cp's OEIS Frontend

Number of parallelograms in an n X n rhombus. - Matti De Craene (Matti.DeCraene(AT)rug.ac.be), May 14 2000

Or, number of orthogonal rectangles in an n X n checkerboard, or rectangles in an n X n array of squares. - Jud McCranie, Feb 28 2003. Compare A085582.

Also number of 2-dimensional cage assemblies (cf. A059827, A059860).

The n-th triangular number T(n) = Sum_{r=1..n} r = n(n+1)/2 satisfies the relations: (i) T(n) + T(n-1) = n^2 and (ii) T(n) - T(n-1) = n by definition, so that n^2*n = n^3 = {T(n)}^2 - {T(n-1)}^2 and by summing on n we have Sum_{ r = 1..n } r^3 = {T(n)}^2 = (1+2+3+...+n)^2 = (n*(n+1)/2)^2. - Lekraj Beedassy, May 14 2004

Number of 4-tuples of integers from {0,1,...,n}, without repetition, whose last component is strictly bigger than the others. Number of 4-tuples of integers from {1,...,n}, with repetition, whose last component is greater than or equal to the others.

Number of ordered pairs of two-element subsets of {0,1,...,n} without repetition.

Number of ordered pairs of 2-element multisubsets of {1,...,n} with repetition.

1^3 + 2^3 + 3^3 + ... + n^3 = (1 + 2 + 3 + ... + n)^2.

a(n) is the number of parameters needed in general to know the Riemannian metric g of an n-dimensional Riemannian manifold (M,g), by knowing all its second derivatives; even though to know the curvature tensor R requires (due to symmetries) (n^2)*(n^2-1)/12 parameters, a smaller number (and a 4-dimensional pyramidal number). - Jonathan Vos Post, May 05 2006

Also number of hexagons with vertices in an hexagonal grid with n points in each side. - Ignacio Larrosa Cañestro, Oct 15 2006

Number of permutations of n distinct letters (ABCD...) each of which appears twice with 4 and n-4 fixed points. - Zerinvary Lajos, Nov 09 2006

With offset 1 = binomial transform of [1, 8, 19, 18, 6, ...]. - Gary W. Adamson, Dec 03 2008

The sequence is related to A000330 by a(n) = n*A000330(n) - Sum_{i=0..n-1} A000330(i): this is the case d=1 in the identity n*(n*(d*n-d+2)/2) - Sum_{i=0..n-1} i*(d*i-d+2)/2 = n*(n+1)*(2*d*n-2*d+3)/6. - Bruno Berselli, Apr 26 2010, Mar 01 2012

From Wolfdieter Lang, Jan 11 2013: (Start)

For sums of powers of positive integers S(k,n) := Sum_{j=1..n}j^k one has the recurrence S(k,n) = (n+1)*S(k-1,n) - Sum_{l=1..n} S(k-1,l), n >= 1, k >= 1.

This was used for k=4 by Ibn al-Haytham in an attempt to compute the volume of the interior of a paraboloid. See the Strick reference where the trick he used is shown, and the W. Lang link.

This trick generalizes immediately to arbitrary powers k. For k=3: a(n) = (n+1)*A000330(n) - Sum_{l=1..n} A000330(l), which coincides with the formula given in the previous comment by Berselli. (End)

Regarding to the previous contribution, see also Matem@ticamente in Links field and comments on this recurrences in similar sequences (partial sums of n-th powers). - Bruno Berselli, Jun 24 2013

A rectangular prism with sides A000217(n), A000217(n+1), and A000217(n+2) has surface area 6*a(n+1). - J. M. Bergot, Aug 07 2013, edited with corrected indices by Antti Karttunen, Aug 09 2013

A formula for the r-th successive summation of k^3, for k = 1 to n, is (6*n^2+r*(6*n+r-1)*(n+r)!)/((r+3)!*(n-1)!), (H. W. Gould). - Gary Detlefs, Jan 02 2014

