example: a(2)=4 because we have uv, vu, uw and wu. - Zerinvary Lajos, Dec 27 2007

A member of the family of sequences defined by a(n) = n*[c(1)*...*c(r)]^(n-1); c(i) integer. This sequence has c(1)=2, A027471 has c(1)=3. - Ctibor O. Zizka, Feb 23 2008

a(n) is the number of ways to split {1,2,...,n-1} into two (possibly empty) complementary intervals {1,2,...,i} and {i+1,i+2,...,n-1} and then select a subset from each interval. - Geoffrey Critzer, Jan 31 2009

Equals the Jacobsthal sequence A001045 convolved with A003945: (1, 3, 6, 12, ...). - Gary W. Adamson, May 23 2009

Starting with offset 1 = A059570: (1, 2, 6, 14, 34, ...) convolved with (1, 2, 2, 2, ...). - Gary W. Adamson, May 23 2009

Equals the first left hand column of A167591. - Johannes W. Meijer, Nov 12 2009

The number of tatami tilings of an n X n square with n monomers is n*2^(n-1). - Frank Ruskey, Sep 25 2010

Under T. D. Noe's variant of the hypersigma function, this sequence gives hypersigma(2^n): a(n) = A191161(A000079(n)). - Alonso del Arte, Nov 04 2011

Number of Dyck (n+2)-paths with exactly one valley at height 1 and no higher valley. - David Scambler, Nov 07 2011

Equals triangle A059260 * A016777 as a vector, where A016777 = (3n + 1): [1, 4, 7, 10, 13, ...]. - Gary W. Adamson, Mar 06 2012

Main transitions in systems of n particles with spin 1/2 (see A212697 with b=2). - Stanislav Sykora, May 25 2012

Let T(n,k) be the triangle with (first column) T(n,1) = 2*n-1 for n >= 1, otherwise T(n,k) = T(n,k-1) + T(n-1,k-1), then a(n) = T(n,n). - J. M. Bergot, Jan 17 2013

Sum of all parts of all compositions (ordered partitions) of n. The equivalent sequence for partitions is A066186. - Omar E. Pol, Aug 28 2013

Starting with a(1)=1: powers of 2 (A000079) self-convolved. - Bob Selcoe, Aug 05 2015

Coefficients of the series expansion of the normalized Schwarzian derivative -S{p(x)}/6 of the polynomial p(x) = -(x-x1)*(x-x2) with x1 + x2 = 1 (cf. A263646). - Tom Copeland, Nov 02 2015

a(n) is the number of North-East lattice paths from (0,0) to (n+1,n+1) that have exactly one east step below y = x-1 and no east steps above y = x+1. Details can be found in Pan and Remmel's link. - Ran Pan, Feb 03 2016

Also the number of maximal and maximum cliques in the n-hypercube graph for n > 0. - Eric W. Weisstein, Dec 01 2017

Let [n]={1,2,...,n}; then a(n-1) is the total number of elements missing in proper subsets of [n] that contain n to form [n]. For example, for n = 3, a(2) = 4 since the proper subsets of [3] that contain 3 are {3}, {1,3}, {2,3} and the total number of elements missing in these subsets to form [3] is 4: 2 in the first subset, 1 in the second, and 1 in the third. - Enrique Navarrete, Aug 08 2020

Number of 3-permutations of n elements avoiding the patterns 132, 231. See Bonichon and Sun. - Michel Marcus, Aug 19 2022

Also the number of partitions of n-1, n >= 2, such that the least part occurs exactly once. See A096373, A097091, A097092, A097093. - Robert G. Wilson v, Jul 24 2004 [Corrected by Wolfdieter Lang, Feb 18 2009]

Number of partitions of n+1 where the number of parts is itself a part. Take a partition of n (with k parts) which does not contain 1, remove 1 from each part and add a new part of size k+1. - Franklin T. Adams-Watters, May 01 2006

Number of partitions where the largest part occurs at least twice. - Joerg Arndt, Apr 17 2011

Row sums of triangle A147768. - Gary W. Adamson, Nov 11 2008

From Lewis Mammel (l_mammel(AT)att.net), Oct 06 2009: (Start)

a(n) is the number of sets of n disjoint pairs of 2n things, called a pairing, disjoint with a given pairing (A053871), that are unique under permutations preserving the given pairing.

Can be seen immediately from a graphical representation which must decompose into even numbered cycles of 4 or more things, as connected by pairs alternating between the pairings. Each thing is in a single cycle, so this is a partition of 2n into even parts greater than 2, equivalent to a partition of n into parts greater than 1. (End)

Convolution product (1, 1, 2, 2, 4, 4, ...) * (1, 2, 3, ...) = A058682 starting (1, 3, 7, 13, 23, 37, ...); with row sums of triangle A171239 = A058682. - Gary W. Adamson, Dec 05 2009

Also the number of 2-regular multigraphs with loops forbidden. - Jason Kimberley, Jan 05 2011

Number of appearances of the multiplicity n, n-1, ..., n-k in all partitions of n, for k < n/2. (Only populated by multiplicities of large numbers of 1's.) - William Keith, Nov 20 2011

Also the number of equivalence classes of n X n binary matrices with exactly 2 1's in each row and column, up to permutations of rows and columns (cf. A133687). - N. J. A. Sloane, Sep 16 2013

The q-Catalan numbers ((1-q)/(1-q^(n+1)))[2n,n]q, where [2n,n]_q are the central q-binomial coefficients, match this sequence in their initial segment of length n. - _William J. Keith, Nov 14 2013

Starting at a(2) this sequence gives the number of vertices on a nim tree created in the game of edge removal for a path P_{n} where n is the number of vertices on the path. This is the number of nonisomorphic graphs that can result from the path when the game of edge removal is played. - Lyndsey Wong, Jul 09 2016

The number of different ways to climb a staircase taking at least two stairs at a time. - Mohammad K. Azarian, Nov 20 2016

Let 1,0,1,1,1,... (offset 0) count unlabeled, connected, loopless 1-regular digraphs. This here is the Euler transform of that sequence, counting unlabeled loopless 1-regular digraphs. A145574 is the associated multiset transformation. A000166 are the labeled loopless 1-regular digraphs. - R. J. Mathar, Mar 25 2019

For n > 1, also the number of partitions with no part greater than the number of ones. - George Beck, May 09 2019 [See A187219 which is the correct sequence for this interpretation for n >= 1. - Spencer Miller, Jan 30 2023]

From Gus Wiseman, May 19 2019: (Start)

Conjecture: Also the number of integer partitions of n - 1 that have a consecutive subsequence summing to each positive integer from 1 to n - 1. For example, (32211) is such a partition because we have consecutive subsequences:

1: (1)

2: (2)

3: (3) or (21)

4: (22) or (211)

5: (32) or (221)

6: (2211)

7: (322)

8: (3221)

9: (32211)

(End)

There is a sufficient and necessary condition to characterize the partitions defined by Gus Wiseman. It is that the largest part must be less than or equal to the number of ones plus one. Hence, the number of partitions of n with no part greater than the number of ones is the same as the number of partitions of n-1 that have a consecutive subsequence summing to each integer from 1 to n-1. Gus Wiseman's conjecture can be proved bijectively. - Andrew Yezhou Wang, Dec 14 2019

From Peter Bala, Dec 01 2024: (Start)

Let P(2, n) denote the set of partitions of n into parts k > 1. Then A000041(n) = - Sum_{parts k in all partitions in P(2, n+2)} mu(k). For example, with n = 5, there are 4 partitions of n + 2 = 7 into parts greater than 1, namely, 7, 5 + 2, 4 + 3, 3 + 2 + 2, and mu(7) + (mu(5) + mu(2)) + (mu(4 ) + mu(3)) + (mu(3) + mu(2) + mu(2)) = -7 = - A000041(5). (End)

These are Hogben's central polygonal numbers denoted by the symbol

...2....

....P...

...2.n..

(P with three attachments).

Also the maximal number of 1's that an n X n invertible {0,1} matrix can have. (See Halmos for proof.) - Felix Goldberg (felixg(AT)tx.technion.ac.il), Jul 07 2001

Maximal number of interior regions formed by n intersecting circles, for n >= 1. - Amarnath Murthy, Jul 07 2001

The terms are the smallest of n consecutive odd numbers whose sum is n^3: 1, 3 + 5 = 8 = 2^3, 7 + 9 + 11 = 27 = 3^3, etc. - Amarnath Murthy, May 19 2001

(n*a(n+1)+1)/(n^2+1) is the smallest integer of the form (n*k+1)/(n^2+1). - Benoit Cloitre, May 02 2002

For n >= 3, a(n) is also the number of cycles in the wheel graph W(n) of order n. - Sharon Sela (sharonsela(AT)hotmail.com), May 17 2002

Let b(k) be defined as follows: b(1) = 1 and b(k+1) > b(k) is the smallest integer such that Sum_{i=b(k)..b(k+1)} 1/sqrt(i) > 2; then b(n) = a(n) for n > 0. - Benoit Cloitre, Aug 23 2002

Drop the first three terms. Then n*a(n) + 1 = (n+1)^3. E.g., 7*1 + 1 = 8 = 2^3, 13*2 + 1 = 27 = 3^3, 21*3 + 1 = 64 = 4^3, etc. - Amarnath Murthy, Oct 20 2002

Arithmetic mean of next 2n - 1 numbers. - Amarnath Murthy, Feb 16 2004

The n-th term of an arithmetic progression with first term 1 and common difference n: a(1) = 1 -> 1, 2, 3, 4, 5, ...; a(2) = 3 -> 1, 3, ...; a(3) = 7 -> 1, 4, 7, ...; a(4) = 13 -> 1, 5, 9, 13, ... - Amarnath Murthy, Mar 25 2004

Number of walks of length 3 between any two distinct vertices of the complete graph K_{n+1} (n >= 1). Example: a(2) = 3 because in the complete graph ABC we have the following walks of length 3 between A and B: ABAB, ACAB and ABCB. - Emeric Deutsch, Apr 01 2004

Narayana transform of [1, 2, 0, 0, 0, ...] = [1, 3, 7, 13, 21, ...]. Let M = the infinite lower triangular matrix of A001263 and let V = the Vector [1, 2, 0, 0, 0, ...]. Then A002061 starting (1, 3, 7, ...) = M * V. - Gary W. Adamson, Apr 25 2006

The sequence 3, 7, 13, 21, 31, 43, 57, 73, 91, 111, ... is the trajectory of 3 under repeated application of the map n -> n + 2 * square excess of n, cf. A094765.

