cp's OEIS Frontend

The number of rounds in a round-robin tournament with n competitors. - A. Timothy Royappa, Aug 13 2011

Diagonal sums of number triangle A113126. - Paul Barry, Oct 14 2005

When partitioning a convex n-gon by all the diagonals, the maximum number of sides in resulting polygons is 2*floor(n/2)+1 = a(n-1) (from Moscow Olympiad problem 1950). - Tanya Khovanova, Apr 06 2008

The inverse values of the coefficients in the series expansion of f(x) = (1/2)*(1+x)*log((1+x)/(1-x)) lead to this sequence; cf. A098557. - Johannes W. Meijer, Nov 12 2009

From Reinhard Zumkeller, Dec 05 2009: (Start)

First differences: A010673; partial sums: A000982;

A059329(n) = Sum_{k = 0..n} a(k)*a(n-k);

A167875(n) = Sum_{k = 0..n} a(k)*A005408(n-k);

A171218(n) = Sum_{k = 0..n} a(k)*A005843(n-k);

A008794(n+2) = Sum_{k = 0..n} a(k)*A059841(n-k). (End)

Dimension of the space of weight 2n+4 cusp forms for Gamma_0(5). - Michael Somos, May 29 2013

For n > 4: a(n) = A230584(n) - A230584(n-2). - Reinhard Zumkeller, Feb 10 2015

The arithmetic function v+-(n,2) as defined in A290988. - Robert Price, Aug 22 2017

For n > 0, also the chromatic number of the (n+1)-triangular (Johnson) graph. - Eric W. Weisstein, Nov 17 2017

a(n-1), for n >= 1, is also the upper bound a_{up}(b), where b = 2*n + 1, in the first (top) row of the complete coach system Sigma(b) of Hilton and Pedersen [H-P]. All odd numbers <= a_{up}(b) of the smallest positive restricted residue system of b appear once in the first rows of the c(2*n+1) = A135303(n) coaches. If b is an odd prime a_{up}(b) is the maximum. See a comment in the proof of the quasi-order theorem of H-P, on page 263 ["Furthermore, every possible a_i < b/2 ..."]. For an example see below. - Wolfdieter Lang, Feb 19 2020

Satisfies the nested recurrence a(n) = a(a(n-2)) + 2*a(n-a(n-1)) with a(0) = a(1) = 1. Cf. A004001. - Peter Bala, Aug 30 2022

The binomial transform is 1, 2, 6, 16, 40, 96, 224, 512, 1152, 2560,.. (see A057711). - R. J. Mathar, Feb 25 2023

a(n) is also the binary rank of the complete graph K(n). - Alessandro Cosentino (cosenal(AT)gmail.com), Feb 07 2009

Let I=I_n be the n X n identity matrix and P=P_n be the incidence matrix of the cycle (1,2,3,...,n). Then, for n >= 6, a(n) is the number of (0,1) n X n matrices A <= P^(-1)+I+P having exactly two 1's in every row and column with perA=2. - Vladimir Shevelev, Apr 12 2010

a(n+2) is the number of symmetry allowed, linearly independent terms at n-th order in the series expansion of the (E+A)xe vibronic perturbation matrix, H(Q) (cf. Eisfeld & Viel). - Bradley Klee, Jul 21 2015

The arithmetic function v_2(n,1) as defined in A289187. - Robert Price, Aug 22 2017

For n > 1, also the chromatic number of the n X n white bishop graph. - Eric W. Weisstein, Nov 17 2017

For n > 2, also the maximum vertex degree of the n-polygon diagonal intersection graph. - Eric W. Weisstein, Mar 23 2018

For n >= 2, a(n+2) gives the minimum weight of a Boolean function of algebraic degree at most n-2 whose support contains n linearly independent elements. - Christof Beierle, Nov 25 2019

First differences of A195014.

The n-th column of the triangle is the binomial transform of the n-th row of A081277, followed by zeros. Example: column 3, (1, 6, 19, 44, ...) = binomial transform of row 3 of A081277: (1, 5, 8, 4, 0, 0, 0, ...). A104698 = reversal by rows of A142978. - Gary W. Adamson, Jul 17 2008

This sequence is jointly generated with A210222 as an array of coefficients of polynomials u(n,x): initially, u(1,x)=v(1,x)=1; for n > 1, u(n,x) = x*u(n-1,x) + v(n-1) + 1 and v(n,x) = 2x*u(n-1,x) + v(n-1,x) + 1. See the Mathematica section at A210222. - Clark Kimberling, Mar 19 2012

This Riordan triangle T appears in a formula for A001100(n, 0) = A002464(n), for n >= 1. - Wolfdieter Lang, May 13 2025

From G. C. Greubel, Jan 18 2025: (Start)

A more general triangle of coefficients may be defined by T(n, k, p, q) = (p^(n-k)*q^k + p^k*q^(n-k))*A007318(n, k). When (p, q) = (2, 1) this sequence is obtained.

Some related triangles are:

(p, q) = (1, 1) : 2*A007318(n,k).

(p, q) = (2, 2) : 2*A038208(n,k).

(p, q) = (3, 2) : A154692(n,k).

(p, q) = (3, 3) : 2*A038221(n,k). (End)

Periodic with period {0,2,3,2,0,5}.

In the square array we have that:

Antidiagonal sums give A168237.

Odd-indexed rows give A010673.

Even-indexed rows give A010684.

Odd-indexed columns give A000035.

Even-indexed columns give A010693.

Odd-indexed antidiagonals give the initial terms of A010674.

Even-indexed antidiagonals give the initial terms of A000034.

Main diagonal gives A010674.

This is also a triangle read by rows in which each new term is the least nonnegative integer distinct from its neighbors.

In the triangle we have that:

Row sums give A168237.

Odd-indexed columns give A000035.

Even-indexed columns give A010693.

Odd-indexed diagonals give A010673.

Even-indexed diagonals give A010684.

Odd-indexed rows give the initial terms of A010674.

Even-indexed rows give the initial terms of A000034.

Odd-indexed antidiagonals give the initial terms of A010673.

Even-indexed antidiagonals give the initial terms of A010684.

From Omar E. Pol, Feb 06 2023: (Start)

The "word" of this cellular automaton is "ab".

Apart from the initial zero the structure of the irregular triangle is as shown below:

a,b;

a,b;

a,b,a,b;

a,b,a,b,a,b,a,b;

a,b,a,b,a,b,a,b,a,b,a,b,a,b,a,b;

...

Columns "a" contain numbers of toothpicks and D-toothpicks when in the top border of the structure there are only toothpicks (of length 1).

Columns "b" contain numbers of toothpicks and D-toothpicks when in the top border of the structure there are only D-toothpicks (of length sqrt(2)).

An associated sound to the animation could be (tick, tock), (tick, tock), ..., the same as the ticking clock sound.

Row lengths are the terms of A011782 multiplied by 2, also the column 2 of A296612.

For further information about the word of cellular automata see A296612.

It appears that the right border of the irregular triangle gives the even powers of 2. (End)