cp's OEIS Frontend

Also triangle read by rows: T(n,k), n>=0, k>=0, in which n appears n+1 times in row n. - Omar E. Pol, Jul 15 2012

The PARI functions t1, t2 can be used to read a triangular array T(n,k) (n >= 0, 0 <= k <= n-1) by rows from left to right: n -> T(t1(n), t2(n)). - Michael Somos, Aug 23 2002

Number of terms in partition of n with greatest number of distinct terms. - Amarnath Murthy, May 20 2001

Summation table for (x+y) = (0+0),(0+1),(1+0),(0+2),(1+1),(2+0), ...

Also the number of triangular numbers less than or equal to n, not counting 0 as triangular. - Robert G. Wilson v, Oct 21 2005

Permutation of A116939: a(n) = A116939(A116941(n)), a(A116942(n)) = A116939(n). - Reinhard Zumkeller, Feb 27 2006

Maximal size of partitions of n into distinct parts, see A000009. - Reinhard Zumkeller, Jun 13 2009

Also number of digits of A000462(n). - Reinhard Zumkeller, Mar 27 2011

Also the maximum number of 1's contained in the list of hook-lengths of a partition of n. E.g., a(4)=2 because hooks of partitions of n=4 comprise {4,3,2,1}, {4,2,1,1}, {3,2,2,1}, {4,1,2,1}, {4,3,2,1} where the number of 1's in each is 1,2,1,2,1. Hence the maximum is 2. - T. Amdeberhan, Jun 03 2012

Fan, Yang, and Yu (2012) prove a conjecture of Amdeberhan on the generating function of a(n). - Jonathan Sondow, Dec 17 2012

Also the number of partitions of n into distinct parts p such that max(p) - min(p) <= length(p). - Clark Kimberling, Apr 18 2014

Also the maximum number of occurrences of any single value among the previous terms. - Ivan Neretin, Sep 20 2015

Where records occur gives A000217. - Omar E. Pol, Nov 05 2015

Also number of peaks in the largest Dyck path of the symmetric representation of sigma(n), n >= 1. Cf. A237593. - Omar E. Pol, Dec 19 2016

T(n,k) is the number of cells in the k-th region of the n-th set of regions in a diagram of the symmetry of sigma(n), see example.

Row n is a palindromic composition of sigma(n).

Row sums give A000203.

Row n has length A237271(n).

In the row 2n-1 of triangle both the first term and the last term are equal to n.

If n is an odd prime then row n is [m, m], where m = (1 + n)/2.

The connection with A196020 is as follows: A196020 --> A236104 --> A235791 --> A237591 --> A237593 --> A239660 --> this sequence.

For the boundary segments in an octant see A237591.

For the boundary segments in a quadrant see A237593.

For the boundary segments in the spiral see also A239660.

For the parts in every quadrant of the spiral see A239931, A239932, A239933, A239934.

We can find the spiral on the terraces of the stepped pyramid described in A244050. - Omar E. Pol, Dec 07 2016

T(n,k) is also the area of the k-th terrace, from left to right, at the n-th level, starting from the top, of the stepped pyramid described in A245092 (see Links section). - Omar E. Pol, Aug 14 2018

The original name was: Triangle read by rows: T(n,k) = A235791(n,k) - A235791(n,k+1), assuming that the virtual right border of triangle A235791 is A000004.

T(n,k) is also the length of the k-th segment in a zig-zag path on the first quadrant of the square grid, connecting the point (n, 0) with the point (m, m), starting with a segment in vertical direction, where m <= n.

Conjecture: the area of the polygon defined by the x-axis, this zig-zag path and the diagonal [(0, 0), (m, m)], is equal to A024916(n)/2, one half of the sum of all divisors of all positive integers <= n. Therefore the reflected polygon, which is adjacent to the y-axis, with the zig-zag path connecting the point (0, n) with the point (m, m), has the same property. And so on for each octant in the four quadrants.

For the representation of A024916 and A000203 we use two octants, for example: the first octant and the second octant, or the 6th octant and the 7th octant, etc., see A237593.

At least up to n = 128, two zig-zag paths never cross (checked by hand).

The finite sequence formed by the n-th row of triangle together with its mirror row gives the n-th row of triangle A237593.

The connection between A196020 and A237271 is as follows: A196020 --> A236104 --> A235791 --> this sequence --> A237593 --> A239660 --> A237270 --> A237271.