Also n^3 mod (n^2+1). - Zak Seidov, Aug 31 2006

Also, omitting the first 1, the main diagonal of A081344. - Zak Seidov, Oct 05 2006

Ignoring the first ones, these are rectangular parallelepipeds with integer dimensions that have integer interior diagonals. Using Pythagoras: sqrt(a^2 + b^2 + c^2) = d, an integer; then this sequence: sqrt(n^2 + (n+1)^2 + (n(n+1))^2) = 2T_n + 1 is the first and most simple example. Problem: Are there any integer diagonals which do not satisfy the following general formula? sqrt((k*n)^2 + (k*(n+(2*m+1)))^2 + (k*(n*(n+(2*m+1)) + 4*T_m))^2) = k*d where m >= 0, k >= 1, and T is a triangular number. - Marco Matosic, Nov 10 2006

Numbers n such that a(n) is prime are listed in A055494. Prime a(n) are listed in A002383. All terms are odd. Prime factors of a(n) are listed in A007645. 3 divides a(3*k-1), 7 divides a(7*k-4) and a(7*k-2), 7^2 divides a(7^2*k-18) and a(7^2*k+19), 7^3 divides a(7^3*k-18) and a(7^3*k+19), 7^4 divides a(7^4*k+1048) and a(7^4*k-1047), 7^5 divides a(7^5*k+1354) and a(7^5*k-1353), 13 divides a(13*k-9) and a(13*k-3), 13^2 divides a(13^2*k+23) and a(13^2*k-22), 13^3 divides a(13^3*k+1037) and a(13^3*k-1036). - Alexander Adamchuk, Jan 25 2007

Complement of A135668. - Kieren MacMillan, Dec 16 2007

From William A. Tedeschi, Feb 29 2008: (Start)

Numbers (sorted) on the main diagonal of a 2n X 2n spiral. For example, when n=2:

.

7---8---9--10

| |

6 1---2 11

| | |

5---4---3 12

|

16--15--14--13

.

Cf. A137928. (End)

a(n) = AlexanderPolynomial[n] defined as Det[Transpose[S]-n S] where S is Seifert matrix {{-1, 1}, {0, -1}}. - Artur Jasinski, Mar 31 2008

Starting (1, 3, 7, 13, 21, ...) = binomial transform of [1, 2, 2, 0, 0, 0]; example: a(4) = 13 = (1, 3, 3, 1) dot (1, 2, 2, 0) = (1 + 6 + 6 + 0). - Gary W. Adamson, May 10 2008

Starting (1, 3, 7, 13, ...) = triangle A158821 * [1, 2, 3, ...]. - Gary W. Adamson, Mar 28 2009

Starting with offset 1 = triangle A128229 * [1,2,3,...]. - Gary W. Adamson, Mar 26 2009

a(n) = k such that floor((1/2)*(1 + sqrt(4*k-3))) + k = (n^2+1), that is A000037(a(n)) = A002522(n) = n^2 + 1, for n >= 1. - Jaroslav Krizek, Jun 21 2009

For n > 0: a(n) = A170950(A002522(n-1)), A170950(a(n)) = A174114(n), A170949(a(n)) = A002522(n-1). - Reinhard Zumkeller, Mar 08 2010

From Emeric Deutsch, Sep 23 2010: (Start)

a(n) is also the Wiener index of the fan graph F(n). The fan graph F(n) is defined as the graph obtained by joining each node of an n-node path graph with an additional node. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph. The Wiener polynomial of the graph F(n) is (1/2)t[(n-1)(n-2)t + 2(2n-1)]. Example: a(2)=3 because the corresponding fan graph is a cycle on 3 nodes (a triangle), having distances 1, 1, and 1.

(End)

For all elements k = n^2 - n + 1 of the sequence, sqrt(4*(k-1)+1) is an integer because 4*(k-1) + 1 = (2*n-1)^2 is a perfect square. Building the intersection of this sequence with A000225, k may in addition be of the form k = 2^x - 1, which happens only for k = 1, 3, 7, 31, and 8191. [Proof: Still 4*(k-1)+1 = 2^(x+2) - 7 must be a perfect square, which has the finite number of solutions provided by A060728: x = 1, 2, 3, 5, or 13.] In other words, the sequence A038198 defines all elements of the form 2^x - 1 in this sequence. For example k = 31 = 6*6 - 6 + 1; sqrt((31-1)*4+1) = sqrt(121) = 11 = A038198(4). - Alzhekeyev Ascar M, Jun 01 2011

a(n) such that A002522(n-1) * A002522(n) = A002522(a(n)) where A002522(n) = n^2 + 1. - Michel Lagneau, Feb 10 2012

Left edge of the triangle in A214661: a(n) = A214661(n, 1), for n > 0. - Reinhard Zumkeller, Jul 25 2012

a(n) = A215630(n, 1), for n > 0; a(n) = A215631(n-1, 1), for n > 1. - Reinhard Zumkeller, Nov 11 2012

Sum_{n > 0} arccot(a(n)) = Pi/2. - Franz Vrabec, Dec 02 2012

If you draw a triangle with one side of unit length and one side of length n, with an angle of Pi/3 radians between them, then the length of the third side of the triangle will be the square root of a(n). - Elliott Line, Jan 24 2013

a(n+1) is the number j such that j^2 = j + m + sqrt(j*m), with corresponding number m given by A100019(n). Also: sqrt(j*m) = A027444(n) = n * a(n+1). - Richard R. Forberg, Sep 03 2013

Let p(x) the interpolating polynomial of degree n-1 passing through the n points (n,n) and (1,1), (2,1), ..., (n-1,1). Then p(n+1) = a(n). - Giovanni Resta, Feb 09 2014

The number of square roots >= sqrt(n) and < n+1 (n >= 0) gives essentially the same sequence, 1, 3, 7, 13, 21, 31, 43, 57, 73, 91, 111, 133, 157, 183, 211, ... . - Michael G. Kaarhus, May 21 2014

For n > 1: a(n) is the maximum total number of queens that can coexist without attacking each other on an [n+1] X [n+1] chessboard. Specifically, this will be a lone queen of one color placed in any position on the perimeter of the board, facing an opponent's "army" of size a(n)-1 == A002378(n-1). - Bob Selcoe, Feb 07 2015

a(n+1) is, for n >= 1, the number of points as well as the number of lines of a finite projective plane of order n (cf. Hughes and Piper, 1973, Theorem 3.5., pp. 79-80). For n = 3, a(4) = 13, see the 'Finite example' in the Wikipedia link, section 2.3, for the point-line matrix. - Wolfdieter Lang, Nov 20 2015

Denominators of the solution to the generalization of the Feynman triangle problem. If each vertex of a triangle is joined to the point (1/p) along the opposite side (measured say clockwise), then the area of the inner triangle formed by these lines is equal to (p - 2)^2/(p^2 - p + 1) times the area of the original triangle, p > 2. For example, when p = 3, the ratio of the areas is 1/7. The numerators of the ratio of the areas is given by A000290 with an offset of 2. [Cook & Wood, 2004.] - Joe Marasco, Feb 20 2017

n^2 equal triangular tiles with side lengths 1 X 1 X 1 may be put together to form an n X n X n triangle. For n>=2 a(n-1) is the number of different 2 X 2 X 2 triangles being contained. - Heinrich Ludwig, Mar 13 2017

For n >= 0, the continued fraction [n, n+1, n+2] = (n^3 + 3n^2 + 4n + 2)/(n^2 + 3n + 3) = A034262(n+1)/a(n+2) = n + (n+2)/a(n+2); e.g., [2, 3, 4] = A034262(3)/a(4) = 30/13 = 2 + 4/13. - Rick L. Shepherd, Apr 06 2017

Starting with b(1) = 1 and not allowing the digit 0, let b(n) = smallest nonnegative integer not yet in the sequence such that the last digit of b(n-1) plus the first digit of b(n) is equal to k for k = 1, ..., 9. This defines 9 finite sequences, each of length equal to a(k), k = 1, ..., 9. (See A289283-A289287 for the cases k = 5..9.) For k = 10, the sequence is infinite (A289288). For example, for k = 4, b(n) = 1,3,11,31,32,2,21,33,12,22,23,13,14. These terms can be ordered in the following array of size k*(k-1)+1:

1 2 3

21 22 23

31 32 33

11 12 13 14

.

The sequence ends with the term 1k, which lies outside the rectangular array and gives the term +1 (see link).- Enrique Navarrete, Jul 02 2017

The central polygonal numbers are the delimiters (in parenthesis below) when you write the natural numbers in groups of odd size 2*n+1 starting with the group {2} of size 1: (1) 2 (3) 4,5,6 (7) 8,9,10,11,12 (13) 14,15,16,17,18,19,20 (21) 22,23,24,25,26,27,28,29,30 (31) 32,33,34,35,36,37,38,39,40,41,42 (43) ... - Enrique Navarrete, Jul 11 2017

Also the number of (non-null) connected induced subgraphs in the n-cycle graph. - Eric W. Weisstein, Aug 09 2017

Since (n+1)^2 - (n+1) + 1 = n^2 + n + 1 then from 7 onwards these are also exactly the numbers that are represented as 111 in all number bases: 111(2)=7, 111(3)=13, ... - Ron Knott, Nov 14 2017

Number of binary 2 X (n-1) matrices such that each row and column has at most one 1. - Dmitry Kamenetsky, Jan 20 2018

Observed to be the squares visited by bishop moves on a spirally numbered board and moving to the lowest available unvisited square at each step, beginning at the second term (cf. A316667). It should be noted that the bishop will only travel to squares along the first diagonal of the spiral. - Benjamin Knight, Jan 30 2019

From Ed Pegg Jr, May 16 2019: (Start)

Bound for n-subset coverings. Values in A138077 covered by difference sets.