Note that this sequence and its formula were known to (and possibly discovered by) Nicomachus, predating Ibn al-Haytham by 800 years. - Charles R Greathouse IV, Apr 23 2014

a(n) is the number of ways to paint the sides of a nonsquare rectangle using at most n colors. Cf. A039623. - Geoffrey Critzer, Jun 18 2014

For n > 0: A256188(a(n)) = A000217(n) and A256188(m) != A000217(n) for m < a(n), i.e., positions of first occurrences of triangular numbers in A256188. - Reinhard Zumkeller, Mar 26 2015

There is no cube in this sequence except 0 and 1. - Altug Alkan, Jul 02 2016

Also the number of chordless cycles in the complete bipartite graph K_{n+1,n+1}. - Eric W. Weisstein, Jan 02 2018

a(n) is the sum of the elements in the multiplication table [0..n] X [0..n]. - Michel Marcus, May 06 2021

Principal diagonal: A002412.

Antidiagonal sums: A002415.

Row 1: (1,2,3,...)**(1,2,3,...) = A000292.

Row 2: (1,2,3,...)**(2,3,4,...) = A005581.

Row 3: (1,2,3,...)**(3,4,5,...) = A006503.

Row 4: (1,2,3,...)**(4,5,6,...) = A060488.

Row 5: (1,2,3,...)**(5,6,7,...) = A096941.

Row 6: (1,2,3,...)**(6,7,8,...) = A096957.

...

In general, the convolution of two infinite sequences is defined from the convolution of two n-tuples: let X(n) = (x(1),...,x(n)) and Y(n)=(y(1),...,y(n)); then X(n)**Y(n) = x(1)*y(n)+x(2)*y(n-1)+...+x(n)*y(1); this sum is the n-th term in the convolution of infinite sequences:(x(1),...,x(n),...)**(y(1),...,y(n),...), for all n>=1.

...

In the following guide to related arrays and sequences, row n of each array T(n,k) is the convolution b**c of the sequences b(h) and c(h+n-1). The principal diagonal is given by T(n,n) and the n-th antidiagonal sum by S(n). In some cases, T(n,n) or S(n) differs in offset from the listed sequence.

b(h)........ c(h)........ T(n,k) .. T(n,n) .. S(n)

