cp's OEIS Frontend

Number of compositions of n into 1's, 3's and 4's. - Len Smiley, May 08 2001

The sum of any two alternating terms (terms separated by one term) produces a number from the Fibonacci sequence. (e.g. 4+9=13, 9+25=34, 6+15=21, etc.) Taking square roots starting from the first term and every other term after, we get the Fibonacci sequence. - Sreyas Srinivasan (sreyas_srinivasan(AT)hotmail.com), May 02 2002

(1 + x + 2*x^2 + x^3)/(1 - x - x^3 - x^4) = 1 + 2*x + 4*x^2 + 6*x^3 + 9*x^4 + 15*x^5 + 25*x^6 + 40*x^7 + ... is the g.f. for the number of binary strings of length where neither 101 nor 111 occur. [Lozansky and Rousseau] Or, equivalently, where neither 000 nor 010 occur.

Equivalently, a(n+2) is the number of length-n binary strings with no two set bits with distance 2; see fxtbook link. - Joerg Arndt, Jul 10 2011

a(n) is the number of words written with the letters "a" and "b", with the following restriction: any "a" must be followed by at least two letters, the second of which is a "b". - Bruno Petazzoni (bpetazzoni(AT)ac-creteil.fr), Oct 31 2005. [This is also equivalent to the previous two conditions.]

Let a(0) = 1, then a(n) = partial products of Product_{n>2} (F(n)/F(n-1))^2 = 1*1*2*2*(3/2)*(3/2)*(5/3)*(5/3)*(8/5)*(8/5)*.... E.g., a(7) = 15 = 1*1*1*2*2*(3/2)*(3/2)*(5/3). - Gary W. Adamson, Dec 13 2009

Number of permutations satisfying -k <= p(i) - i <= r and p(i)-i not in I, i=1..n, with k=1, r=3, I={1}. - Vladimir Baltic, Mar 07 2012

The 2-dimensional version, which counts sets of pairs no two of which are separated by graph-distance 2, is A273461. - Gus Wiseman, Nov 27 2019

a(n+1) is the number of multus bitstrings of length n with no runs of 4 ones. - Steven Finch, Mar 25 2020

Coefficients of the (left, normalized) shifted cyclotomic polynomial. Or, coefficients of the basic n-th q-series for q=-2. Indeed, let Y_n(x) = Sum_{k=0..n} x^k, having as roots all the n-th roots of unity except for 0; then coefficients in x of (-1)^n Y_n(-x-1) give exactly the n-th row of A059260 and a practical way to compute it. - Olivier Gérard, Jul 30 2002

The maximum in the (2n)-th row is T(n,n), which is A026641; also T(n,n) ~ (2/3)*binomial(2n,n). The maximum in the (2n-1)-th row is T(n-1,n), which is A014300 (but T does not have the same definition as in A026637); also T(n-1,n) ~ (1/3)*binomial(2n,n). Here is a generalization of the formula given in A026641: T(i,j) = Sum_{k=0..j} binomial(i+k-x,j-k)*binomial(j-k+x,k) for all x real (the proof is easy by induction on i+j using T(i,j) = T(i-1,j) + T(i,j-1)). - Claude Morin, May 21 2002

The second greatest term in the (2n)-th row is T(n-1,n+1), which is A014301; the second greatest term in the (2n+1)-th row is T(n+1,n) = 2*T(n-1,n+1), which is 2*A014301. - Claude Morin

Diagonal sums give A008346. - Paul Barry, Sep 23 2004

Riordan array (1/(1-x^2), x/(1-x)). As a product of Riordan arrays, factors into the product of (1/(1+x),x) and (1/(1-x),1/(1-x)) (binomial matrix). - Paul Barry, Oct 25 2004

Signed version is A239473 with relations to partial sums of sequences. - Tom Copeland, Mar 24 2014

From Robert Coquereaux, Oct 01 2014: (Start)

Columns of the triangle (cf. Example below) give alternate partial sums along nw-se diagonals of the Pascal triangle, i.e., sequences A000035, A004526, A002620 (or A087811), A002623 (or A173196), A001752, A001753, A001769, A001779, A001780, A001781, A001786, A001808, etc.

The dimension of the space of closed currents (distributional forms) of degree p on Gr(n), the Grassmann algebra with n generators, equivalently, the dimension of the space of Gr(n)-valued symmetric multilinear forms with vanishing graded divergence, is V(n,p) = 2^n T(p,n-1) - (-1)^p.

If p is odd V(n,p) is also the dimension of the cyclic cohomology group of order p of the Z2 graded algebra Gr(n).

If p is even the dimension of this cohomology group is V(n,p)+1.

Cf. A193844. (End)

From Peter Bala, Feb 07 2024: (Start)

The following remarks assume the row indexing starts at n = 1.