C(7,3,2), {1,2,4}

C(13,4,2), {0,1,3,9}

C(21,5,2), {3,6,7,12,14}

C(31,6,2), {1,5,11,24,25,27}

C(43,7,2), existence unresolved

C(57,8,2), {0,1,6,15,22,26,45,55}

Next unresolved cases are C(111,11,2) and C(157,13,2). (End)

"In the range we explored carefully, the optimal packings were substantially irregular only for n of the form n = k(k+1)+1, k = 3, 4, 5, 6, 7, i.e., for n = 13, 21, 31, 43, and 57." (cited from Lubachevsky, Graham link, Introduction). - Rainer Rosenthal, May 27 2020

From Bernard Schott, Dec 31 2020: (Start)

For n >= 1, a(n) is the number of solutions x in the interval 1 <= x <= n of the equation x^2 - [x^2] = (x - [x])^2, where [x] = floor(x). For n = 3, the a(3) = 7 solutions in the interval [1, 3] are 1, 3/2, 2, 9/4, 5/2, 11/4 and 3.

This sequence is the answer to the 4th problem proposed during the 20th British Mathematical Olympiad in 1984 (see link B.M.O 1984. and Gardiner reference). (End)

Called "Hogben numbers" after the British zoologist, statistician and writer Lancelot Thomas Hogben (1895-1975). - Amiram Eldar, Jun 24 2021

Minimum Wiener index of 2-degenerate graphs with n+1 vertices (n>0). A maximal 2-degenerate graph can be constructed from a 2-clique by iteratively adding a new 2-leaf (vertex of degree 2) adjacent to two existing vertices. The extremal graphs are maximal 2-degenerate graphs with diameter at most 2. - Allan Bickle, Oct 14 2022

a(n) is the number of parking functions of size n avoiding the patterns 123, 213, and 312. - Lara Pudwell, Apr 10 2023

Repeated iteration of a(k) starting with k=2 produces Sylvester's sequence, i.e., A000058(n) = a^n(2), where a^n is the n-th iterate of a(k). - Curtis Bechtel, Apr 04 2024

a(n) is the maximum number of triangles that can be traversed by starting from a triangle and moving to adjacent triangles via an edge, without revisiting any triangle, in an n X n X n equilateral triangular grid made up of n^2 unit equilateral triangles. - Kiran Ananthpur Bacche, Jan 16 2025

Rayes et al. prove that the a(n)-th Chebyshev-T polynomial, divided by x, is irreducible over the integers.

Odd primes can be written as a sum of no more than two consecutive positive integers. Powers of 2 do not have a representation as a sum of k consecutive positive integers (other than the trivial n=n, for k=1). See A111774. - Jaap Spies, Jan 04 2007

Intersection of A005408 and A000040. - Reinhard Zumkeller, Oct 14 2008

Primes which are the sum of two consecutive numbers. - Juri-Stepan Gerasimov, Nov 07 2009

The arithmetic mean of divisors of p^3, (1+p)(1+p^2)/4, for odd primes p is an integer. - Ctibor O. Zizka, Oct 20 2009

Primes == -+ 1 (mod 4). - Juri-Stepan Gerasimov, Apr 27 2010

a(n) = A053670(A179675(n)) and a(n) <> A053670(m) for m < A179675(n). - Reinhard Zumkeller, Jul 23 2010

Triads of the form <2*a(n+1), a(n+1), 3*a(n+1)> like <6,3,9>, <10,5,15>, <14,7,21> appear in the EKG sequence A064413, see Theorem (3) there. - Paul Curtz, Feb 13 2011.

Complement of A065090; abs(A151763(a(n))) = 1. - Reinhard Zumkeller, Oct 06 2011

Right edge of the triangle in A065305. - Reinhard Zumkeller, Jan 30 2012

Numbers with two odd divisors. - Omar E. Pol, Mar 24 2012

Odd prime p divides some (2^k + 1) or (2^k - 1), (k>0, minimal, cf. A003558) depending on the parity of A179480((p+1)/2) = r. This is a consequence of the Quasi-order theorem and corollaries, [Hilton and Pederson, pp. 260-264]: 2^k == (-1)^r mod b, b odd; and b divides 2^k - (-1)^r, where p is a subset of b. - Gary W. Adamson, Aug 26 2012

Subset of the arithmetic numbers (A003601). - Wesley Ivan Hurt, Sep 27 2013

Odd primes p satisfy the identity: p = (product(2*cos((2*k+1)*Pi/(2*p)), k=0..(p-3)/2))^2. This follows from C(2*p, 0) = (-1)^((p-1)/2)*p, n>=2, with the minimal polynomial C(k,x) of rho(k) := 2*cos(Pi/k). See A187360 for C and the W. Lang link on the field Q(rho(n)), eqs. (20) and (37). - Wolfdieter Lang, Oct 23 2013

Numbers m > 1 such that m^2 divides (2m-1)!! + m. - Thomas Ordowski, Nov 28 2014

Numbers m such that m divides 2*(m-3)! + 1. - Thomas Ordowski, Jun 20 2015

Numbers m such that (2m-3)!! == m (mod m^2). - Thomas Ordowski, Jul 24 2016

Odd numbers m such that ((m-3)!!)^2 == +-1 (mod m). - Thomas Ordowski, Jul 27 2016

Primes of the form x^2 - y^2. - Thomas Ordowski, Feb 27 2017

Conjecture: a(n) is the smallest odd number m > prime(n) such that Sum_{k=1..prime(n)-1} k^(m-1) == prime(n)-1 (mod m). This is an extension of the Agoh-Giuga conjecture. - Thomas Ordowski, Aug 01 2018

Numbers k > 1 such that either Phi(k,x) == 1 (mod k) or Phi(k,x) == k (mod k^2) holds, where Phi(k,x) is the k-th cyclotomic polynomial. - Jianing Song, Aug 02 2018

Row sums: A083329

Alternating row sums: 1,0,-1,-1,-1,-1,-1,-1,-1,-1,...

Antidiagonal sums: A000071 (-1+Fibonacci numbers)

col 1: A000012

col 2: A005408

col 3: A000290

col 4: A000330

col 5: A002415

col 6: A005585

col 7: A040977

col 8: A050486

col 9: A053347

col 10: A054333

col 11: A054334

col 12: A057788

col 2n-1 of A208510 is column n of A208508

col 2n of A208510 is column n of A208509.

...

GENERAL DISCUSSION:

A208510 typifies arrays generated by paired recurrence equations of the following form:

u(n,x)=a(n,x)*u(n-1,x)+b(n,x)*v(n-1,x)+c(n,x)

v(n,x)=d(n,x)*u(n-1,x)+e(n,x)*v(n-1,x)+f(n,x).

...

These first-order recurrences imply separate second-order recurrences. In order to show them, the six functions a(n,x),...,f(n,x) are abbreviated as a,b,c,d,e,f.

Then, starting with initial values u(1,x)=1 and u(2,x)=a+b+c: u(n,x) = (a+e)u(n-1,x) + (bd-ae)u(n-2,x) + bf-ce+c.

With initial values v(1,x)=1 and v(2,x)=d+e+f: v(n,x) = (a+e)v(n-1,x) + (bd-ae)v(n-2,x) + cd-af+f.

...

In the guide below, the last column codes certain sequences that occur in one of these ways: row, column, edge, row sum, alternating row sum. Coding:

A: 1,-1,1,-1,1,-1,1.... A033999

B: 1,2,4,8,16,32,64,... powers of 2

C: 1,1,1,1,1,1,1,1,.... A000012

D: 2,2,2,2,2,2,2,2,.... A007395

E: 2,4,6,8,10,12,14,... even numbers

F: 1,1,2,3,5,8,13,21,.. Fibonacci numbers

N: 1,2,3,4,5,6,7,8,.... A000027

O: 1,3,5,7,9,11,13,.... odd numbers

P: 1,3,9,27,81,243,.... powers of 3

S: 1,4,9,16,25,36,49,.. squares

T: 1,3,6,10,15,21,38,.. triangular numbers

Z: 1,0,0,0,0,0,0,0,0,.. A000007

*: (eventually) periodic alternating row sums

^: has a limiting row; i.e., the polynomials "approach" a power series

This coding includes indirect and repeated occurrences; e.g. F occurs thrice at A094441: in column 1 directly as Fibonacci numbers, in row sums as odd-indexed Fibonacci numbers, and in alternating row sums as signed Fibonacci numbers.