h .......... h .......... A213500 . A002412 . A002415

h .......... h^2 ........ A212891 . A213436 . A024166

h^2 ........ h .......... A213503 . A117066 . A033455

h^2 ........ h^2 ........ A213505 . A213546 . A213547

h .......... h*(h+1)/2 .. A213548 . A213549 . A051836

h*(h+1)/2 .. h .......... A213550 . A002418 . A005585

h*(h+1)/2 .. h*(h+1)/2 .. A213551 . A213552 . A051923

h .......... h^3 ........ A213553 . A213554 . A101089

h^3 ........ h .......... A213555 . A213556 . A213547

h^3 ........ h^3 ........ A213558 . A213559 . A213560

h^2 ........ h*(h+1)/2 .. A213561 . A213562 . A213563

h*(h+1)/2 .. h^2 ........ A213564 . A213565 . A101094

2^(h-1) .... h .......... A213568 . A213569 . A047520

2^(h-1) .... h^2 ........ A213573 . A213574 . A213575

h .......... Fibo(h) .... A213576 . A213577 . A213578

Fibo(h) .... h .......... A213579 . A213580 . A053808

Fibo(h) .... Fibo(h) .... A067418 . A027991 . A067988

Fibo(h+1) .. h .......... A213584 . A213585 . A213586

Fibo(n+1) .. Fibo(h+1) .. A213587 . A213588 . A213589

h^2 ........ Fibo(h) .... A213590 . A213504 . A213557

Fibo(h) .... h^2 ........ A213566 . A213567 . A213570

h .......... -1+2^h ..... A213571 . A213572 . A213581

-1+2^h ..... h .......... A213582 . A213583 . A156928

-1+2^h ..... -1+2^h ..... A213747 . A213748 . A213749

h .......... 2*h-1 ...... A213750 . A007585 . A002417

2*h-1 ...... h .......... A213751 . A051662 . A006325

2*h-1 ...... 2*h-1 ...... A213752 . A100157 . A071238

2*h-1 ...... -1+2^h ..... A213753 . A213754 . A213755

-1+2^h ..... 2*h-1 ...... A213756 . A213757 . A213758

2^(n-1) .... 2*h-1 ...... A213762 . A213763 . A213764

2*h-1 ...... Fibo(h) .... A213765 . A213766 . A213767

Fibo(h) .... 2*h-1 ...... A213768 . A213769 . A213770

Fibo(h+1) .. 2*h-1 ...... A213774 . A213775 . A213776

Fibo(h) .... Fibo(h+1) .. A213777 . A001870 . A152881

h .......... 1+[h/2] .... A213778 . A213779 . A213780

1+[h/2] .... h .......... A213781 . A213782 . A005712

1+[h/2] .... [(h+1)/2] .. A213783 . A213759 . A213760

h .......... 3*h-2 ...... A213761 . A172073 . A002419

3*h-2 ...... h .......... A213771 . A213772 . A132117

3*h-2 ...... 3*h-2 ...... A213773 . A214092 . A213818

h .......... 3*h-1 ...... A213819 . A213820 . A153978

3*h-1 ...... h .......... A213821 . A033431 . A176060

3*h-1 ...... 3*h-1 ...... A213822 . A213823 . A213824

3*h-1 ...... 3*h-2 ...... A213825 . A213826 . A213827

3*h-2 ...... 3*h-1 ...... A213828 . A213829 . A213830

2*h-1 ...... 3*h-2 ...... A213831 . A213832 . A212560

3*h-2 ...... 2*h-1 ...... A213833 . A130748 . A213834

h .......... 4*h-3 ...... A213835 . A172078 . A051797

4*h-3 ...... h .......... A213836 . A213837 . A071238

4*h-3 ...... 2*h-1 ...... A213838 . A213839 . A213840

2*h-1 ...... 4*h-3 ...... A213841 . A213842 . A213843

2*h-1 ...... 4*h-1 ...... A213844 . A213845 . A213846

4*h-1 ...... 2*h-1 ...... A213847 . A213848 . A180324

[(h+1)/2] .. [(h+1)/2] .. A213849 . A049778 . A213850

h .......... C(2*h-2,h-1) A213853

...

Suppose that u = (u(n)) and v = (v(n)) are sequences having generating functions U(x) and V(x), respectively. Then the convolution u**v has generating function U(x)*V(x). Accordingly, if u and v are homogeneous linear recurrence sequences, then every row of the convolution array T satisfies the same homogeneous linear recurrence equation, which can be easily obtained from the denominator of U(x)*V(x). Also, every column of T has the same homogeneous linear recurrence as v.

The row polynomials R(n,x) := Sum_{m=0..n} a(n,m)*x^m have been called Chebyshev C_n(x) polynomials in the Abramowitz-Stegun handbook, p. 778, 22.5.11 (see A049310 for the reference, and note that on p. 774 the S and C polynomials have been mixed up in older printings). - Wolfdieter Lang, Jun 03 2011

This is a signed version of triangle A114525.

The unsigned column sequences (without zeros) are, for m=1..11: A005408, A000290, A000330, A002415, A005585, A040977, A050486, A053347, A054333, A054334, A057788.

The row polynomials R(n,x) := Sum_{m=0..n} a(n,m)*x*m, give for n=2,3,...,floor(N/2) the positive zeros of the Chebyshev S(N-1,x)-polynomial (see A049310) in terms of its largest zero rho(N):= 2*cos(Pi/N) by putting x=rho(N). The order of the positive zeros is falling: n=1 corresponds to the largest zero rho(N) and n=floor(N/2) to the smallest positive zero. Example N=5: rho(5)=phi (golden section), R(2,phi)= phi^2-2 = phi-1, the second largest (and smallest) positive zero of S(4,x). - Wolfdieter Lang, Dec 01 2010

The row polynomial R(n,x), for n >= 1, factorizes into minimal polynomials of 2*cos(Pi/k), called C(k,x), with coefficients given in A187360, as follows.

