cp's OEIS Frontend

Sum of i-th and j-th triangular numbers, where i=A096029(n), j=A096030(n); i.e., a(n) = A000217(A096029(n)) + A000217(A096030(n)). - Lekraj Beedassy, Jun 16 2004

For every k in the sequence, there is exactly 1 square number that can be subtracted to leave a pronic (A002378). E.g., 27 - 25 = 2, 99 - 9 = 90. - Jon Perry, Nov 06 2010

See A208295 for details concerning the preceding Jon Perry comment. - Wolfdieter Lang, Mar 29 2012

a(k) appears in the o.g.f. for floor(A002144(k)*j^2/4), j >= 0, for k >= 1: x*(a(k)*(1 + x^2) + b(k)*x)/((1 - x)^3*(1 + x)), together with b(k) = (A002144(k) + 1)/2 = A119681(k). - Wolfdieter Lang, Aug 07 2013

This generalizes A032527, which uses 5, to 13 (the next prime 1 (mod 4)). The figures in A032527 use n/2 concentric dotted pentagons for even n, and (n-1)/2 concentric dotted pentagons plus an extra dot in the middle if n is odd. In the present case one can take n/2 concentric dotted 13-gons (the dot numbers of each side for these 13-gons are 2, 4, 6, ..., n) for even n>=2. There is no figure for n = 0. For odd n one has (n-1)/2 concentric dotted 13-gons (the dot numbers of each side for these 13-gons are 3, 5, 7, ..., n) and an extra dotted 3-gon in the middle. See the example section below for the counting.

a(n) = -N(-floor(n/2),n) with the N(a,b) = ((2*a+b)^2 - b^2*13)/4, the norm for integers a + b*omega(13), a, b rational integers, in the quadratic number field Q(sqrt(13)), where omega(13) = (1 + sqrt(13))/2.

a(n) = max({|N(a,n)|,a = -n..+n}) = |N(-floor(n/2),n)| = 3*n^2 + n*floor(n/2) - floor(n/2)^2 = floor(13*n^2/4) (the last eq. checks for even and odd n).

In the general case one has for primes 1 (mod 4), p(k) = A002144(k), k >= 1, a(p(k);n) = floor(p(k)*n^2/4) with o.g.f. G(p(k);x) = x*(A(k)*(1+x^2) + B(k)*x)/((1-x)^3*(1+x)), where A(k) = A005098(k) = (p(k)-1)/4 and B(k) = A119681(k) = (p(k)+1)/2. This follows from the alternative formula a(p(k),n) = p(k)*n^2/4 + ((-1)^n-1)/8, n >= 0 (which checks for even and odd n). Because the denominator of the o.g.f. is 1-2*x+2*x^3-x^4 the recurrence given by Bruno Berselli below holds for all a(k;n) sequences with inputs for n = -1, 0, 1, 2 given by (p(k)-1)/4, 0, (p(k)-1)/4, p(k), respectively.

The dot counting in the concentric p(k)-gons is similar to the one described for p = 5 in A032527 and for p=13 here. For odd n one puts an additional dotted A(k)-gon into the center. - Wolfdieter Lang, Aug 08 2013

Conjecture: Every prime of the form 4*k+1 (A002144) is contained in the sequence {2*a(n)-1}.

The author's former conjecture that, for n>=2 the numbers {2*a(n)-1} are consecutive primes of the form 4*k+1, was disproved at n = 553 by Peter J. C. Moses. (553*2 - 1 = 1105 is the smallest term which is a product of three distinct (4*k+1)-primes). - Vladimir Shevelev, Sep 27 2017

553 is also (after 1) the smallest number which is missing from A119681 but is present here. - R. J. Mathar, Sep 29 2017