cp's OEIS Frontend

a(n) as illustrated is related to the following sequences: The row sum values are A001400. The column sums are A000292. The row lengths are the stuttering sequence A037915 (stutter values in A016777). The column lengths are the sequence A016777. The max values in each column are A001971. - Alford Arnold, Aug 16 2004

The Gaussian polynomial (or Gaussian binomial) [n,3]q is an example of a q-binomial coefficient (see the link) and may be defined for n >= 3 by [n,3]_q = ([n]_q * [n-1]_q * [n-2]_q)/([1]_q * [2]_q * [3]_q), where [n]_q := q^n - 1. The first few values are: [3,3]_q = 1; [4,3]_q = 1 + q + q^2 + q^3; [5,3]_q = 1 + q + 2q^2 + 2q^3 + 2q^4 + q^5 + q^6. - _Peter Bala, Sep 23 2007

The entry a(p,w), p >= 0, w = 0,1,...,3*p, of this irregular triangle is the number of nonnegative solutions of m_0 + m_1 + m_2 + m_3 = p and 1*m_1 + 2*m_2 + 3*m_3 = w. See the Hawkins reference given in A008967, p. 264, (4,7),(4.8), concerning Cayley's counting problem. N(p,3,w) there equals a(p,w). The o.g.f. has been given in the formula section by Peter Bala. See also the Cayley reference given in A008967, p. 110, 35. with m = 3, Theta = p and q = w. - Wolfdieter Lang, Dec 02 2012

The entry a(p,w) p >= 0, w = 0,1,...,3*p, of this array gives the number of partitions of w into at most p parts, each at most 3. This follows from the preceding comment with the two Diophantine equations. From Andrews, p. 33 and p. 35, a(p,w) (called there p(N,M,n) with N=p, M=3, n=w) gives also the number of partitions of w into at most 3 parts, each at most p. This is in accordance with the symmetry of the q-binomials [p+3,p] = [p+3,3]. - Wolfdieter Lang, Dec 04 2012

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. The sequence (1+[3n/4]) is a subsequence ofy A037915.

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. Every pair of rows eventually intersperse. As a sequence, A194998 is a permutation of the positive integers, with inverse A195099.

The row length sequence for this array is A037915(k-4)= floor(3*(k-4)/4)+1, k>=4: [1, 1, 2, 3, 4, 4, 5, 6, 7, 7, 8, 9, 10, 10, 11, ...].

The g.f. G(k,x) for the k-th column (with leading zeros) of array A090214 is given there. The recurrence is G(k,x) = x*sum(binomial(k-r,4-r)*fallfac(4,4-r)*G(k-r,x),r=1..4))/(1-fallfac(k,4)*x), k>=4, with inputs G(k,x)=0 for k=1,2,3 and G(4,x)=x/(1-4!*x); where fallfac(n,m) := A008279(n,m) (falling factorials with fallfac(n,0) := 1). Computed from the Blasiak et al. reference, eqs. (20) and (21) with r=4: recurrence for S_{4,4}(n,k).

A contraction in the complement of any set of paths reduces the total number of edges in the complement by at most 4. This gives an upper bound for the Hadwiger number which is obtainable for all path lengths except 4 and 5. In particular, for n >= 6, the complement of a P_n reduces to the complement of a P_{n-4} union 3 universal nodes by contracting the second and second to last nodes of the path. With P_8 and P_9 the 2nd and 6th nodes should be contracted (instead of reducing to P_4 or P_5 respectively). - Andrew Howroyd, Jun 18 2025

The graph of this sequence is a special case of de Rham's fractal curve. In general, the graph of any sequence of the form a(n)=sum_(1..n) [Sk(n)mod m - floor(p*Sk(n)/q)mod m], where Sk(n) is the digit sum of n, n in k-ary notation, p,q,m integers, gives a de Rham fractal curve. The self-symmetries of all of de Rham curves are given by the monoid that describes the symmetries of the infinite binary tree or Cantor set. This so-called period-doubling monoid is a subset of the modular group.

Inspired by A276669.

Two other sequences in the same vein could be constructed, one with points from 0 to n-1 and the other with points from 0 to n. The latter would only insert a zero before n.

Column 1 appears to be A037915 and the last column appears to be A002265.