R(n,x) = Product_{d|oddpart(n)} C(2*n/d,x)

= Product_{d|oddpart(n)} C(2^(k+1)*d,x),

with oddpart(n)=A000265(n), and 2^k is the largest power of 2 dividing n, where k=0,1,2,...

(Proof: R and C are monic, the degree on both sides coincides, and the zeros of R(n,x) appear all on the r.h.s.) - Wolfdieter Lang, Jul 31 2011 [Theorem 1B, eq. (43) in the W. Lang link. - Wolfdieter Lang, Apr 13 2018]

The zeros of the row polynomials R(n,x) are 2*cos(Pi*(2*k+1)/(2*n)), k=0,1, ..., n-1; n>=1 (from those of the Chebyshev T-polynomials). - Wolfdieter Lang, Sep 17 2011

The discriminants of the row polynomials R(n,x) are found under A193678. - Wolfdieter Lang, Aug 27 2011

The determinant of the N X N matrix M(N) with entries M(N;n,m) = R(m-1,x[n]), 1 <= n,m <= N, N>=1, and any x[n], is identical with twice the Vandermondian Det(V(N)) with matrix entries V(N;n,m) = x[n]^(m-1). This is an instance of the general theorem given in the Vein-Dale reference on p. 59. Note that R(0,x) = 2 (not 1). See also the comments from Aug 26 2013 under A049310 and from Aug 27 2013 under A000178. - Wolfdieter Lang, Aug 27 2013

This triangle a(n,m) is also used to express in the regular (2*(n+1))-gon, inscribed in a circle of radius R, the length ratio side/R, called s(2*(n+1)), as a polynomial in rho(2*(n+1)), the length ratio (smallest diagonal)/side. See the bisections ((-1)^(k-s))*A111125(k,s) and A127677 for comments and examples. - Wolfdieter Lang, Oct 05 2013

From Tom Copeland, Nov 08 2015: (Start)

These are the characteristic polynomials a_n(x) = 2*T_n(x/2) for the adjacency matrix of the Coxeter simple Lie algebra B_n, related to the Cheybshev polynomials of the first kind, T_n(x) = cos(n*q) with x = cos(q) (see p. 20 of Damianou). Given the polynomial (x - t)*(x - 1/t) = 1 - (t + 1/t)*x + x^2 = e2 - e1*x + x^2, the symmetric power sums p_n(t,1/t) = t^n + t^(-n) of the zeros of this polynomial may be expressed in terms of the elementary symmetric polynomials e1 = t + 1/t = y and e2 = t*1/t = 1 as p_n(t,1/t) = a_n(y) = F(n,-y,1,0,0,...), where F(n,b1,b2,...,bn) are the Faber polynomials of A263916.

The partial sum of the first n+1 rows given t and y = t + 1/t is PS(n,t) = Sum_{k=0..n} a_n(y) = (t^(n/2) + t^(-n/2))*(t^((n+1)/2) - t^(-(n+1)/2)) / (t^(1/2) - t^(-1/2)). (For n prime, this is related simply to the cyclotomic polynomials.)

Then a_n(y) = PS(n,t) - PS(n-1,t), and for t = e^(iq), y = 2*cos(q), and, therefore, a_n(2*cos(q)) = PS(n,e^(iq)) - PS(n-1,e^(iq)) = 2*cos(nq) = 2*T_n(cos(q)) with PS(n,e^(iq)) = 2*cos(nq/2)*sin((n+1)q/2) / sin(q/2).

(End)

R(45, x) is the famous polynomial used by Adriaan van Roomen (Adrianus Romanus) in his Ideae mathematicae from 1593 to pose four problems, solved by Viète. See, e.g., the Havil reference, pp. 69-74. - Wolfdieter Lang, Apr 28 2018

From Wolfdieter Lang, May 05 2018: (Start)

Some identities for the row polynomials R(n, x) following from the known ones for Chebyshev T-polynomials (A053120) are:

(1) R(-n, x) = R(n, x).