......... a....b....c....d....e....f....code

A034839 u 1....1....0....1....x....0....CCOT

A034867 v 1....1....0....1....x....0....CEN

A210221 u 1....1....0....1....2x...0....BBFF

A210596 v 1....1....0....1....2x...0....BBFF

A105070 v 1....2x...0....1....1....0....BN

A207605 u 1....1....0....1....x+1..0....BCFFN

A106195 v 1....1....0....1....x+1..0....BCFFN

A207606 u 1....1....0....x....x+1..0....DNT

A207607 v 1....1....0....x....x+1..0....DNT

A207608 u 1....1....0....2x...x+1..0....N

A207609 v 1....1....0....2x...x+1..0....C

A207610 u 1....1....0....1....x....1....CF

A207611 v 1....1....0....1....x....1....BCF

A207612 u 1....1....0....1....2x...1....BF

A207613 v 1....1....0....1....2x...1....BF

A207614 u 1....1....0....1....x+1..1....CN

A207615 v 1....1....0....1....x+1..1....CFN

A207616 u 1....1....0....x....1....1....CE

A207617 v 1....1....0....x....1....1....CNO

A029638 u 1....1....0....x....x....1....CDNO

A029635 v 1....1....0....x....x....1....CDNOZ

A207618 u 1....1....0....x....2x...1....N

A207619 v 1....1....0....x....2x...1....CFN

A207620 u 1....1....0....x....x+1..1....DET

A207621 v 1....1....0....x....x+1..1....DNO

A207622 u 1....1....0....2x...1....1....BT

A207623 v 1....1....0....2x...1....1....BN

A207624 u 1....1....0....2x...x....1....N

A102662 v 1....1....0....2x...x....1....CO

A207625 u 1....1....0....2x...x+1..1....T

A207626 v 1....1....0....2x...x+1..1....N

A207627 u 1....1....0....2x...2x...1....BN

A207628 v 1....1....0....2x...2x...1....BCE

A207629 u 1....1....0....x+1..1....1....CET

A207630 v 1....1....0....x+1..1....1....CO

A207631 u 1....1....0....x+1..x....1....DF

A207632 v 1....1....0....x+1..x....1....DEF

A207633 u 1....1....0....x+1..2x...1....F

A207634 v 1....1....0....x+1..2x...1....F

A207635 u 1....1....0....x+1..x+1..1....DN

A207636 v 1....1....0....x+1..x+1..1....CD

A160232 u 1....x....0....1....2x...0....BCFN

A208341 v 1....x....0....1....2x...0....BCFFN

A085478 u 1....x....0....1....x+1..0....CCOFT*

A078812 v 1....x....0....1....x+1..0....CEFN*

A208342 u 1....x....0....x....x....0....CCFNO

A208343 v 1....x....0....x....x....0....BBCDFZ

A208344 u 1....x....0....x....2x...0....CCFN

A208345 v 1....x....0....x....2x...0....CFZ

A094436 u 1....x....0....x....x+1..0....CFFN

A094437 v 1....x....0....x....x+1..0....CEFF

A117919 u 1....x....0....2x...1....0....BCNT

A135837 v 1....x....0....2x...1....0....BCET

A208328 u 1....x....0....2x...x....0....CCOP

A208329 v 1....x....0....2x...x....0....DPZ

A208330 u 1....x....0....2x...x+1..0....CNPT

A208331 v 1....x....0....2x...x+1..0....CN

A208332 u 1....x....0....2x...2x...0....CCE

A208333 v 1....x....0....2x...2x...0....DZ

A208334 u 1....x....0....x+1..1....0....CCNT

A208335 v 1....x....0....x+1..1....0....CCN*

A208336 u 1....x....0....x+1..x....0....CFNT*

A208337 v 1....x....0....x+1..x....0....ACFN*

A208338 u 1....x....0....x+1..2x...0....CNP

A208339 v 1....x....0....x+1..2x...0....BCNP

A202390 u 1....x....0....x+1..x+1..0....CFPTZ*

A208340 v 1....x....0....x+1..x+1..0....FNPZ*

A208508 u 1....x....0....1....1....1....CCES

A208509 v 1....x....0....1....1....1....BCO

A208510 u 1....x....0....1....x....1....CCCNOS*

A029653 v 1....x....0....1....x....1....BCDOSZ*

A208511 u 1....x....0....1....2x...1....BCFO

A208512 v 1....x....0....1....2x...1....BDFO

A208513 u 1....x....0....1....x+1..1....CCES*

A111125 v 1....x....0....1....x+1..1....COO*

A133567 u 1....x....0....x....1....1....CCOTT

A133084 v 1....x....0....x....1....1....BBCEN

A208514 u 1....x....0....x....x....1....CEFN

A208515 v 1....x....0....x....x....1....BCDFN

A208516 u 1....x....0....x....2x...1....CNN

A208517 v 1....x....0....x....2x...1....CCN

A208518 u 1....x....0....x....x+1..1....CFNT

A208519 v 1....x....0....x....x+1..1....NFFT

A208520 u 1....x....0....2x...1....1....BCTT

A208521 v 1....x....0....2x...1....1....BEN

A208522 u 1....x....0....2x...x....1....CCN

A208523 v 1....x....0....2x...x....1....CCO

A208524 u 1....x....0....2x...x+1..1....CT*

A208525 v 1....x....0....2x...x+1..1....ACNP*

A208526 u 1....x....0....2x...2x...1....CEN

A208527 v 1....x....0....2x...2x...1....CCE

A208606 u 1....x....0....x+1..1....1....CCS

A208607 v 1....x....0....x+1..1....1....CNO

A208608 u 1....x....0....x+1..x....1....CFOT

A208609 v 1....x....0....x+1..x....1....DEN*

A208610 u 1....x....0....x+1..2x...1....CO

A208611 v 1....x....0....x+1..2x...1....DE

A208612 u 1....x....0....x+1..x+1..1....CFNS

A208613 v 1....x....0....x+1..x+1..1....CFN*

A105070 u 1....2x...0....1....1....0....BN

A207536 u 1....2x...0....1....1....0....BCT

A208751 u 1....2x...0....1....x+1..0....CDPT

A208752 v 1....2x...0....1....x+1..0....CNP

A135837 u 1....2x...0....x....1....0....BCNT

A117919 v 1....2x...0....x....1....0....BCNT

A208755 u 1....2x...0....x....x....0....BCDEP

A208756 v 1....2x...0....x....x....0....BCCOZ

A208757 u 1....2x...0....x....2x...0....CDEP

A208758 v 1....2x...0....x....2x...0....CCEPZ

A208763 u 1....2x...0....2x...x....0....CDOP

A208764 v 1....2x...0....2x...x....0....CCCP

A208765 u 1....2x...0....2x...x+1..0....CE

A208766 v 1....2x...0....2x...x+1..0....CC

A208747 u 1....2x...0....2x...2x...0....CDE

A208748 v 1....2x...0....2x...2x...0....CCZ

A208749 u 1....2x...0....x+1..1....0....BCOPT

A208750 v 1....2x...0....x+1..1....0....BCNP*

A208759 u 1....2x...0....x+1..2x....0...CE

A208760 v 1....2x...0....x+1..2x....0...BCO

A208761 u 1....2x...0....x+1..x+1...0...BCCT*

A208762 v 1....2x...0....x+1..x+1...0...BNZ*

A208753 u 1....2x...0....1....1.....1...BCS

A208754 v 1....2x...0....1....1.....1...BO

A105045 u 1....2x...0....1....2x....1...BCCOS*

A208659 v 1....2x...0....1....2x....1...BDOSZ*

A208660 u 1....2x...0....1....x+1...1...CDS

A208904 v 1....2x...0....1....x+1...1...CNO

A208905 u 1....2x...0....x....1.....1...BCT

A208906 v 1....2x...0....x....1.....1...BNN

A208907 u 1....2x...0....x....x.....1...BCN

A208756 v 1....2x...0....x....x.....1...BCCE

A208755 u 1....2x...0....x....2x....1...CEN

A208910 v 1....2x...0....x....2x....1...CCE

A208911 u 1....2x...0....x....x+1...1...BCT

A208912 v 1....2x...0....x....x+1...1...BNT

A208913 u 1....2x...0....2x...1.....1...BCT

A208914 v 1....2x...0....2x...1.....1...BEN

A208915 u 1....2x...0....2x...x.....1...CE

A208916 v 1....2x...0....2x...x.....1...CCO

A208919 u 1....2x...0....2x...x+1...1...CT

A208920 v 1....2x...0....2x...x+1...1...N

A208917 u 1....2x...0....2x...2x....1...CEN

A208918 v 1....2x...0....2x...2x....1...CCNP

A208921 u 1....2x...0....x+1..1.....1...BC

A208922 v 1....2x...0....x+1..1.....1...BON

A208923 u 1....2x...0....x+1..x.....1...BCNO

A208908 v 1....2x...0....x+1..x.....1...BDN*

A208909 u 1....2x...0....x+1..2x....1...BN

A208930 v 1....2x...0....x+1..2x....1...DN

A208931 u 1....2x...0....x+1..x+1...1...BCOS

A208932 v 1....2x...0....x+1..x+1...1...BCO*

A207537 u 1....x+1..0....1....1.....0...BCO

A207538 v 1....x+1..0....1....1.....0...BCE

A122075 u 1....x+1..0....1....x.....0...CCFN*

A037027 v 1....x+1..0....1....x.....0...CCFN*

A209125 u 1....x+1..0....1....2x....0...BCFN*

A164975 v 1....x+1..0....1....2x....0...BF

A209126 u 1....x+1..0....x....x.....0...CDFO*

A209127 v 1....x+1..0....x....x.....0...DFOZ*

A209128 u 1....x+1..0....x....2x....0...CDE*

A209129 v 1....x+1..0....x....2x....0...DEZ

A102756 u 1....x+1..0....x....x+1...0...CFNP*

A209130 v 1....x+1..0....x....x+1...0...CCFNP*

A209131 u 1....x+1..0....2x...x.....0...CDEP*

A209132 v 1....x+1..0....2x...x.....0...CNPZ*

A209133 u 1....x+1..0....2x...2x....0...CDN

A209134 v 1....x+1..0....2x...2x....0...CCN*

A209135 u 1....x+1..0....2x...x+1...0...CN*

A209136 v 1....x+1..0....2x...x+1...0...CCS*

A209137 u 1....x+1..0....x+1..x.....0...CFFP*

A209138 v 1....x+1..0....x+1..x.....0...AFFP*

