cp's OEIS Frontend

Number of edges in the line graph of the complete bipartite graph of order 2n, L(K_n,n). - Roberto E. Martinez II, Jan 07 2002

Number of edges of the Cartesian product of two complete graphs K_n X K_n. - Roberto E. Martinez II, Jan 07 2002

That is, number of edges in the n X n rook graph. - Eric W. Weisstein, Jun 20 2017

n such that x^3 + x^2 + n factors over the integers. - James R. Buddenhagen, Apr 19 2005

Also the number of triangles in a 2 X n grid of points and therefore also (n choose 2) * (n choose 1) * 2, or (2n choose 3) - 2*(n choose 3). - Joshua Zucker, Jan 11 2006

Nonnegative X values of solutions to the equation (X-Y)^3-XY=0. To find Y values: b(n)=(n+1)*n^2 (see A011379). I proved that, if(X,Y) is different from (0,0) and m=2, 4, 6, 8, 10, 12,..., then the equation (X-Y)^m-XY=0,... has no solution. - Mohamed Bouhamida, May 10 2006

For n>=1, a(n) is equal to the number of functions f:{1,2,3}->{1,2,...,n} such that for a fixed x in {1,2,3} and a fixed y in {1,2,...,n} we have f(x)<>y. - Aleksandar M. Janjic and Milan Janjic, Mar 13 2007

a(n) equals the coefficient of log(2) in 2F1(n-1,n-1,n+1,-1). - John M. Campbell, Jul 16 2011

Define the infinite square array m(n,k) = (n-k)^2 for 1<=k<=n below the diagonal and m(n,k) = (k+n)(k-n) for 1<=n<=k above the diagonal. Then a(n) = Sum_{k=1..n} m(n,k) + Sum_{r=1..n} m(r,n), the "hook sum" of the terms left from m(n,n) and above m(n,n). - J. M. Bergot, Aug 16 2013

Partial sums of A049451. - Bruno Berselli, Feb 10 2014

Volume of an extruded rectangular brick with side lengths n, n and n-1. - Luciano Ancora, Jun 24 2015

(1) a(n) = sum of second string of n triangular numbers - sum of first n triangular numbers, or the 2n-th partial sum of triangular numbers (A000217) - the n-th partial sum of triangular numbers (A000217). The same for natural numbers gives squares. (2) a(n) = (n-th triangular number)*(the n-th even number) = n(n+1)/2 * (2n). - Amarnath Murthy, Nov 05 2002

Let M(n) be the n X n matrix m(i,j)=1/(i+j+x), let P(n,x) = (Product_{i=0..n-1} i!^2)/det(M(n)). Then P(n,x) is a polynomial with integer coefficients of degree n^2 and a(n) is the coefficient of x^(n^2-1). - Benoit Cloitre, Jan 15 2003

Y values of solutions of the equation: (X-Y)^3-X*Y=0. X values are a(n)=n*(n+1)^2 (see A045991) - Mohamed Bouhamida, May 09 2006

a(2d-1) is the number of self-avoiding walk of length 3 in the d-dimensional hypercubic lattice. - Michael Somos, Sep 06 2006

a(n) mod 10 is periodic 5: repeat [0, 2, 2, 6, 0]. - Mohamed Bouhamida, Sep 05 2009

This sequence is related to A005449 by a(n) = n*A005449(n)-sum(A005449(i), i=0..n-1), and this is the case d=3 in the identity n^2*(d*n+d-2)/2 - Sum_{k=0..n-1} k*(d*k+d-2)/2 = n*(n+d)*(2*d*n+d-3)/6. - Bruno Berselli, Nov 18 2010

Using (n, n+1) to generate a primitive Pythagorean triangle, the sides will be 2*n+1, 2*(n^2+n), and 2*n^2+2*n+1. Inscribing the largest rectangle with integral sides will have sides of length n and n^2+n. Side n is collinear to side 2*n+1 of the triangle and side n^2+n is collinear to side 2*(n^2+n) of the triangle. The areas of theses rectangles are a(n). - J. M. Bergot, Sep 22 2011

a(n+1) is the sum of n-th row of the triangle in A195437. - Reinhard Zumkeller, Nov 23 2011

Partial sums of A049450. - Omar E. Pol, Jan 12 2013

From Jon Perry, May 11 2013: (Start)

Define a 'stable brick triangle' as:

-----

| c |

---------

| a | | b |

----------

with a, b, c > 0 and c <= a + b. This can be visualized as two bricks with a third brick on top. The third brick can only be as strong as a+b, otherwise the wall collapses - for example, (1,2,4) is unstable.

a(n) gives the number of stable brick triangles that can be formed if the two supporting bricks are 1 <= a <= n and 1 <= b <= n: a(n) = Sum_{a=1..n} Sum_{b=1..n} Sum_c 1 = n^3 + n^2 as given in the Adamchuk formula.

