cp's OEIS Frontend

Consider the quadratic cyclotomic polynomial f(x) = x^2+x+1 and the quotients f(x + n*f(x))/f(x), as in A168235 and A168240. a(n) is the real part of the quotient at x = 1+sqrt(-5).

The imaginary part of the quotient is sqrt(5)*A045944(n).

As stated in short description of A168244 the quotient is in two parts: rational integers (cf. A168244) and rational integer multiples of sqrt(-5). It so happens that the sequence of rational integer coefficients of sqrt(-5) is A045944. - A.K. Devaraj, Nov 22 2009

This sequence contains half of all integers m such that -8*m +17 is an odd square. The other half are found in A091823 multiplied by -1. The squares resulting from A168244 are (4*n - 3)^2, those from A091823 are (4*n + 3)^2. - Klaus Purath, Jul 11 2021

Consider the quadratic cyclotomic polynomial f(x) = x^2+x+1 and the quotients defined by f(x + n*f(x))/f(x). a(n) is the quotient at x=3.

Let P be a polygon with vertices (0,0), (0,2), (1,1) and (0,3/2). The number of integer points in nP is counted by this quasi-polynomial (nP is the n-fold dilation of P). See Wikipedia in Links section.

From Bob Selcoe, Sep 10 2016: (Start)

a(n) = the number of partitions in reverse lexicographic order starting with n 3's followed by n 2's; i.e., the number of partitions summing to 5n such that no part > 3 and the number of 3's digits <= the number of 2's digits.

First differences are A047346(n+1); second differences are 4 when n is even and 3 when n is odd (i.e., A010702(n+1)); third differences are 1 when n is even and -1 when n is odd. (End)