A209139 u 1....x+1..0....x+1..2x....0...CF*

A209140 v 1....x+1..0....x+1..2x....0...BF

A209141 u 1....x+1..0....x+1..x+1...0...BCF*

A209142 v 1....x+1..0....x+1..x+1...0...BFZ*

A209143 u 1....x+1..0....1....1.....1...CCE*

A209144 v 1....x+1..0....1....1.....1...COO*

A209145 u 1....x+1..0....1....x.....1...CCFN*

A122075 v 1....x+1..0....1....x.....1...CCFN*

A209146 u 1....x+1..0....1....2x....1...BCF*

A209147 v 1....x+1..0....1....2x....1...BF

A209148 u 1....x+1..0....1....x+1...1...CCO*

A209149 v 1....x+1..0....1....x+1...1...CDO*

A209150 u 1....x+1..0....x....1.....1...CCNT*

A208335 v 1....x+1..0....x....1.....1...CDNN*

A209151 u 1....x+1..0....x....x.....1...CFN*

A208337 v 1....x+1..0....x....x.....1...ACFN*

A209152 u 1....x+1..0....x....2x....1...CN*

A208339 v 1....x+1..0....x....x.....1...BCN

A209153 u 1....x+1..0....x....x+1...1...CFT*

A208340 v 1....x+1..0....x....x.....1...FNZ*

A209154 u 1....x+1..0....2x...1.....1...BCT*

A209157 v 1....x+1..0....2x...1.....1...BNN

A209158 u 1....x+1..0....2x...x.....1...CN*

A209159 v 1....x+1..0....2x...x.....1...CO*

A209160 u 1....x+1..0....2x...2x....1...CN*

A209161 v 1....x+1..0....2x...2x....1...CE

A209162 u 1....x+1..0....2x...x+1...1...CT*

A209163 v 1....x+1..0....2x...x+1...1...CO*

A209164 u 1....x+1..0....x+1..1.....1...CC*

A209165 v 1....x+1..0....x+1..1.....1...CCN

A209166 u 1....x+1..0....x+1..x.....1...CFF*

A209167 v 1....x+1..0....x+1..x.....1...FF*

A209168 u 1....x+1..0....x+1..2x....1...CF*

A209169 v 1....x+1..0....x+1..2x....1...CF

A209170 u 1....x+1..0....x+1..x+1...1...CF*

A209171 v 1....x+1..0....x+1..x+1...1...CF*

A053538 u x....1....0....1....1.....0...BBCCFN

A076791 v x....1....0....1....1.....0...BBCDF

A209172 u x....1....0....1....2x....0...BCCFF

A209413 v x....1....0....1....2x....0...BCCFF

A094441 u x....1....0....1....x+1...0...CFFFN

A094442 v x....1....0....1....x+1...0...CEFFF

A054142 u x....1....0....x....x+1...0...CCFOT*

A172431 v x....1....0....x....x+1...0...CEFN*

A008288 u x....1....0....2x...1.....0...CCOO*

A035607 v x....1....0....2x...1.....0...ACDE*

A209414 u x....1....0....2x...x+1...0...CCS

A112351 v x....1....0....2x...x+1...0...CON

A209415 u x....1....0....x+1..x.....0...CCTN

A209416 v x....1....0....x+1..x.....0...ACN*

A209417 u x....1....0....x+1..2x....0...CC

A209418 v x....1....0....x+1..2x....0...BBC

A209419 u x....1....0....x+1..x+1...0...CFTZ*

A209420 v x....1....0....x+1..x+1...0...FNZ*

A209421 u x....1....0....1....1.....1...CCN

A209422 v x....1....0....1....1.....1...CD

A209555 u x....1....0....1....x.....1...CNN

A209556 v x....1....0....1....x.....1...CNN

A209557 u x....1....0....1....2x....1...BCN

A209558 v x....1....0....1....2x....1...BN

A209559 u x....1....0....1....x+1...1...CN

A209560 v x....1....0....1....x+1...1...CN

A209561 u x....1....0....x....1.....1...CCNNT*

A209562 v x....1....0....x....1.....1...CDNNT*

A209563 u x....1....0....x....x.....1...CCFT^

A209564 v x....1....0....x....x.....1...CFN^

A209565 u x....1....0....x....2x....1...CC^

A209566 v x....1....0....x....2x....1...BC^

A209567 u x....1....0....x....x+1...1...CNT*

A209568 v x....1....0....x....x+1...1...NNS*

A209569 u x....1....0....2x...1.....1...CNO*

A209570 v x....1....0....2x...1.....1...DNN*

A209571 u x....1....0....2x...x.....1...CCS^

A209572 v x....1....0....2x...x.....1...CN^

A209573 u x....1....0....2x...x+1...1...CNS

A209574 v x....1....0....2x...x+1...1...NO

A209575 u x....1....0....2x...2x....1...CC

A209576 v x....1....0....2x...2x....1...C

A209577 u x....1....0....x+1..1.....1...CNNT

A209578 v x....1....0....x+1..1.....1...CNN

A209579 u x....1....0....x+1..x.....1...CNNT

A209580 v x....1....0....x+1..x.....1...NN*

A209581 u x....1....0....x+1..2x....1...CN

A209582 v x....1....0....x+1..2x....1...BN

A209583 u x....1....0....x+1..x+1...1...CT*

A209584 v x....1....0....x+1..x+1...1...CN*

A121462 u x....x....0....x....x+1...0...BCFFNZ

A208341 v x....x....0....x....x+1...0...BCFFN

A209687 u x....x....0....2x...x+1...0...BCNZ

A208339 v x....x....0....2x...x+1...0...BCN

A115241 u x....x....0....1....1.....1...CDNZ*

A209688 v x....x....0....1....1.....1...DDN*

A209689 u x....x....0....1....x.....1...FNZ^

A209690 v x....x....0....1....x.....1...FN^

A209691 u x....x....0....1....2x....1...BCZ^

A209692 v x....x....0....1....2x....1...BCC^

A209693 u x....x....0....1....x+1...1...NNZ*

A209694 v x....x....0....1....x+1...1...CN*

A209697 u x....x....0....x....x+1...1...BNZ

A209698 v x....x....0....x....x+1...1...BNT

A209699 u x....x....0....2x...1.....1...BNNZ

A209700 v x....x....0....2x...1.....1...BDN

A209701 u x....x....0....2x...x+1...1...NZ

A209702 v x....x....0....2x...x+1...1...N

A209703 u x....x....0....x+1..1.....1...FNTZ

A209704 v x....x....0....x+1..1.....1...FNNT

A209705 u x....x....0....x+1..x+1...1...BNZ*

A209706 v x....x....0....x+1..x+1...1...BCN*

A209695 u x....x+1..0....2x...x+1...0...ACN*

A209696 v x....x+1..0....2x...x+1...0...CDN*

A209830 u x....x+1..0....x+1..2x....0...ACF

A209831 v x....x+1..0....x+1..2x....0...BCF*

A209745 u x....x+1..0....x+1..x+1...0...ABF*

A209746 v x....x+1..0....x+1..x+1...0...BFZ*

A209747 u x....x+1..0....1....1.....1...ADE*

A209748 v x....x+1..0....1....1.....1...DEO

A209749 u x....x+1..0....1....x.....1...ANN*

A209750 v x....x+1..0....1....x.....1...CNO

A209751 u x....x+1..0....1....2x....1...ABN*

A209752 v x....x+1..0....1....2x....1...BN

A209753 u x....x+1..0....1....x+1...1...AN*

A209754 v x....x+1..0....1....x+1...1...NT*

A209755 u x....x+1..0....x....1.....1...AFN

A209756 v x....x+1..0....x....1.....1...FNO*

A209759 u x....x+1..0....x....2x....1...ACF^

A209760 v x....x+1..0....x....2x....1...CF^*

A209761 u x....x+1..0....x.....x+1..1...ABNS*

A209762 v x....x+1..0....x.....x+1..1...BNS*

A209763 u x....x+1..0....2x....1....1...ABN*

A209764 v x....x+1..0....2x....1....1...BNN

A209765 u x....x+1..0....2x....x....1...ACF^*

A209766 v x....x+1..0....2x....x....1...CF^

A209767 u x....x+1..0....2x....x+1..1...AN*

A209768 v x....x+1..0....2x....x+1..1...N*

A209769 u x....x+1..0....x+1...1....1...AF*

A209770 v x....x+1..0....x+1...1....1...FN

A209771 u x....x+1..0....x+1...x....1...ABN*

A209772 v x....x+1..0....x+1...x....1...BN*

A209773 u x....x+1..0....x+1...2x...1...AF

A209774 v x....x+1..0....x+1...2x...1...FN*

A209775 u x....x+1..0....x+1...x+1..1...AB*

A209776 v x....x+1..0....x+1...x+1..1...BC*

A210033 u 1....1....1....1.....x....1...BCN

A210034 v 1....1....1....1.....x....1...BCDFN

A210035 u 1....1....1....1.....2x...1...BBF

A210036 v 1....1....1....1.....2x...1...BBFF

A210037 u 1....1....1....1.....x+1..1...BCFFN

A210038 v 1....1....1....1.....x+1..1...BCFFN

A210039 u 1....1....1....x.....1....1...BCOT

A210040 v 1....1....1....x.....1....1...BCEN

A210042 u 1....1....1....x.....x....1...BCDEOT*

A124927 v 1....1....1....x.....x....1...BCDET*

A210041 u 1....1....1....x.....2x...1...BFO

A209758 v 1....1....1....x.....2x...1...BCFO

A210187 u 1....1....1....x.....x+1..1...DTF*

A210188 v 1....1....1....x.....x+1..1...DNF*

A210189 u 1....1....1....2x....1....1...BT

A210190 v 1....1....1....2x....1....1...BN

A210191 u 1....1....1....2x....x....1...CO*

A210192 v 1....1....1....2x....x....1...CCO*

A210193 u 1....1....1....2x....x+1..1...CPT

A210194 v 1....1....1....2x....x+1..1...CN

A210195 u 1....1....1....2x....2x...1...BOPT*

A210196 v 1....1....1....2x....2x...1...BCC*

A210197 u 1....1....1....x+1...1....1...BCOT

A210198 v 1....1....1....x+1...1....1...BCEN

A210199 u 1....1....1....x+1...x....1...DFT

A210200 v 1....1....1....x+1...x....1...DFO*

A210201 u 1....1....1....x+1...2x...1...BFP

A210202 v 1....1....1....x+1...2x...1...BF

A210203 u 1....1....1....x+1...x+1..1...BDOP