The sequence of row polynomials R(n,x), beginning R(1,x) = 1, R(2,x) = x, R(3,x) = 1 + x + x^2 , ..., is a strong divisibility sequence of polynomials in the ring Z[x]; that is, for all positive integers n and m, poly_gcd( R(n,x), R(m,x)) = R(gcd(n, m), x) - apply Norfleet (2005), Theorem 3. Consequently, the polynomial sequence {R(n,x): n >= 1} is a divisibility sequence; that is, if n divides m then R(n,x) divides R(m,x) in Z[x]. (End)

From Miquel A. Fiol, Oct 04 2024: (Start)

For j>=1, T(i,j) is the independence number of the (i-j)-supertoken graph FF_(i-j)(S_j) of the star graph S_j with j points.

(Given a graph G on n vertices and an integer k>=1, the k-supertoken (or reduced k-th power) FF_k(G) of G has vertices representing configurations of k indistinguishable tokens in the (not necessarily different) vertices of G, with two configurations being adjacent if one can be obtained from the other by moving one token along an edge. See an example below.)

Following the suggestion of Peter Munn, the k-supertoken graph FF_k(S_j) can also be defined as follows: Consider the Lattice graph L(k,j), whose vertices are the k^j j-vectors with elements in the set {0,..,k-1}, two being adjacent if they differ in just one coordinate by one unity. Then, FF_k(S_j) is the subgraph of L(k+1,j) induced by the vertices at distance at most k from (0,..,0). (End)

From Johannes W. Meijer, Jul 20 2011: (Start)

The triangle sums, see A180662 for their definitions, link this "Races with Ties" triangle with several sequences, see the crossrefs. Observe that the Kn4 sums lead to the golden rectangle numbers A001654 and that the Fi1 and Fi2 sums lead to the Jacobsthal sequence A001045.

The series expansion of G(x, y) = 1/((y*x-1)*(y*x+1)*((y+1)*x-1)) as function of x leads to this sequence, see the second Maple program. (End)

T(2n,k) = the number of hatted frog arrangements with k frogs on the 2xn grid. See the linked paper "Frogs, hats and common subsequences". - Chris Cox, Apr 12 2024

Sum_{k=0..n} T(n,k) = A078008(n);

Sum_{k=0..n} abs(T(n,k)) = A052953(n-1) for n > 0;

T(n,1) = n - 2 for n > 1;

T(n,2) = A000124(n-3) for n > 2;

T(n,3) = A003600(n-4) for n > 4;

T(n,n-6) = A001753(n-6) for n > 6;

T(n,n-5) = A001752(n-5) for n > 5;

T(n,n-4) = A002623(n-4) for n > 4;

T(n,n-3) = A002620(n-1) for n > 3;

T(n,n-2) = A008619(n-2) for n > 2;

T(n,n-1) = n mod 2 for n > 0;

T(2*n,n) = A072547(n+1).

Sum_{k=0..n} T(n,k)*x^k = A232015(n), A078008(n), A000012(n), A040000(n), A001045(n+2), A140725(n+1) for x = 2, 1, 0, -1, -2, -3 respectively. - Philippe Deléham, Nov 17 2013, Nov 19 2013

(1,a^n) Pascal triangle with a = -1. - Philippe Deléham, Dec 27 2013

T(n,k) = A112465(n,n-k). - Reinhard Zumkeller, Jan 03 2014

Also the triangle of the coefficients of the squares of the Fibonacci polynomials. Row n has 1+2*floor(n/2) terms. Sum of terms in row n = (Fibonacci(n+1))^2 (A007598).

From Michael A. Allen, Jun 24 2020: (Start)

T(n,k) is the number of tilings of an n-board (a board with dimensions n X 1) using k (1/2, 1/2)-fence tiles and 2*(n-k) half-squares (1/2 X 1 pieces, always placed so that the shorter sides are horizontal). A (1/2, 1/2)-fence is a tile composed of two 1/2 X 1 pieces separated by a gap of width 1/2.

T(n,k) is the (n, (n-k))-th entry of the (1/(1-x^2), x/(1-x)^2) Riordan array.

(-1)^(n+k)*T(n,k) is the (n, (n-k))-th entry of the (1/(1-x^2), x/(1+x)^2) Riordan array (A158454). (End)

T(n, k) is the number of tilings of an n-board that use k (1/2, 1)-fences and n-k squares. A (1/2, 1)-fence is a tile composed of two pieces of width 1/2 separated by a gap of width 1. (Result proved in paper by K. Edwards - see the links section.) - Michael A. Allen, Apr 28 2019

T(n, k) is the (n, n-k)-th entry in the (1/(1-x^3), x*(1+x)/(1-x^3)) Riordan array. - Michael A. Allen, Mar 11 2021