Comments from Franklin T. Adams-Watters on sequences related to the "symmetric representation of sigma" in A235791 and related sequences, Mar 31 2014. (Start)

The place to start is with A235791, which is very simple. Then go to A237591, also very simple, and A237593, still very simple.

You then need to interpret the rows of A237593 as Dyck paths. This interpretation is in terms of run lengths, so 2,1,1,2 means up twice, down once, up once, and down twice. Because the rows of A237593 are symmetric and of even length, this path will always be symmetric.

Now the surprising fact is that the areas enclosed by the Dyck path for n (laid on its side) always includes the area enclosed for n-1; and the number of squares added is sigma(n).

Finally, look at the connected areas enclosed by n but not by n-1; the size of these areas is the symmetric representation of sigma. (End)

From Hartmut F. W. Hoft, Apr 07 2014: (Start)

The row sum is A235791(n,1) - A235791(n,floor((sqrt(8n+1)-1)/2)+1) = n - 0.

Mathematica function has been written to check the conjecture as well as non-crossing zig-zag paths (Dyck paths rotated by 90 degrees) up through n=30000 (same applies to A237593). (End)

The n-th zig-zag path ending at the point (m, m), where m = A240542(n). - Omar E. Pol, Apr 16 2014

From Omar E. Pol, Aug 23 2015: (Start)

n is an odd prime if and only if T(n,2) = 1 + T(n-1,2) and T(n,k) = T(n-1,k) for the rest of the values of k.

The elements of the n-th row of triangle together with the elements of the n-th row of triangle A261350 give the n-th row of triangle A237593.

T(n,k) is also the area (or the number of cells) of the k-th vertical side at the n-th level (starting from the top) in the left hand part of the front view of the stepped pyramid described in A245092, see Example section.

(End)

From Omar E. Pol, Nov 19 2015: (Start)

T(n,k) is also the number of cells between the k-th and the (k+1)st line segments (from left to right) in the n-th row of the diagram as shown in Example section.

Note that the number of horizontal line segments in the n-th row of the diagram equals A001227(n), the number of odd divisors of n. (End)

Conjecture: the values f(n,k) in the n-th row of the triangle are either 1 or 2 for all k with ceiling((sqrt(4*n+1)-1)/2) <= k <= floor((sqrt(8*n+1)-1)/2) = r(n), the length of the n-th row, though the lower bound need not be minimal; tested through 2500000. See also A285356. - Hartmut F. W. Hoft, Apr 17 2017

Conjecture: T(n,k) is the difference between the total number of partitions of all positive integers <= n into exactly k consecutive parts, and the total number of partitions of all positive integers <= n into exactly k+1 consecutive parts. - Omar E. Pol, Apr 30 2017

From Omar E. Pol, Aug 31 2021: (Start)

It appears that T(n,2)/T(n,1) converges to 1/3.

It appears that T(n,3)/T(n,2) converges to 1/2.

It appears that T(n,4)/T(n,3) converges to 3/5.

It appears that T(n,5)/T(n,4) converges to 2/3. (End)

In other words: T(n,k) is the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(n). - Omar E. Pol, Sep 08 2021

For n >= 1, a(n) is the maximal number of pieces that can be obtained by cutting an annulus with n cuts. See illustration. - Robert G. Wilson v

n(n-3)/2 (n >= 3) is the number of diagonals of an n-gon. - Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr)

n(n-3)/2 (n >= 4) is the degree of the third-smallest irreducible presentation of the symmetric group S_n (cf. James and Kerber, Appendix 1).

a(n) is also the multiplicity of the eigenvalue (-2) of the triangle graph Delta(n+1). (See p. 19 in Biggs.) - Felix Goldberg (felixg(AT)tx.technion.ac.il), Nov 25 2001

For n > 3, a(n-3) = dimension of the traveling salesman polytope T(n). - Benoit Cloitre, Aug 18 2002

Also counts quasi-dominoes (quasi-2-ominoes) on an n X n board. Cf. A094170-A094172. - Jon Wild, May 07 2004

Coefficient of x^2 in (1 + x + 2*x^2)^n. - Michael Somos, May 26 2004

a(n) is the number of "prime" n-dimensional polyominoes. A "prime" n-polyomino cannot be formed by connecting any other n-polyominoes except for the n-monomino and the n-monomino is not prime. E.g., for n=1, the 1-monomino is the line of length 1 and the only "prime" 1-polyominoes are the lines of length 2 and 3. This refers to "free" n-dimensional polyominoes, i.e., that can be rotated along any axis. - Bryan Jacobs (bryanjj(AT)gmail.com), Apr 01 2005

