A055523 Longest other leg of a Pythagorean triangle with n as length of a leg.

4, 3, 12, 8, 24, 15, 40, 24, 60, 35, 84, 48, 112, 63, 144, 80, 180, 99, 220, 120, 264, 143, 312, 168, 364, 195, 420, 224, 480, 255, 544, 288, 612, 323, 684, 360, 760, 399, 840, 440, 924, 483, 1012, 528, 1104, 575, 1200, 624, 1300, 675, 1404, 728, 1512, 783

Offset: 3

Author

Henry Bottomley, May 22 2000

Keywords

• nonn
• easy

Links

• Todd Silvestri, Table of n, a(n) for n = 3..1002
• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Crossrefs

Cf. A009112, A046079, A046080, A046081, A054435, A054436, A055522, A055524, A055525, A055526, A055527.

Programs

Maple

seq(`if`(n::even, (n/2-1)*(n/2+1), (n-1)*(n+1)/2), n=3..100); # Robert Israel, Dec 16 2014

Mathematica

a[n_Integer/;n>=3]:=(3 (n^2-2)+(-1)^(n+1) (n^2+2))/8 (* Todd Silvestri, Dec 16 2014 *)

PARI

Vec(x^3*(x^3-3*x-4)/((x-1)^3*(x+1)^3) + O(x^100)) \\ Colin Barker, Sep 15 2014

Formula

a(n) = 2*A055522(n)/n = sqrt(A055524(n)^2-n^2).

a(2k) = (k-1)*(k+1), a(2k+1) = 2k*(k+1).

a(n) = 3*a(n-2)-3*a(n-4)+a(n-6). G.f.: x^3*(x^3-3*x-4) / ((x-1)^3*(x+1)^3). - Colin Barker, Sep 15 2014

a(n) = (3*(n^2-2)+(-1)^(n+1)*(n^2+2))/8. - Todd Silvestri, Dec 16 2014

E.g.f.: 1 + (3*x^2/8 + 3*x/8 - 3/4)*exp(x) + (-x^2/8 + x/8 - 1/4)*exp(-x). - Robert Israel, Dec 16 2014