A210204 v 1....1....1....x+1...x+1..1...BCDN*

A210211 u x....1....1....1.....2x...1...BCFN

A210212 v x....1....1....1.....2x...1...BFN

A210213 u x....1....1....1.....x+1..1...CFFN

A210214 v x....1....1....1.....x+1..1...CFFO

A210215 u x....1....1....x.....x....1...BCDFT^

A210216 v x....1....1....x.....x....1...BCFO^

A210217 u x....1....1....x.....2x...1...CDF^

A210218 v x....1....1....x.....2x...1...BCF^

A210219 u x....1....1....x.....x+1..1...CNSTF*

A210220 v x....1....1....x.....x+1..1...FNNT*

A104698 u x....1....1....2x......1..1...CENS*

A210220 v x....1....1....2x....x+1..1...DNNT*

A210223 u x....1....1....2x....x....1...CD^

A210224 v x....1....1....2x....x....1...CO^

A210225 u x....1....1....2x....x+1..1...CNP

A210226 v x....1....1....2x....x+1..1...NOT

A210227 u x....1....1....2x....2x...1...CDP^

A210228 v x....1....1....2x....2x...1...C^

A210229 u x....1....1....x+1...1....1...CFNN

A210230 v x....1....1....x+1...1....1...CCN

A210231 u x....1....1....x+1...x....1...CNT

A210232 v x....1....1....x+1...x....1...NN*

A210233 u x....1....1....x+1...2x...1...CNP

A210234 v x....1....1....x+1...2x...1...BN

A210235 u x....1....1....x+1...x+1..1...CCFPT*

A210236 v x....1....1....x+1...x+1..1...CFN*

A124927 u x....x....1....1.....1....1...BCDEET*

A210042 v x....1....1....x+1...x+1..1...BDEOT*

A210216 u x....x....1....1.....x....1...BCFO^

A210215 v x....x....1....1.....x....1...BCDFT^

A210549 u x....x....1....1.....2x...1...BCF^

A210550 v x....x....1....1.....2x...1...BDF^

A172431 u x....x....1....1.....x+1..1...CEFN*

A210551 v x....x....1....1.....x+1..1...CFOT*

A210552 u x....x....1....x.....1....1...BBCFNO

A210553 v x....x....1....x.....1....1...BNNFB

A208341 u x....x....1....x.....x+1..1...BCFFN

A210554 v x....x....1....x.....x+1..1...BNFFT

A210555 u x....x....1....2x....1....1...BCNN

A210556 v x....x....1....2x....1....1...BENP

A210557 u x....x....1....2x....x+1..1...CNP

A210558 v x....x....1....2x....x+1..1...N

A210559 u x....x....1....x+1...1....1...CEF

A210560 v x....x....1....x+1...1....1...OFNS

A210561 u x....x....1....x+1...x....1...BCNP^

A210562 v x....x....1....x+1...x....1...BDP*^

A210563 u x....x....1....x+1...2x...1...CFP^

A210564 v x....x....1....x+1...2x...1...DF^

A013609 u x....x....1....x+1...x+1..1...BCEPT*

A209757 v x....x....1....x+1...x+1..1...BCOS*

A209819 u x....2x...1....x+1...x....1...CFN^

A209820 v x....2x...1....x+1...x....1...DF^

A209996 u x....2x...1....x+1...2x...1...CP^

A209998 v x....2x...1....x+1...2x...1...DP^

A209999 u x....x+1..1....1.....x+1..1...FN*

A210287 v x....x+1..1....1.....x+1..1...CFT*

A210565 u x....x+1..1....x.....1....1...FNT*

A210595 v x....x+1..1....x.....1....1...FNNT

A210598 u x....x+1..1....x+1...2x...1...FN*

A210599 v x....x+1..1....x+1...2x...1...FN

A210600 u x....x+1..1....x+1...x+1..1...BF*

A210601 v x....x+1..1....x+1...x+1..1...BF*

A210597 u 2x...1....1....x+1...1....1...BF

A210601 v 2x...1....1....x+1...1....1...BFN*

A210603 u 2x...1....1....x+1...x+1..1...BF

A210738 v 2x...1....1....x+1...x+1..1...CBF*

A210739 u 2x...x....1....x+1...x....1...CF^

A210740 v 2x...x....1....x+1...x....1...DF*^

A210741 u 2x...x....1....x+1...x+1..1...BCFO

A210742 v 2x...x....1....x+1...x+1..1...CFO*

A210743 u 2x...x+1..1....x+1...1....1...F

A210744 v 2x...x+1..1....x+1...1....1...FN

A210747 u 2x...x+1..1....x+1...x+1..1...FF

A210748 v 2x...x+1..1....x+1...x+1..1...CFF*

A210749 u x+1..1....1....x+1...2x...1...BCF

A210750 v x+1..1....1....x+1...2x...1...BF

A210751 u x+1..x....1....x+1...2x...1...FNT

A210752 v x+1..x....1....x+1...2x...1...FN

A210753 u x+1..x....1....x+1...x+1..1...BNZ*

A210754 v x+1..x....1....x+1...x+1..1...BCT*

A210755 u x+1..2x...1....x+1...x+1..1...N*

A210756 v x+1..2x...1....x+1...x+1..1...CT*

A210789 u 1....x....0....x+2...x-1..0...CFFN

A210790 v 1....x....0....x+2...x-1..0...CEFF

A210791 u 1....x....0....x-1...x+2..0...CFNP

A210792 v 1....x....0....x-1...x+2..0...CF

A210793 u 1....x+1..0....x+2...x-1..0...CFNP

A210794 v 1....x+1..0....x+2...x-1..0...FPP

A210795 u 1....x....1....x+2...x-1..0...FN

A210796 v 1....x....1....x+2...x-1..0...FO

A210797 u 1....x....0....x+2...x-1..1...CF

A210798 v 1....x....0....x+2...x-1..1...F

A210799 u 1....x+1..1....x+2...x-1..0...FN

A210800 v 1....x+1..1....x+2...x-1..0...F

A210801 u 1....x+1..1....x+2...x-1..1...FN

A210802 v 1....x+1..1....x+2...x-1..1...F

A210803 u 1....x....0....x-1...x+3..0...F*

A210804 v 1....x....0....x-1...x+3..0...F*

A210805 u 1....x....0....x+2...x-1.-1...CFFN

A210806 v 1....x....0....x+2...x-1.-1...FF

A210858 u 1....x....0....x+n...x....0...CFT*

A210859 v 1....x....0....x+n...x....0...FN*

A210860 u 1....x+1..0....x+n...x....0...F

A210861 v 1....x+1..0....x+n...x....0...F*

A210862 u 1....x....1....x+n-1.x....0...FN

A210863 v 1....x....1....x+n-1.x....0...FS

A210864 u 1....x....1....x+n...x....0...FN

A210865 v 1....x....1....x+n...x....0...FT

A210866 u 1....x....0....x+n...x...-x...CFT

A210867 v 1....x....0....x+n...x...-x...FN

A210868 u 1....x....0....x+1...x-1..0...BCFN

A210869 v 1....x....0....x+1...x-1..0...BBCFNZ

A210870 u 1....x....0....x+1...x-1..1...CFFN

A210871 v 1....x....0....x+1...x-1..1...CFF

A210872 u x....1...-1....x.....x....1...BDFZ^

A210873 v x....1...-1....x.....x....1...BCFN^

A210876 u x....1....1....x.....x....x...BCCF^

A210877 v x....1....1....x.....x....x...BDFNZ^

A210878 u x....2x...0....x+1...x....1...DFZ^

A210879 v x....2x...0....x+1...x....1...FC*^

Some of these triangles have irregular row lengths, making it difficult to retrieve individual rows/columns/diagonals without actually computing the recurrence. - Georg Fischer, Sep 04 2021

These are Hogben's central polygonal numbers denoted by

...2...

....P..

...4.n.

Numbers of the form (k^2+1)/2 for k odd.

(y(2x+1))^2 + (y(2x^2+2x))^2 = (y(2x^2+2x+1))^2. E.g., let y = 2, x = 1; (2(2+1))^2 + (2(2+2))^2 = (2(2+2+1))^2, (2(3))^2 + (2(4))^2 = (2(5))^2, 6^2 + 8^2 = 10^2, 36 + 64 = 100. - Glenn B. Cox (igloos_r_us(AT)canada.com), Apr 08 2002

a(n) is also the number of 3 X 3 magic squares with sum 3(n+1). - Sharon Sela (sharonsela(AT)hotmail.com), May 11 2002

For n > 0, a(n) is the smallest k such that zeta(2) - Sum_{i=1..k} 1/i^2 <= zeta(3) - Sum_{i=1..n} 1/i^3. - Benoit Cloitre, May 17 2002

Number of convex polyominoes with a 2 X (n+1) minimal bounding rectangle.

The prime terms are given by A027862. - Lekraj Beedassy, Jul 09 2004

First difference of a(n) is 4n = A008586(n). Any entry k of the sequence is followed by k + 2*(1 + sqrt(2k - 1)). - Lekraj Beedassy, Jun 04 2006

Integers of the form 1 + x + x^2/2 (generating polynomial is Schur's polynomial as in A127876). - Artur Jasinski, Feb 04 2007

If X is an n-set and Y and Z disjoint 2-subsets of X then a(n-4) is equal to the number of 4-subsets of X intersecting both Y and Z. - Milan Janjic, Aug 26 2007

Row sums of triangle A132778. - Gary W. Adamson, Sep 02 2007

Binomial transform of [1, 4, 4, 0, 0, 0, ...]; = inverse binomial transform of A001788: (1, 6, 24, 80, 240, ...). - Gary W. Adamson, Sep 02 2007

Narayana transform (A001263) of [1, 4, 0, 0, 0, ...]. Equals A128064 (unsigned) * [1, 2, 3, ...]. - Gary W. Adamson, Dec 29 2007

k such that the Diophantine equation x^3 - y^3 = x*y + k has a solution with y = x-1. If that solution is (x,y) = (m+1,m) then m^2 + (m+1)^2 = k. Note that this Diophantine equation is an elliptic curve and (m+1,m) is an integer point on it. - James R. Buddenhagen, Aug 12 2008

Numbers k such that (k, k, 2*k-2) are the sides of an isosceles triangle with integer area. Also, k such that 2*k-1 is a square. - James R. Buddenhagen, Oct 17 2008

a(n) is also the least weight of self-conjugate partitions having n+1 different odd parts. - Augustine O. Munagi, Dec 18 2008