Solutions to the quadratic equation q(m, r) = (-3 +- sqrt(9 + 8(m - r))) / 2, where m - r is included in a(n). Let t(m) = the triangular number (A000217) less than some number k and r = k - t(m). If k is neither prime nor a power of two and m - r is included in A000096, then m - q(m, r) will produce a value that shares a divisor with k. - Andrew S. Plewe, Jun 18 2005

Sum_{k=2..n+1} 4/(k*(k+1)*(k-1)) = ((n+3)*n)/((n+2)*(n+1)). Numerator(Sum_{k=2..n+1} 4/(k*(k+1)*(k-1))) = (n+3)*n/2. - Alexander Adamchuk, Apr 11 2006

Number of rooted trees with n+3 nodes of valence 1, no nodes of valence 2 and exactly two other nodes. I.e., number of planted trees with n+2 leaves and exactly two branch points. - Theo Johnson-Freyd (theojf(AT)berkeley.edu), Jun 10 2007

If X is an n-set and Y a fixed 2-subset of X then a(n-2) is equal to the number of (n-2)-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007

For n >= 1, a(n) is the number of distinct shuffles of the identity permutation on n+1 letters with the identity permutation on 2 letters (12). - Camillia Smith Barnes, Oct 04 2008

If s(n) is a sequence defined as s(1) = x, s(n) = kn + s(n-1) + p for n > 1, then s(n) = a(n-1)*k + (n-1)*p + x. - Gary Detlefs, Mar 04 2010

The only primes are a(1) = 2 and a(2) = 5. - Reinhard Zumkeller, Jul 18 2011

a(n) = m such that the (m+1)-th triangular number minus the m-th triangular number is the (n+1)-th triangular number: (m+1)(m+2)/2 - m(m+1)/2 = (n+1)(n+2)/2. - Zak Seidov, Jan 22 2012

For n >= 1, number of different values that Sum_{k=1..n} c(k)*k can take where the c(k) are 0 or 1. - Joerg Arndt, Jun 24 2012

On an n X n chessboard (n >= 2), the number of possible checkmate positions in the case of king and rook versus a lone king is 0, 16, 40, 72, 112, 160, 216, 280, 352, ..., which is 8*a(n-2). For a 4 X 4 board the number is 40. The number of positions possible was counted including all mirror images and rotations for all four sides of the board. - Jose Abutal, Nov 19 2013

If k = a(i-1) or k = a(i+1) and n = k + a(i), then C(n, k-1), C(n, k), C(n, k+1) are three consecutive binomial coefficients in arithmetic progression and these are all the solutions. There are no four consecutive binomial coefficients in arithmetic progression. - Michael Somos, Nov 11 2015

a(n-1) is also the number of independent components of a symmetric traceless tensor of rank 2 and dimension n >= 1. - Wolfdieter Lang, Dec 10 2015

Numbers k such that 8k + 9 is a square. - Juri-Stepan Gerasimov, Apr 05 2016

Let phi_(D,rho) be the average value of a generic degree D monic polynomial f when evaluated at the roots of the rho-th derivative of f, expressed as a polynomial in the averaged symmetric polynomials in the roots of f. [See the Wojnar et al. link] The "last" term of phi_(D,rho) is a multiple of the product of all roots of f; the coefficient is expressible as a polynomial h_D(N) in N:=D-rho. These polynomials are of the form h_D(N)= ((-1)^D/(D-1)!)*(D-N)*N^chi*g_D(N) where chi = (1 if D is odd, 0 if D is even) and g_D(N) is a monic polynomial of degree (D-2-chi). Then a(n) are the negated coefficients of the next to the highest order term in the polynomials N^chi*g_D(N), starting at D=3. - Gregory Gerard Wojnar, Jul 19 2017

For n >= 2, a(n) is the number of summations required to solve the linear regression of n variables (n-1 independent variables and 1 dependent variable). - Felipe Pedraza-Oropeza, Dec 07 2017

For n >= 2, a(n) is the number of sums required to solve the linear regression of n variables: 5 for two variables (sums of X, Y, X^2, Y^2, X*Y), 9 for 3 variables (sums of X1, X2, Y1, X1^2, X1*X2, X1*Y, X2^2, X2*Y, Y^2), and so on. - Felipe Pedraza-Oropeza, Jan 11 2018

a(n) is the area of a triangle with vertices at (n, n+1), ((n+1)*(n+2)/2, (n+2)*(n+3)/2), ((n+2)^2, (n+3)^2). - J. M. Bergot, Jan 25 2018

