cp's OEIS Frontend

Coordination sequence for infinite tree with valency 4.

The n-th term of the coordination sequence of the infinite tree with valency 2m is the same as the number of reduced words of size n in the free group on m generators. In the five sequences A003946, A003948, A003950, A003952, A003954 m is 2, 3, 4, 5, 6. - Avi Peretz (njk(AT)netvision.net.il), Feb 23 2001 and Ola Veshta (olaveshta(AT)my-deja.com), Mar 30 2001

a(n) is the number of nonreversing random walks of the length of n edges on a two-dimensional square lattice, all beginning at a fixed point P. - Pawel P. Mazur (Pawel.Mazur(AT)pwr.wroc.pl), Apr 06 2005

Binomial transform of {1, 3, 5, 11, 21, 43, ...}, see A001045. Binomial transform is {1, 5, 21, 85, 341, 1365, ...}, see A002450. - Philippe Deléham, Jul 22 2005

For n >= 2, a(n) is equal to the number of functions f:{1,2,...,n+1}->{1,2,3} such that for fixed, different x_1, x_2 in {1,2,...,n} and fixed y_1, y_2 in {1,2,3} we have f(x_1) <> y_1 and f(x_2) <> y_2. - Milan Janjic, Apr 19 2007

Equals row sums of triangle A143865. - Gary W. Adamson, Sep 04 2008

Equals INVERT transform of the odd integers = 1/(1 - x - 3x^2 - 5x^3 - ...). - Gary W. Adamson, Jul 27 2009

a(n) is the number of generalized compositions of n+1 when there are 2 *i-1 different types of the part i, (i=1,2,...). - Milan Janjic, Aug 26 2010

Number of length-n strings of 4 letters with no two adjacent letters identical. The general case (strings of r letters) is the sequence with g.f. (1+x)/(1-(r-1)*x). - Joerg Arndt, Oct 11 2012

The sequence is the INVERTi transform of A015448: (1, 5, 21, 89, 377, ...). - Gary W. Adamson, Aug 06 2016

Let D(m) = {d(m,i)}, i = 1..q, denote the set of the q divisors of a number m, and consider s1(m) and s2(m) the sums of the divisors that are congruent to 1 and 2 (mod 3) respectively. For n > 0, the sequence a(n) lists the numbers m such that s1(m) = 5 and s2(m) = 2. - Michel Lagneau, Feb 09 2017

a(n) is the number of quaternary sequences of length n such that no two consecutive terms have distance 2. - David Nacin, May 31 2017

Also the number of maximal cliques in the n-Sierpinski gasket graph. - Eric W. Weisstein, Dec 01 2017

Number of 3-permutations of n elements avoiding the patterns 231, 321. See Bonichon and Sun. - Michel Marcus, Aug 19 2022

Coordination sequence for infinite tree with valency 6.

The n-th term of the coordination sequence of the infinite tree with valency 2m is the same as the number of reduced words of size n in the free group on m generators. In the five sequences A003946, A003948, A003950, A003952, A003954, m is 2, 3, 4, 5, 6. - Avi Peretz (njk(AT)netvision.net.il), Feb 23 2001 and Ola Veshta (olaveshta(AT)my-deja.com), Mar 30 2001

Hamiltonian path in S_4 X P_2n.

For n>=1, a(n+1) is equal to the number of functions f:{1,2,...,n+1}->{1,2,3,4,5,6} such that for fixed, different x_1, x_2,...,x_n in {1,2,...,n+1} and fixed y_1, y_2,...,y_n in {1,2,3,4,5,6} we have f(x_i)<>y_i, (i=1..n). - Milan Janjic, May 10 2007

For n>=1, a(n) equals the numbers of words of length n over the alphabet {0..5} with no two adjacent letters identical. - Milan Janjic, Jan 31 2015 [Corrected by David Nacin, May 30 2017]

a(n) equals the numbers of sequences of length n on {0,...,5} where no two adjacent terms differ by three. - David Nacin, May 30 2017

It appears that these are the only n>1 for which alpha(n)=2n, where alpha(n) is the entry point of n in the Fibonacci sequence, see A001177. - Philippe Schnoebelen, Apr 11 2024

Coordination sequence for infinite tree with valency 8.

The n-th term of the coordination sequence of the infinite tree with valency 2m is the same as the number of reduced words of size n in the free group on m generators. In the five sequences A003946, A003948, A003950, A003952, A003954 m is 2, 3, 4, 5, 6 . - Avi Peretz (njk(AT)netvision.net.il), Feb 23 2001 and Ola Veshta (olaveshta(AT)my-deja.com), Mar 30 2001.