So for i=j=n=2 we have 4:

1 2 3 4

2 2 2 2 2 2 2 2

For example, n=2 gives 2 from [a=1,b=1], 3 from both [a=1,b=2] and [a=2,b=1] and 4 from [a=2,b=2] so a(2) = 2 + 3 + 3 + 4 = 12. (End)

Define the infinite square array m(n,k) by m(n,k) = (n-k)^2 if n >= k >= 0 and by m(n,k) = (k+n)*(k-n) if 0 <= n <= k. This contains A120070 below the diagonal. Then a(n) = Sum_{k=0..n} m(n,k) + Sum_{r=0..n} m(r,n), the "hook sum" of the terms to the left of m(n,n) and above m(n,n) with irrelevant (vanishing) terms on the diagonal. - J. M. Bergot, Aug 16 2013

a(n) is the sum of all pairs with repetition drawn from the set of odd numbers 2*n-3. This is similar to A027480 but using the odd integers instead. Example using n=3 gives the odd numbers 1,3,5: 1+1, 1+3, 1+5, 3+3, 3+5,5+5 having a total of 36=a(3). - J. M. Bergot, Apr 05 2016

a(n) is the first Zagreb index of the complete graph K[n+1]. The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternately, it is the sum of the degree sums d(i)+d(j) over all edges ij of the graph. - Emeric Deutsch, Nov 07 2016

a(n-2) is the maximum sigma irregularity over all trees with n vertices. The extremal graphs are stars. (The sigma irregularity of a graph is the sum of squares of the differences between the degrees over all edges of the graph.) - Allan Bickle, Jun 14 2023

Without the leading zero, Poincaré series [or Poincare series] P(C_{2,2}; t).

Starting (1, 2, 6, ...) = partial sums of the tetrahedral numbers, A000292 with repeats: (1, 1, 4, 4, 10, 10, 20, 20, 35, 35, ...). - Gary W. Adamson, Mar 30 2009

Starting with 1 = [1, 2, 3, ...] convolved with the aerated triangular series, [1, 0, 3, 0, 6, ...]. - Gary W. Adamson, Jun 11 2009

From Alford Arnold, Oct 14 2009: (Start)

a(n) is also related to Dyck Paths. Note that

0 1 2 6 10 20 30 50 70 105 ...

minus

0 0 0 0 1 2 6 10 20 30 ...

equals

0 1 2 6 9 18 24 40 50 75 ... A028724

(End)

The Kn11, Kn12, Kn13, Fi1 and Ze1 triangle sums of A139600 are related to the sequence given above; e.g., Ze1(n) = 3*A096338(n-1) - 3*A096338(n-3) + 9*A096338(n-4), with A096338(n) = 0 for n <= -1. For the definition of these triangle sums, see A180662. - Johannes W. Meijer, Apr 29 2011

Rows 2, 4, 6, ... give A088459.

Diagonal sums are in A088518(n-1). - Philippe Deléham, Jan 04 2009

Row sums are in A001405(n). - Philippe Deléham, Jan 04 2009

Subtriangle (1 <= k <= n) of triangle T(n,k), 0 <= k <= n, read by rows, given by A101455 DELTA A056594 := [0,1,0,-1,0,1,0,-1,0,1,0,-1,0,...] DELTA [1,0,-1,0,1,0,-1,0,1,0,-1,0,1,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 03 2009

Also, number of symmetric noncrossing partitions of an n-set with k blocks. - Andrew Howroyd, Nov 15 2017

From Roger Ford, Oct 17 2018: (Start)

T(n,k) = t(n+2,d) where t(n,d) is the number of different semi-meander arch depth listings with n top arches and with d the depth of the deepest embedded arch.

Examples: /\ semi-meander with 5 top arches

//\\ /\ 2 arches are at depth=0 (no covering arches)

///\\\ //\\ 2 arches are at depth=1 (1 covering arch)

(0)(1)(2) 1 arch is at depth=2 (2 covering arches)

2, 2, 1 is the listing for this t(5,2)

/\ semi-meander with 5 top arches

/ \ (0)(1)

/\ /\ //\/\\ 3, 2 is the listing for this t(5,1)

a(6,5) = t(8,5)= 3 {2,1,1,1,2,1; 2,1,2,1,1,1; 3,1,1,1,1,1} (End)

Let the triangle = M as an infinite lower triangular matrix.

M * (1, 2, 3, ...) = A002620: (1, 2, 4, 6, 9, 12, 16, 20, ...);

M * (1, 3, 5, ...) = A084265: (1, 2, 6, 9, 15, 20, 28, 35, ...);

M * (1, 3, 6, ...) = A028724: (1, 2, 6, 9, 18, 24, 40, 50, ...);

Limit_{n->infinity} M^n = A171609: (1, 2, 4, 6, 12, 16, 24, 30, ...).

The third diagonal is the generalized pentagonal sequence A001318

Euler transform of length 3 sequence [2,k,-1] with k=4 (cf. A028724 for k=3). - Georg Fischer, Nov 28 2020

Number of symmetric Dyck paths of semilength n with k peaks.