Number of terms less than 10^k: 1, 4, 13, 44, 140, 446, 1413, 4471, 14141, 44720, 141420, 447213, ... - Muniru A Asiru, Jan 25 2018

a(n) is also the number of irredundant sets in the (n+1)-path complement graph for n > 2. - Eric W. Weisstein, Apr 11 2018

a(n) is also the largest number k such that the largest Dyck path of the symmetric representation of sigma(k) has exactly n peaks, n >= 1. (Cf. A237593.) - Omar E. Pol, Sep 04 2018

For n > 0, a(n) is the number of facets of associahedra. Cf. A033282 and A126216 and their refinements A111785 and A133437 for related combinatorial and analytic constructs. See p. 40 of Hanson and Sha for a relation to projective spaces and string theory. - Tom Copeland, Jan 03 2021

For n > 0, a(n) is the number of bipartite graphs with 2n or 2n+1 edges, no isolated vertices, and a stable set of cardinality 2. - Christian Barrientos, Jun 13 2022

For n >= 2, a(n-2) is the number of permutations in S_n which are the product of two different transpositions of adjacent points. - Zbigniew Wojciechowski, Mar 31 2023

a(n) represents the optimal stop-number to achieve the highest running score for the Greedy Pig game with an (n-1)-sided die with a loss on a 1. The total at which one should stop is a(s-1), e.g. for a 6-sided die, one should pass the die at 20. See Sparks and Haran. - Nicholas Stefan Georgescu, Jun 09 2024

These are the squares of the entries of the triangle in A235791: T(n,k) = (A235791(n,k))^2.

Row n has length A003056(n) hence the first element of column k is in row A000217(k).

Columns 1-3 (including the initial zeros) are A000290, A008794, A211547.

Also column k lists the partial sums of the k-th column of triangle A196020 which gives an identity for sigma.

Since all the elements of this sequence are squares, we can draw an illustration of the alternating sum of row n step by step, and a symmetric diagram for A000203, A024916, A004125; see example.

For more information about the diagram see A237593.

The alternating sum of the squares of the elements of the n-th row equals the sum of all divisors of all positive integers <= n, i.e., Sum_{k=1..A003056(n)} (-1)^(k-1)*(T(n,k))^2 = A024916(n).

Row n has length A003056(n) hence the first element of column k is in row A000217(k).

For more information see A236104.

The sum of row n gives A060831(n), the sum of the number of odd divisors of all positive integers <= n. - Omar E. Pol, Mar 01 2014. [An equivalent assertion is that the sum of row n of A237048 is the number of odd divisors of n, and this was proved by Hartmut F. W. Hoft in a comment in A237048. - N. J. A. Sloane, Dec 07 2020]

Comments from Franklin T. Adams-Watters on sequences related to the "symmetric representation of sigma" in A235791 and related sequences, Mar 31 2014: (Start)

The place to start is with A235791, which is very simple. Then go to A237591, also very simple, and A237593, still very simple.

You then need to interpret the rows of A237593 as Dyck paths. This interpretation is in terms of run lengths, so 2,1,1,2 means up twice, down once, up once, and down twice. Because the rows of A237593 are symmetric and of even length, this path will always be symmetric.

Now the surprising fact is that the areas enclosed by the Dyck path for n (laid on its side) always includes the area enclosed for n-1; and the number of squares added is sigma(n).

Finally, look at the connected areas enclosed by n but not by n-1; the size of these areas is the symmetric representation of sigma. (End)

From Hartmut F. W. Hoft, Apr 07 2014: (Start)

Mathematica function has been written to check the first property up to n = 20000.

T(n,(sqrt(8n+1)-1)/2+1) = 0 for all n >= 1, which is useful for formulas for A237591 and A237593. (End)

Alternating row sums give A240542. - Omar E. Pol, Apr 16 2014

Conjecture: T(n,k) is also the total number of partitions of all positive integers <= n into exactly k consecutive parts, i.e., the partial column sum of A285898, or in accordance with the triangles of the same family: the partial column sum of A237048. - Omar E. Pol, Apr 28 2017, Nov 24 2020

The above conjecture is true. The proof will be added soon (it uses the generating function for the columns). - N. J. A. Sloane, Nov 24 2020

T(n,k) is also the total length of all line segments between the k-th vertex and the central vertex of the largest Dyck path of the symmetric representation of sigma(n). In other words: T(n,k) is the sum of the last (A003056(n)-k+1) terms of the n-th row of A237591. - Omar E. Pol, Sep 07 2021

T(n,k) is also the Manhattan distance between the k-th vertex and the central vertex of the Dyck path described in the n-th row of the triangle A237593. - Omar E. Pol, Jan 11 2023

Gives an identity for sigma(n): alternating sum of row n equals the sum of divisors of n. For proof see Max Alekseyev link.

