A057721 a(n) = n^4 + 3*n^2 + 1.

1, 5, 29, 109, 305, 701, 1405, 2549, 4289, 6805, 10301, 15005, 21169, 29069, 39005, 51301, 66305, 84389, 105949, 131405, 161201, 195805, 235709, 281429, 333505, 392501, 459005, 533629, 617009, 709805, 812701, 926405, 1051649

Offset: 0

Author

N. J. A. Sloane, Oct 27 2000

Keywords

• nonn
• easy
• changed

Comments

Longest possible side c of a triangle with integer sides a <= b < c and inradius n. Triangle has sides (n^2+2, n^4+2n^2+1, n^4+3n^2+1). Proved by Joseph Myers, Jun 11 2006.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michelle Rudolph-Lilith, On the Product Representation of Number Sequences, with Application to the Fibonacci Family, arXiv:1508.07894 [math.NT], 2015
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

See A120062 for sequences related to integer-sided triangles with integer inradius n.

Cf. A120062 [triangles with integer inradius], A120063 [minimum of their longest sides].

Programs

GAP

List([0..40], n-> n^4+3*n^2+1); # G. C. Greubel, Aug 12 2019

Magma

[n^4+3*n^2+1: n in [0..40]]; // G. C. Greubel, Aug 12 2019

Mathematica

a[n_]:=n^4+3n^2+1; Array[a,40,0] (* Vladimir Joseph Stephan Orlovsky, Nov 03 2009 *)

PARI

vector(40, n, n--; n^4+3*n^2+1) \\ G. C. Greubel, Aug 12 2019

Formula

a(n) = denominator of Integral_{x=0..infinity} sin(n*x)/exp((n^2+1)*x). - Francesco Daddi, Jul 07 2013