(2) R(n*m, x) = R(n, R(m, x)) = R(m, R(n, x)).

(3) R(2*k+1, x) = (-1)^k*x*S(2*k, sqrt(4-x^2)), k >= 0, with the S row polynomials of A049310.

(4) R(2*k, x) = R(k, x^2-2), k >= 0.

(End)

For y = z^n + z^(-n) and x = z + z^(-1), Hirzebruch notes that y(z) = R(n,x) for the row polynomial of this entry. - Tom Copeland, Nov 09 2019

This is also called Vieta's array. - N. J. A. Sloane, Nov 22 2017

Dropping the first term and changing the boundary conditions to T(n,1)=n, T(n,n-1)=2 (n>=2), T(n,n)=1 yields the number of nonterminal symbols (which generate strings of length k) in a certain context-free grammar in Chomsky normal form that generates all permutations of n symbols. Summation over k (1<=k<=n) results in A003945. For the number of productions of this grammar: see A090327. Example: 1; 2, 1; 3, 2, 1; 4, 5, 2, 1; 5, 9, 7, 2, 1; 6, 14, 16, 9, 2, 1; In addition to the example of A090327 we have T(3,3)=#{S}=1, T(3,2)=#{D,E}=2 and T(3,1)=#{A,B,C}=3. - Peter R. J. Asveld, Jan 29 2004

Much as the original Pascal triangle gives the Fibonacci numbers as sums of its diagonals, this triangle gives the Lucas numbers (A000032) as sums of its diagonals; see Posamentier & Lehmann (2007). - Alonso del Arte, Apr 09 2012

For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 04 2013

It appears that for the infinite set of (1,N) Pascal's triangles, the binomial transform of the n-th row (n>0), followed by zeros, is equal to the n-th partial sum of (1, N, N, N, ...). Example: for the (1,2) Pascal's triangle, the binomial transform of the second row followed by zeros, i.e., of (1, 3, 2, 0, 0, 0, ...), is equal to the second partial sum of (1, 2, 2, 2, ...) = (1, 4, 9, 16, ...). - Gary W. Adamson, Aug 11 2015

Given any (1,N) Pascal triangle, let the binomial transform of the n-th row (n>1) followed by zeros be Q(x). It appears that the binomial transform of the (n-1)-th row prefaced by a zero is Q(n-1). Example: In the (1,2) Pascal triangle the binomial transform of row 3: (1, 4, 5, 2, 0, 0, 0, ...) is A000330 starting with 1: (1, 5, 14, 30, 55, 91, ...). The binomial transform of row 2 prefaced by a zero and followed by zeros, i.e., of (0, 1, 3, 2, 0, 0, 0, ...) is (0, 1, 5, 14, 30, 55, ...). - Gary W. Adamson, Sep 28 2015

It appears that in the array accompanying each (1,N) Pascal triangle (diagonals of the triangle), the binomial transform of (..., 1, N, 0, 0, 0, ...) preceded by (n-1) zeros generates the n-th row of the array (n>0). Then delete the zeros in the result. Example: in the (1,2) Pascal triangle, row 3 (1, 5, 14, 30, ...) is the binomial transform of (0, 0, 1, 2, 0, 0, 0, ...) with the resulting zeros deleted. - Gary W. Adamson, Oct 11 2015

Read as a square array (similar to the Example section Sq(m,j), but with Sq(0,0)=0 and Sq(m,j)=P(m+1,j) otherwise), P(n,k) are the multiplicities of the eigenvalues, lambda_n = n(n+k-1), of the Laplacians on the unit k-hypersphere, given by Teo (and Choi) as P(n,k) = (2n-k+1)(n+k-2)!/(n!(k-1)!). P(n,k) is also the numerator of a Dirichlet series for the Minakashisundarum-Pleijel zeta function for the sphere. Also P(n,k) is the dimension of the space of homogeneous, harmonic polynomials of degree k in n variables (Shubin, p. 169). For relations to Chebyshev polynomials and simple Lie algebras, see A034807. - Tom Copeland, Jan 10 2016

For a relation to a formulation for a universal Lie Weyl algebra for su(1,1), see page 16 of Durov et al. - Tom Copeland, Jan 15 2016

These polynomials arise in the following setup. Suppose G and H are power series satisfying G + H = G*H = 1/x. Then G^n + H^n = (1/x^n)*L_n(-x).