Row n has length A003056(n) hence column k starts in row A000217(k).

The number of positive terms in row n is A001227(n), the number of odd divisors of n.

If n = 2^j then the only positive integer in row n is T(n,1) = 2^(j+1) - 1.

If n is an odd prime then the only two positive integers in row n are T(n,1) = 2n - 1 and T(n,2) = n - 2.

If T(n,k) = 3 then T(n+1,k+1) = 1, the first element of the column k+1.

The partial sums of column k give the column k of A236104.

The connection with the symmetric representation of sigma is as follows: A236104 --> A235791 --> A237591 --> A237593 --> A239660 --> A237270.

Alternating sum of row n equals the number of units cubes that protrude from the n-th level of the stepped pyramid described in A245092. - Omar E. Pol, Oct 28 2015

Conjecture: T(n,k) is the difference between the square of the total number of partitions of all positive integers <= n into exactly k consecutive parts, and the square of the total number of partitions of all positive integers < n into exactly k consecutive parts. - Omar E. Pol, Feb 14 2018

From Omar E. Pol, Nov 24 2020: (Start)

T(n,k) is also the number of steps in the first n levels of the k-th double-staircase that has at least one step in the n-th level of the "Double- staircases" diagram, otherwise T(n,k) = 0, (see the Example section).

For the connection with A280851 see also the algorithm of A280850 and the conjecture of A296508. (End)

The number of zeros in the n-th row equals A238005(n). - Omar E. Pol, Sep 11 2021

Apart from the alternating row sums and the sum of divisors function A000203 another connection with Euler's pentagonal theorem is that in the irregular triangle of A238442 the k-th column starts in the row that is the k-th generalized pentagonal number A001318(k) while here the k-th column starts in the row that is the k-th generalized hexagonal number A000217(k). Both A001318 and A000217 are successive members of the same family: the generalized polygonal numbers. - Omar E. Pol, Sep 23 2021

Other triangle with the same row lengths and alternating row sums equals sigma(n) is A252117. - Omar E. Pol, May 03 2022

The diagram of the symmetry of sigma has been via A196020 --> A236104 --> A235791 --> A237591 --> A237593.

For more information see A237270.

a(n) is also the number of terraces at n-th level (starting from the top) of the stepped pyramid described in A245092. - Omar E. Pol, Apr 20 2016

a(n) is also the number of subparts in the first layer of the symmetric representation of sigma(n). For the definion of "subpart" see A279387. - Omar E. Pol, Dec 08 2016

Note that the number of subparts in the symmetric representation of sigma(n) equals A001227(n), the number of odd divisors of n. (See the second example). - Omar E. Pol, Dec 20 2016

From Hartmut F. W. Hoft, Dec 26 2016: (Start)

Using odd prime number 3, observe that the 1's in the 3^k-th row of the irregular triangle of A237048 are at index positions

3^0 < 2*3^0 < 3^1 < 2*3^1 < ... < 2*3^((k-1)/2) < 3^(k/2) < ...

the last being 2*3^((k-1)/2) when k is odd and 3^(k/2) when k is even. Since odd and even index positions alternate, each pair (3^i, 2*3^i) specifies one part in the symmetric representation with a center part present when k is even. A straightforward count establishes that the symmetric representation of 3^k, k>=0, has k+1 parts. Since this argument is valid for any odd prime, every positive integer occurs infinitely many times in the sequence. (End)

a(n) = number of runs of consecutive nonzero terms in row n of A262045. - N. J. A. Sloane, Jan 18 2021

Indices of odd terms give A071562. Indices of even terms give A071561. - Omar E. Pol, Feb 01 2021

a(n) is also the number of prisms in the three-dimensional version of the symmetric representation of k*sigma(n) where k is the height of the prisms, with k >= 1. - Omar E. Pol, Jul 01 2021

With a(1) = 0; a(n) is also the number of parts in the symmetric representation of A001065(n), the sum of aliquot parts of n. - Omar E. Pol, Aug 04 2021