For n>=1, a(n) equals the number of words of length n on the alphabet {0,1,...,7} with no two adjacent letters identical. - Milan Janjic, Jan 31 2015 [Corrected by David Nacin, May 31 2017]

a(n) is the number of octonary sequences of length n such that no two consecutive terms have distance 4. - David Nacin, May 31 2017

Coordination sequence for infinite tree with valency 10.

The n-th term of the coordination sequence of the infinite tree with valency 2m is the same as the number of reduced words of size n in the free group on m generators. In the five sequences A003946, A003948, A003950, A003952, A003954 m is 2, 3, 4, 5, 6 . - Avi Peretz (njk(AT)netvision.net.il), Feb 23 2001 and Ola Veshta (olaveshta(AT)my-deja.com), Mar 30 2001

Except 1, all terms are in A033583. - Vincenzo Librandi, May 26 2014

For n>=1, a(n) equals the number of words of length n on alphabet {0,1,...,9} with no two adjacent letters identical. - Milan Janjic, Jan 31 2015 [Corrected by David Nacin, May 31 2017]

a(n) is the number of sequences over the alphabet {0,1,...,9} of length n such that no two consecutive terms have distance 5. - David Nacin, May 31 2017

Reverse of A029635. Row sums are A003945. Diagonal sums are Fibonacci(n+2) = Sum_{k=0..floor(n/2)} (2n-3k)*C(n-k,n-2k)/(n-k). - Paul Barry, Jan 30 2005

Riordan array ((1+x)/(1-x), x/(1-x)). The signed triangle (-1)^(n-k)T(n,k) or ((1-x)/(1+x), x/(1+x)) is the inverse of A055248. Row sums are A003945. Diagonal sums are F(n+2). - Paul Barry, Feb 03 2005

Row sums = A003945: (1, 3, 6, 12, 24, 48, 96, ...) = (1, 3, 7, 15, 31, 63, 127, ...) - (0, 0, 1, 3, 7, 15, 31, ...); where (1, 3, 7, 15, ...) = A000225. - Gary W. Adamson, Apr 22 2007

Triangle T(n,k), read by rows, given by (2,-1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 17 2011

A029653 is jointly generated with A208510 as an array of coefficients of polynomials v(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+x*v(n-1)x and v(n,x)=u(n-1,x)+x*v(n-1,x)+1. See the Mathematica section. - Clark Kimberling, Feb 28 2012

For a closed-form formula for arbitrary left and right borders of Pascal like triangle, see A228196. - Boris Putievskiy, Aug 18 2013

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

The n-th row polynomial is (2 + x)*(1 + x)^(n-1) for n >= 1. More generally, the n-th row polynomial of the Riordan array ( (1-a*x)/(1-b*x), x/(1-b*x) ) is (b - a + x)*(b + x)^(n-1) for n >= 1. - Peter Bala, Feb 25 2018

For n >= 1, a(n) equals the number of words of length n-1 on the alphabet {0,1,...,12} with no two adjacent letters identical. - Milan Janjic, Jan 31 2015

For n>=1, a(n) equals the numbers of words of length n-1 on alphabet {0,1,...,13} with no two adjacent letters identical. - Milan Janjic, Jan 31 2015

For n>=1, a(n) equals the numbers of words of length n-1 on alphabet {0,1,...,14} with no two adjacent letters identical. -Milan Janjic, Jan 31 2015

The initial terms coincide with those of A003954, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

Consider the k-fold Cartesian products CP(n,k) of the vector A(n) = [1, 2, 3, ..., n].

An element of CP(n,k) is a n-tuple T_t of the form T_t = [i_1, i_2, i_3, ..., i_k] with t=1, .., n^k.

We count members T of CP(n,k) which satisfy some condition delta(T_t), so delta(.) is an indicator function which attains values of 1 or 0 depending on whether T_t is to be counted or not; the summation sum_{CP(n,k)} delta(T_t) over all elements T_t of CP produces the count.

For the triangle here we have delta(T_t) = 0 if for any two i_j, i_(j+1) in T_t one has i_j = i_(j+1): T(n,k) = Sum_{CP(n,k)} delta(T_t) = Sum_{CP(n,k)} delta(i_j = i_(j+1)).

The test on i_j > i_(j+1) generates A158498. One gets the Pascal triangle A007318 if the indicator function tests whether for any two i_j, i_(j+1) in T_t one has i_j >= i_(j+1).

Use of other indicator functions can also calculate the Bell numbers A000110, A000045 or A000108.