Apart from signs, triangle of coefficients when 2*cos(nt) is expanded in terms of x = 2*cos(t). For example, 2*cos(2t) = x^2 - 2, 2*cos(3t) = x^3 - 3x and 2*cos(4t) = x^4 - 4x^2 + 2. - Anthony C Robin, Jun 02 2004

Triangle of coefficients of expansion of Z_{nk} in terms of Z_k.

Row n has 1 + floor(n/2) terms. - Emeric Deutsch, Dec 25 2004

T(n,k) = number of k-matchings of the cycle C_n (n > 1). Example: T(6,2)=9 because the 2-matchings of the hexagon with edges a, b, c, d, e, f are ac, ad, ae, bd, be, bf, ce, cf and df. - Emeric Deutsch, Dec 25 2004

An example for the first comment: G=c(x), H=1/(x*c(x)) with c(x) the o.g.f. Catalan numbers A000108: (x*c(x))^n + (1/c(x))^n = L(n,-x)= Sum_{k=0..floor(n/2)} T(n,k)*(-x)^k.

This triangle also supplies the absolute values of the coefficients in the multiplication formulas for the Lucas numbers A000032.

From L. Edson Jeffery, Mar 19 2011: (Start)

This sequence is related to rhombus substitution tilings. A signed version of it (see A132460), formed as a triangle with interlaced zeros extending each row to n terms, begins as

{2}

{1, 0}

{1, 0, -2}

{1, 0, -3, 0}

{1, 0, -4, 0, 2}

{1, 0, -5, 0, 5, 0}

....

For the n X n tridiagonal unit-primitive matrix G_(n,1) (n >= 2) (see the L. E. Jeffery link below), defined by

G_(n,1) =

(0 1 0 ... 0)

(1 0 1 0 ... 0)

(0 1 0 1 0 ... 0)

...

(0 ... 0 1 0 1)

(0 ... 0 2 0),

Row n (i.e., {T(n,k)}, k=0..n) of the signed table gives the coefficients of its characteristic function: c_n(x) = Sum_{k=0..n} T(n,k)*x^(n-k) = 0. For example, let n=3. Then

G_(3,1) =

(0 1 0)

(1 0 1)

(0 2 0),

and row 3 of the table is {1,0,-3,0}. Hence c_3(x) = x^3 - 3*x = 0. G_(n,1) has n distinct eigenvalues (the solutions of c_n(x) = 0), given by w_j = 2*cos((2*j-1)*Pi/(2*n)), j=1..n. (End)

For n > 0, T(n,k) is the number of k-subsets of {1,2,...,n} which contain neither consecutive integers nor both 1 and n. Equivalently, T(n,k) is the number of k-subsets without neighbors of a set of n points on a circle. - José H. Nieto S., Jan 17 2012

With the first column omitted, this gives A157000. - Philippe Deléham, Mar 17 2013

The number of necklaces of k black and n - k white beads with no adjacent black beads (Kaplansky 1943). Coefficients of the Dickson polynomials D(n,x,-a). - Peter Bala, Mar 09 2014

From Tom Copeland, Nov 07 2015: (Start)

This triangular array is composed of interleaved rows of reversed, unsigned A127677 (cf. A156308, A217476, A263916) and reversed A111125 (cf. A127672).

See also A113279 for another connection to symmetric and Faber polynomials.

The difference of consecutive rows gives the previous row shifted.