Prefaced with a "1": (1, 1, 5, 13, 25, 41, ...) = A153869 * (1, 2, 3, ...). - Gary W. Adamson, Jan 03 2009

Prefaced with a "1": (1, 1, 5, 13, 25, 41, ...) where a(n) = 2n*(n-1)+1, all tuples of square numbers (X-Y, X, X+Y) are produced by ((m*(a(n)-2n))^2, (m*a(n))^2, (m*(a(n)+2n-2))^2) where m is a whole number. - Doug Bell, Feb 27 2009

Equals (1, 2, 3, ...) convolved with (1, 3, 4, 4, 4, ...). E.g., a(3) = 25 = (1, 2, 3, 4) dot (4, 4, 3, 1) = (4 + 8 + 9 + 4). - Gary W. Adamson, May 01 2009

The running sum of squares taken two at a time. - Al Hakanson (hawkuu(AT)gmail.com), May 18 2009

Equals the odd integers convolved with (1, 2, 2, 2, ...). - Gary W. Adamson, May 25 2009

Equals the triangular numbers convolved with [1, 2, 1, 0, 0, 0, ...]. - Gary W. Adamson & Alexander R. Povolotsky, May 29 2009

When the positive integers are written in a square array by diagonals as in A038722, a(n) gives the numbers appearing on the main diagonal. - Joshua Zucker, Jul 07 2009

The finite continued fraction [n,1,1,n] = (2n+1)/(2n^2 + 2n + 1) = (2n+1)/a(n); and the squares of the first two denominators of the convergents = a(n). E.g., the convergents and value of [4,1,1,4] = 1/4, 1/5, 2/9, 9/41 where 4^2 + 5^2 = 41. - Gary W. Adamson, Jul 15 2010

From Keith Tyler, Aug 10 2010: (Start)

Running sum of A008574.

Square open pyramidal number; that is, the number of elements in a square pyramid of height (n) with only surface and no bottom nodes. (End)

For k>0, x^4 + x^2 + k factors over the integers iff sqrt(k) is in this sequence. - James R. Buddenhagen, Aug 15 2010

Create the simple continued fraction from Pythagorean triples to get [2n + 1; 2n^2 + 2n, 2n^2 + 2n + 1]; its value equals the rational number 2n + 1 + a(n) / (4n^4 + 8n^3 + 6n^2 + 2n + 1). - J. M. Bergot, Sep 30 2011

a(n), n >= 1, has in its prime number factorization only primes of the form 4*k+1, i.e., congruent to 1 (mod 4) (see A002144). This follows from the fact that a(n) is a primitive sum of two squares and odd. See Theorem 3.20, p. 164, in the given Niven-Zuckerman-Montgomery reference. E.g., a(3) = 25 = 5^2, a(6) = 85 = 5*17. - Wolfdieter Lang, Mar 08 2012

From Ant King, Jun 15 2012: (Start)

a(n) is congruent to 1 (mod 4) for all n.

The digital roots of the a(n) form a purely periodic palindromic 9-cycle 1, 5, 4, 7, 5, 7, 4, 5, 1.

The units' digits of the a(n) form a purely periodic palindromic 5-cycle 1, 5, 3, 5, 1.

(End)

Number of integer solutions (x,y) of |x| + |y| <= n. Geometrically: number of lattice points inside a square with vertices (n,0), (0,-n), (-n,0), (0,n). - César Eliud Lozada, Sep 18 2012

(a(n)-1)/a(n) = 2*x / (1+x^2) where x = n/(n+1). Note that in this form, this is the velocity-addition formula according to the special theory of relativity (two objects traveling at 1/(n+1) slower than c relative to each other appear to travel at 1/a(n) less than c to a stationary observer). - Christian N. K. Anderson, May 20 2013 [Corrected by Rémi Guillaume, May 22 2025]

A geometric curiosity: the envelope of the circles x^2 + (y-a(n)/2)^2 = ((2n+1)/2)^2 is the parabola y = x^2, the n=0 circle being the osculating circle at the parabola vertex. - Jean-François Alcover, Dec 02 2013

Draw n ellipses in the plane (n>0), any 2 meeting in 4 points; a(n-1) gives the number of internal regions into which the plane is divided (cf. A051890, A046092); a(n-1) = A051890(n) - 1 = A046092(n-1) + 1. - Jaroslav Krizek, Dec 27 2013

a(n) is also, of course, the scalar product of the 2-vector (n, n+1) (or (n+1, n)) with itself. The unique inverse of (n, n+1) as vector in the Clifford algebra over the Euclidean 2-space is (1/a(n))(0, n, n+1, 0) (similarly for the other vector). In general the unique inverse of such a nonzero vector v (odd element in Cl_2) is v^(-1) = (1/|v|^2) v. Note that the inverse with respect to the scalar product is not unique for any nonzero vector. See the P. Lounesto reference, sects. 1.7 - 1.12, pp. 7-14. See also the Oct 15 2014 comment in A147973. - Wolfdieter Lang, Nov 06 2014

Subsequence of A004431, for n >= 1. - Bob Selcoe, Mar 23 2016

Numbers k such that 2k - 1 is a perfect square. - Juri-Stepan Gerasimov, Apr 06 2016

The number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 574", based on the 5-celled von Neumann neighborhood. - Robert Price, May 13 2016

a(n) is the first integer in a sum of (2*n + 1)^2 consecutive integers that equals (2*n + 1)^4. - Patrick J. McNab, Dec 24 2016

Central elements of odd-length rows of the triangular array of positive integers. a(n) is the mean of the numbers in the (2*n + 1)-th row of this triangle. - David James Sycamore, Aug 01 2018

Intersection of A000982 and A080827. - David James Sycamore, Aug 07 2018

An off-diagonal of the array of Delannoy numbers, A008288, (or a row/column when the array is shown as a square). As such, this is one of the crystal ball sequences. - Jack W Grahl, Feb 15 2021 and Shel Kaphan, Jan 18 2023

a(n) appears as a solution to a "Riddler Express" puzzle on the FiveThirtyEight website. The Jan 21 2022 issue (problem) and the Jan 28 2022 issue (solution) present the following puzzle and include a proof. - Fold a square piece of paper in half, obtaining a rectangle. Fold again to obtain a square with 1/4 the size of the original square. Then make n cuts through the folded paper. a(n) is the greatest number of pieces of the unfolded paper after the cutting. - Manfred Boergens, Feb 22 2022

a(n) is (1/6) times the number of 2 X 2 triangles in the n-th order hexagram with 12*n^2 cells. - Donghwi Park, Feb 06 2024

If k is a centered square number, its index in this sequence is n = (sqrt(2k-1)-1)/2. - Rémi Guillaume, Mar 30 2025.

Row sums of the symmetric triangle of odd numbers [1]; [1, 3, 1]; [1, 3, 5, 3, 1]; [1, 3, 5, 7, 5, 3, 1]; .... - Marco Zárate, Jun 15 2025

The brown rat (rattus norwegicus) breeds very quickly. It can give birth to other rats 7 times a year, starting at the age of three months. The average number of pups is 8. The present sequence gives the total number of rats, when the intervals are 12/7 of a year and a young rat starts having offspring at 24/7 of a year. - Hans Isdahl, Jan 26 2008

Numbers n such that tau(n) is odd where tau(x) denotes the Ramanujan tau function (A000594). - Benoit Cloitre, May 01 2003

If Y is a fixed 2-subset of a (2n+1)-set X then a(n-1) is the number of 3-subsets of X intersecting Y. - Milan Janjic, Oct 21 2007

Binomial transform of [1, 8, 8, 0, 0, 0, ...]; Narayana transform (A001263) of [1, 8, 0, 0, 0, ...]. - Gary W. Adamson, Dec 29 2007

All terms of this sequence are of the form 8k+1. For numbers 8k+1 which aren't squares see A138393. Numbers 8k+1 are squares iff k is a triangular number from A000217. And squares have form 4n(n+1)+1. - Artur Jasinski, Mar 27 2008

Sequence arises from reading the line from 1, in the direction 1, 25, ... and the line from 9, in the direction 9, 49, ..., in the square spiral whose vertices are the squares A000290. - Omar E. Pol, May 24 2008

Equals the triangular numbers convolved with [1, 6, 1, 0, 0, 0, ...]. - Gary W. Adamson & Alexander R. Povolotsky, May 29 2009

First differences: A008590(n) = a(n) - a(n-1) for n>0. - Reinhard Zumkeller, Nov 08 2009

Central terms of the triangle in A176271; cf. A000466, A053755. - Reinhard Zumkeller, Apr 13 2010

Odd numbers with odd abundance. Odd numbers with even abundance are in A088828. Even numbers with odd abundance are in A088827. Even numbers with even abundance are in A088829. - Jaroslav Krizek, May 07 2011

Appear as numerators in the non-simple continued fraction expansion of Pi-3: Pi-3 = K_{k>=1} (1-2*k)^2/6 = 1/(6+9/(6+25/(6+49/(6+...)))), see also the comment in A007509. - Alexander R. Povolotsky, Oct 12 2011

Ulam's spiral (SE spoke). - Robert G. Wilson v, Oct 31 2011

All terms end in 1, 5 or 9. Modulo 100, all terms are among { 1, 9, 21, 25, 29, 41, 49, 61, 69, 81, 89 }. - M. F. Hasler, Mar 19 2012

Right edge of both triangles A214604 and A214661: a(n) = A214604(n+1,n+1) = A214661(n+1,n+1). - Reinhard Zumkeller, Jul 25 2012

Also: Odd numbers which have an odd sum of divisors (= sigma = A000203). - M. F. Hasler, Feb 23 2013

Consider primitive Pythagorean triangles (a^2 + b^2 = c^2, gcd(a, b) = 1) with hypotenuse c (A020882) and respective even leg b (A231100); sequence gives values c-b, sorted with duplicates removed. - K. G. Stier, Nov 04 2013