The parity of this sequence is also the characteristic function of numbers that have middle divisors. - Omar E. Pol, Sep 30 2021

a(n) is also the number of polycubes in the 3D-version of the ziggurat of order n described in A347186. - Omar E. Pol, Jun 11 2024

Conjecture 1: a(n) is the number of odd divisors of n except the "e" odd divisors described in A005279. Thus a(n) is the length of the n-th row of A379288. - Omar E. Pol, Dec 21 2024

The conjecture 1 was checked up n = 10000 by Amiram Eldar. - Omar E. Pol, Dec 22 2024

The conjecture 1 is true. For a proof see A379288. - Hartmut F. W. Hoft, Jan 21 2025

From Omar E. Pol, Jul 31 2025: (Start)

Conjecture 2: a(n) is the number of 2-dense sublists of divisors of n.

We call "2-dense sublists of divisors of n" to the maximal sublists of divisors of n whose terms increase by a factor of at most 2.

In a 2-dense sublist of divisors of n the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of n.

Example: for n = 10 the list of divisors of 10 is [1, 2, 5, 10]. There are two 2-dense sublists of divisors of 10, they are [1, 2], [5, 10], so a(10) = 2.

The conjecture 2 is essentially the same as the second conjecture in the Comments of A384149. See also Peter Munn's formula in A237270.

The indices where a(n) = 1 give A174973 (2-dense numbers). See the proof there. (End)

Conjecture 3: a(n) is the number of divisors p of n such that p is greater than twice the adjacent previous divisor of n. The divisors p give the n-th row of A379288. - Omar E. Pol, Aug 02 2025

Consider an irregular stepped pyramid with n steps. The base of the pyramid is equal to the symmetric representation of A024916(n), the sum of all divisors of all positive integers <= n. Two of the faces of the pyramid are the same as the representation of the n-th triangular numbers as a staircase. The total area of the pyramid is equal to 2*A024916(n) + A046092(n). The volume is equal to A175254(n). By definition a(2n-1) is A000203(n), the sum of divisors of n. Starting from the top a(2n-1) is also the total area of the horizontal part of the n-th step of the pyramid. By definition, a(2n) = A005843(n) = 2n. Starting from the top, a(2n) is also the total area of the irregular vertical part of the n-th step of the pyramid.

On the other hand the sequence also has a symmetric representation in two dimensions, see Example.

From Omar E. Pol, Dec 31 2016: (Start)

We can find the pyramid after the following sequences: A196020 --> A236104 --> A235791 --> A237591 --> A237593.

The structure of this infinite pyramid arises after the 90-degree-zig-zag folding of the diagram of the isosceles triangle A237593 (see the links).

The terraces at the m-th level of the pyramid are also the parts of the symmetric representation of sigma(m), m >= 1, hence the sum of the areas of the terraces at the m-th level equals A000203(m).

Note that the stepped pyramid is also one of the 3D-quadrants of the stepped pyramid described in A244050.

For more information about the pyramid see A237593 and all its related sequences. (End)

Row sums of triangle A128489. E.g., a(5) = 15 = (10 + 3 + 1 + 1), sum of row 4 terms of triangle A128489. - Gary W. Adamson, Jun 03 2007

Row sums of triangle A134867. - Gary W. Adamson, Nov 14 2007

a(10^4) = 82256014, a(10^5) = 8224740835, a(10^6) = 822468118437, a(10^7) = 82246711794796; see A072692. - M. F. Hasler, Nov 22 2007

Equals row sums of triangle A158905. - Gary W. Adamson, Mar 29 2009

n is prime if and only if a(n) - a(n-1) - 1 = n. - Omar E. Pol, Dec 31 2012

Also the alternating row sums of A236104. - Omar E. Pol, Jul 21 2014

a(n) is also the total number of parts in all partitions of the positive integers <= n into equal parts. - Omar E. Pol, Apr 30 2017

a(n) is also the total area of the terraces of the stepped pyramid with n levels described in A245092. - Omar E. Pol, Nov 04 2017

a(n) is also the area under the Dyck path described in the n-th row of A237593 (see example). - Omar E. Pol, Sep 17 2018

From Omar E. Pol, Feb 17 2020: (Start)

Convolution of A340793 and A000027.

Convolved with A340793 gives A000385. (End)

a(n) is also the number of cubic cells (or cubes) in the n-th level starting from the top of the stepped pyramid described in A245092. - Omar E. Pol, Jan 12 2022