For relations among the characteristic polynomials of Cartan matrices of the Coxeter root groups, Chebyshev polynomials, cyclotomic polynomials, and the polynomials of this entry, see Damianou (p. 12, 20, and 21) and Damianou and Evripidou (p. 7). (End)

Diagonals are related to multiplicities of eigenvalues of the Laplacian on hyperspheres through A029635. - Tom Copeland, Jan 10 2016

For n>=3, also the independence and matching polynomials of the n-cycle graph C_n. See also A284966. - Eric W. Weisstein, Apr 06 2017

Apparently, with the rows aerated and then the 2s on the diagonal removed, this matrix becomes the reverse, or mirror, of unsigned A117179. See also A114525 - Tom Copeland, May 30 2017

Briggs's (1633) table with an additional column of 2s on the right can be used to generate this table. See p. 69 of the Newton reference. - Tom Copeland, Jun 03 2017

From Liam Solus, Aug 23 2018: (Start)

For n>3 and k>0, T(n,k) equals the number of Markov equivalence classes with skeleton the cycle on n nodes having exactly k immoralities. See Theorem 2.1 of the article by A. Radhakrishnan et al. below.

For n>2 odd and r = floor(n/2)-1, the n-th row is the coefficient vector of the Ehrhart h*-polynomial of the r-stable (n,2)-hypersimplex. See Theorem 4.14 in the article by B. Braun and L. Solus below.

(End)

Conjecture: If a(n) = H(a,b,c,d,n) is a second-order linear recurrence with constant coefficients defined as a(0) = a, a(1)= b, a(n) = c*a(n-1) + d*a(n-2) then a(m*n) = H(a, H(a,b,c,d,m), Sum_{k=0..floor(m/2)} T(m,k)*c^(m-2*k)*d^k, (-1)^(m+1)*d^m, n) (Wolfdieter Lang). - Gary Detlefs, Feb 06 2023

For the proof of the preceding conjecture see the Detlefs and Lang link. There also proofs for several properties of this table are found. - Wolfdieter Lang, Apr 25 2023

From Mohammed Yaseen, Nov 09 2024: (Start)

Let m - 1/m = x, then

m^2 + 1/m^2 = x^2 + 2,

m^3 - 1/m^3 = x^3 + 3*x,

m^4 + 1/m^4 = x^4 + 4*x^2 + 2,

m^5 - 1/m^5 = x^5 + 5*x^3 + 5*x,

m^6 + 1/m^6 = x^6 + 6*x^4 + 9*x^2 + 2,

m^7 - 1/m^7 = x^7 + 7*x^5 + 14*x^3 + 7*x, etc. (End)

a(n+1) is the number of 2 X n binary matrices with no zero rows or columns, up to row and column permutation.

[ (4th elementary symmetric function of S(n))/(3rd elementary symmetric function of S(n)) ], where S(n) = {first n+3 odd positive integers}.

First differences are 1, 2, 2, 3, 3, 4, 4, 5, 5, ... .

Let M_n denotes the n X n matrix m(i,j) = 1 if i =j; m(i,j) = 1 if (i+j) is odd; m(i,j) = 0 if i+j is even, then a(n) = -det M_(n+1) - Benoit Cloitre, Jun 19 2002

a(n) is the number of squares with corners on an n X n grid, distinct up to translation. See also A002415, A108279.

Starting (1, 3, 5, 8, 11, ...), = row sums of triangle A135841. - Gary W. Adamson, Dec 01 2007

Number of solutions to x+y >= n-1 in integers x,y with 1 <= x <= y <= n-1. - Franz Vrabec, Feb 22 2008

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=5, a(n-4)=-coeff(charpoly(A,x),x^2). - Milan Janjic, Jan 26 2010

Equals row sums of a triangle with alternate columns of (1,2,3,...) and (1,1,1,...). - Gary W. Adamson, May 21 2010

Conjecture: if a(n) = p#(primorial)-1 for some prime number p, then q=(n+1) is also a prime number where p#=floor(q^2/4). Tested up to n=10^100000 no counterexamples are found. It seems that the subsequence is very scattered. So far the triples (p,q,a(q-1)) are {(2,3,1), (3,5,5), (5,11,29), (7,29,209), (17,1429,510509)}. - David Morales Marciel, Oct 02 2015