For n>1 a(n) is twice the area of the irregular quadrilateral created by the points ((n-2)*(n-1),(n-1)*n/2), ((n-1)*n/2,n*(n+1)/2), ((n+1)*(n+2)/2,n*(n+1)/2), and ((n+2)*(n+3)/2,(n+1)*(n+2)/2). - J. M. Bergot, May 27 2014

Number of pairs (x, y) of Z^2, such that max(abs(x), abs(y)) <= n. - Michel Marcus, Nov 28 2014

Except for a(1)=4, the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 737", based on the 5-celled von Neumann neighborhood. - Robert Price, May 23 2016

a(n) is the sum of 2n+1 consecutive numbers, the first of which is n+1. - Ivan N. Ianakiev, Dec 21 2016

a(n) is the number of 2 X 2 matrices with all elements in {0..n} with determinant = 2*permanent. - Indranil Ghosh, Dec 25 2016

Engel expansion of Pi*StruveL_0(1)/2 where StruveL_0(1) is A197037. - Benedict W. J. Irwin, Jun 21 2018

Consider all Pythagorean triples (X,Y,Z=Y+1) ordered by increasing Z; the segments on the hypotenuse {p = a(n)/A001844(n), q = A060300(n)/A001844(n) = A001844(n) - p} and their ratio p/q = a(n)/A060300(n) are irreducible fractions in Q\Z. X values are A005408, Y values are A046092, Z values are A001844. - Ralf Steiner, Feb 25 2020

a(n) is the number of large or small squares that are used to tile primitive squares of type 2 (A344332). - Bernard Schott, Jun 03 2021

Also, positive odd integers with an odd number of odd divisors (for similar sequence with 'even', see A348005). - Bernard Schott, Nov 21 2021

a(n) is the least odd number k = x + y, with 0 < x < y, such that there are n distinct pairs (x,y) for which x*y/k is an integer; for example, a(2) = 25 and the two corresponding pairs are (5,20) and (10,15). The similar sequence with 'even' is A016742 (see Comment of Jan 26 2018). - Bernard Schott, Feb 24 2023

From Peter Bala, Jan 03 2024: (Start)

The sequence terms are the exponents of q in the series expansions of the following infinite products:

1) q*Product_{n >= 1} (1 - q^(16*n))*(1 + q^(8*n)) = q + q^9 + q^25 + q^49 + q^81 + q^121 + q^169 + ....

2) q*Product_{n >= 1} (1 + q^(16*n))*(1 - q^(8*n)) = q - q^9 - q^25 + q^49 + q^81 - q^121 - q^169 + + - - ....

3) q*Product_{n >= 1} (1 - q^(8*n))^3 = q - 3*q^9 + 5*q^25 - 7*q^49 + 9*q^81 - 11*q^121 + 13*q^169 - + ....

4) q*Product_{n >= 1} ( (1 + q^(8*n))*(1 - q^(16*n))/(1 + q^(16*n)) )^3 = q + 3*q^9 - 5*q^25 - 7*q^49 + 9*q^81 + 11*q^121 - 13*q^169 - 15*q^225 + + - - .... (End)

This sequence and A001969 give the unique solution to the problem of splitting the nonnegative integers into two classes in such a way that sums of pairs of distinct elements from either class occur with the same multiplicities [Lambek and Moser]. Cf. A000028, A000379.

In French: les nombres impies.

Has asymptotic density 1/2, since exactly 2 of the 4 numbers 4k, 4k+1, 4k+2, 4k+3 have an even sum of bits, while the other 2 have an odd sum. - Jeffrey Shallit, Jun 04 2002

Nim-values for game of mock turtles played with n coins.

A115384(n) = number of odious numbers <= n; A000120(a(n)) = A132680(n). - Reinhard Zumkeller, Aug 26 2007

Indices of 1's in the Thue-Morse sequence A010060. - Tanya Khovanova, Dec 29 2008

For any positive integer m, the partition of the set of the first 2^m positive integers into evil ones E and odious ones O is a fair division for any polynomial sequence p(k) of degree less than m, that is, Sum_{k in E} p(k) = Sum_{k in O} p(k) holds for any polynomial p with deg(p) < m. - Pietro Majer, Mar 15 2009

For n>1 let b(n) = a(n-1). Then b(b(n)) = 2b(n). - Benoit Cloitre, Oct 07 2010

Lexicographically earliest sequence of distinct nonnegative integers with no term being the binary exclusive OR of any terms. The equivalent sequence for addition or for subtraction is A005408 (the odd numbers) and for multiplication is A026424. - Peter Munn, Jan 14 2018

Numbers of the form m XOR (2*m+1) for some m >= 0. - Rémy Sigrist, Apr 14 2022

Numbers k such that the concatenation of the first k natural numbers is not divisible by 3. E.g., 16 is in the sequence because we have 123456789101111213141516 == 1 (mod 3).

Ignoring the first term, this sequence represents the number of bonds in a hydrocarbon: a(#of carbon atoms) = number of bonds. - Nathan Savir (thoobik(AT)yahoo.com), Jul 03 2003

n such that Sum_{k=0..n} (binomial(n+k,n-k) mod 2) is even (cf. A007306). - Benoit Cloitre, May 09 2004

Hilbert series for twisted cubic curve. - Paul Barry, Aug 11 2006

If Y is a 3-subset of an n-set X then, for n >= 3, a(n-3) is the number of 3-subsets of X having at least two elements in common with Y. - Milan Janjic, Nov 23 2007

a(n) = A144390 (1, 9, 23, 43, 69, ...) - A045944 (0, 5, 16, 33, 56, ...). From successive spectra of hydrogen atom. - Paul Curtz, Oct 05 2008

Number of monomials in the n-th power of polynomial x^3+x^2+x+1. - Artur Jasinski, Oct 06 2008

A145389(a(n)) = 1. - Reinhard Zumkeller, Oct 10 2008

Union of A035504, A165333 and A165336. - Reinhard Zumkeller, Sep 17 2009

Hankel transform of A076025. - Paul Barry, Sep 23 2009

From Jaroslav Krizek, May 28 2010: (Start)

a(n) = numbers k such that the antiharmonic mean of the first k positive integers is an integer.

A169609(a(n-1)) = 1. See A146535 and A169609. Complement of A007494.

See A005408 (odd positive integers) for corresponding values A146535(a(n)). (End)

Apart from the initial term, A180080 is a subsequence; cf. A180076. - Reinhard Zumkeller, Aug 14 2010

Also the maximum number of triangles that n + 2 noncoplanar points can determine in 3D space. - Carmine Suriano, Oct 08 2010

A089911(4*a(n)) = 3. - Reinhard Zumkeller, Jul 05 2013

The number of partitions of 6*n into at most 2 parts. - Colin Barker, Mar 31 2015

For n >= 1, a(n)/2 is the proportion of oxygen for the stoichiometric combustion reaction of hydrocarbon CnH2n+2, e.g., one part propane (C3H8) requires 5 parts oxygen to complete its combustion. - Kival Ngaokrajang, Jul 21 2015

Exponents n > 0 for which 1 + x^2 + x^n is reducible. - Ron Knott, Oct 13 2016

Also the number of independent vertex sets in the n-cocktail party graph. - Eric W. Weisstein, Sep 21 2017

Also the number of (not necessarily maximal) cliques in the n-ladder rung graph. - Eric W. Weisstein, Nov 29 2017

Also the number of maximal and maximum cliques in the n-book graph. - Eric W. Weisstein, Dec 01 2017

For n>=1, a(n) is the size of any snake-polyomino with n cells. - Christian Barrientos and Sarah Minion, Feb 27 2018

The sum of two distinct terms of this sequence is never a square. See Lagarias et al. p. 167. - Michel Marcus, May 20 2018

It seems that, for any n >= 1, there exists no positive integer z such that digit_sum(a(n)*z) = digit_sum(a(n)+z). - Max Lacoma, Sep 18 2019

For n > 2, a(n-2) is the number of distinct values of the magic constant in a normal magic triangle of order n (see formula 5 in Trotter). - Stefano Spezia, Feb 18 2021

Number of 3-permutations of n elements avoiding the patterns 132, 231, 312. See Bonichon and Sun. - Michel Marcus, Aug 20 2022

Erdős & Sárközy conjecture that a set of n positive integers with property P must have some element at least a(n-1) = 3n - 2. Property P states that, for x, y, and z in the set and z < x, y, z does not divide x+y. An example of such a set is {2n-1, 2n, ..., 3n-2}. Bedert proves this for large enough n. (This is an upper bound, and is exact for all known n; I have verified it for n up to 12.) - Charles R Greathouse IV, Feb 06 2023

a(n-1) = 3*n-2 is the dimension of the vector space of all n X n tridiagonal matrices, equals the number of nonzero coefficients: n + 2*(n-1) (see Wikipedia link). - Bernard Schott, Mar 03 2023

Denoted by Delta(n) or Delta_1(n) in Glaisher 1907. - Michael Somos, May 17 2013

A069289(n) <= a(n). - Reinhard Zumkeller, Apr 05 2015

A000203, A001227 and this sequence have the same parity: A053866. - Omar E. Pol, May 14 2016

For the g.f.s given below by Somos Oct 29 2005, Jovovic, Oct 11 2002 and Arndt, Nov 09 2010, see the Hardy-Wright reference, proof of Theorem 382, p. 312, with x^2 replaced by x. - Wolfdieter Lang, Dec 11 2016

a(n) is also the total number of parts in all partitions of n into an odd number of equal parts. - Omar E. Pol, Jun 04 2017

It seems that a(n) divides A000203(n) for every n. - Ivan N. Ianakiev, Nov 25 2017 [Yes, see the formula dated Dec 14 2017].

Also, alternating row sums of A126988. - Omar E. Pol, Feb 11 2018

Where a(n) shows the number of equivalence classes of Hurwitz quaternions with norm n (equivalence defined by right multiplication with one of the 24 Hurwitz units as in A055672), A046897(n) seems to give the number of equivalence classes of Lipschitz quaternions with norm n (equivalence defined by right multiplication with one of the 8 Lipschitz units). - R. J. Mathar, Aug 03 2025