Numbers of an Ulam spiral starting at 0 in which the shape of the spiral is exactly a rectangle. E.g., a(4)=5 the Ulam spiral is including at that moment only the elements 0,1,2,3,4,5 and the shape is a rectangle. The area is always a(n)+1. E.g., for a(4) the area of the rectangle is 2(rows) X 3(columns) = 6 = a(4) + 1. - David Morales Marciel, Apr 05 2016

Numbers of different quadratic forms (quadrics) in the real projective space P^n(R). - Serkan Sonel, Aug 26 2020

a(n+1) is the number of one-dimensional subspaces of (F_3)^n, counted up to coordinate permutation. E.g.: For n=4, there are five one-dimensional subspaces in (F_3)^3 up to coordinate permutation: [1 2 2] [0 2 2] [1 0 2] [0 0 2] [1 1 1]. This example suggests a bijection (which has to be adjusted for the all-ones matrix) with the binary matrices of the first comment. - Álvar Ibeas, Sep 21 2021

Number of Dyck n-paths with exactly k peaks. - Peter Luschny, May 10 2014

Convolution of triangular numbers (A000217) and squares (A000290) (n>=1). - Graeme McRae, Jun 07 2006

p^k divides a(p^k-3), a(p^k-2), a(p^k-1) and a(p^k) for prime p > 5 and integer k > 0. p^k divides a((p^k-3)/2) for prime p > 5 and integer k > 0. - Alexander Adamchuk, May 08 2007

If a 2-set Y and an (n-3)-set Z are disjoint subsets of an n-set X then a(n-5) is the number of 6-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 08 2007

5-dimensional square numbers, fourth partial sums of binomial transform of [1,2,0,0,0,...]. a(n) = Sum_{i=0..n} binomial(n+4, i+4)*b(i), where b(i)=[1,2,0,0,0,...]. - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009

Antidiagonal sums of the convolution array A213550. - Clark Kimberling, Jun 17 2012

Binomial transform of (1, 6, 14, 16, 9, 2, 0, 0, 0, ...). - Gary W. Adamson, Jul 28 2015

2*a(n) is number of ways to place 4 queens on an (n+3) X (n+3) chessboard so that they diagonally attack each other exactly 6 times. The maximal possible attack number, p=binomial(k,2)=6 for k=4 queens, is achievable only when all queens are on the same diagonal. In graph-theory representation they thus form a corresponding complete graph. - Antal Pinter, Dec 27 2015

While adjusting for offsets, add A000389 to find the next in series A000389, A005585, A051836, A034263, A027800, A051843, A051877, A051878, A051879, A051880, A056118, A271567. (See Bruno Berselli's comments in A271567.) - Bruce J. Nicholson, Jun 21 2018

Coefficients in the terminating series identity 1 - 7*n/(n + 6) + 27*n*(n - 1)/((n + 6)*(n + 7)) - 77*n*(n - 1)*(n - 2)/((n + 6)*(n + 7)*(n + 8)) + ... = 0 for n = 1,2,3,.... Cf. A002415 and A040977. - Peter Bala, Feb 18 2019

Triangular numbers t_n as n runs through the squares.

Partial sums of A055112: If one generated Pythagorean primitive triangles from n, n+1, then the collective areas of n of them would be equal to the numbers in this sequence. The sum of the first three triangles is 6+30+84 = 120 which is the third nonzero term of the sequence. - J. M. Bergot, Jul 14 2011

Second leg of Pythagorean triangles with smallest side a cube: A000578(n)^2 + a(n)^2 = A037270(n)^2. - Martin Renner, Nov 12 2011

a(n) is the number of segments on an n X n grid or geoboard. - Martin Renner, Apr 17 2014

Consider the partitions of 2n into two parts (p,q). Then a(n) is the total volume of the family of rectangular prisms with dimensions p, q and |q-p|. - Wesley Ivan Hurt, Apr 15 2018