A247588 Number of integer-sided acute triangles with largest side n.

1, 2, 3, 5, 6, 8, 11, 13, 15, 17, 21, 25, 27, 31, 34, 39, 43, 48, 52, 56, 63, 67, 73, 80, 84, 90, 96, 104, 111, 117, 126, 132, 140, 147, 154, 165, 172, 183, 189, 198, 210, 219, 229, 237, 247, 260, 270, 282, 292, 302

Offset

1,2

Links

Table of n, a(n) for n=1..50.

Vladimir Letsko, Mathematical Marathon, problem 192 (in Russian).

Formula

a(n) = Sum_{j=0..floor(n*(1 - sqrt(2)/2))} (n - j - floor(sqrt(2*j*n - j^2))). - Anton Nikonov, Oct 06 2014

a(n) = (1/8)*(-4*ceiling((n - 1)/sqrt(2)) + 4*n^2 - A000328(n) + 1), n > 1. - Mats Granvik, May 23 2015

Example

a(3) = 3 because there are 3 integer-sided acute triangles with largest side 3: (1,3,3); (2,3,3); (3,3,3).

Maple

tr_a:=proc(n) local a, b, t, d; t:=0:
for a to n do
for b from max(a, n+1-a) to n do
d:=a^2+b^2-n^2:
if d>0 then t:=t+1 fi
od od;
t; end;

Mathematica

a[ n_] := Length @ FindInstance[ n >= b >= a >= 1 && n < b + a && n^2 < b^2 + a^2, {a, b}, Integers, 10^9]; (* Michael Somos, May 24 2015 *)

Prog

(PARI) a(n) = sum(j=0, n*(1 - sqrt(2)/2), n - j - floor(sqrt(2*j*n - j^2))); \\ Michel Marcus, Oct 07 2014
(PARI) {a(n) = sum(j=0, n - sqrtint(n*n\2) - 1, n - j - sqrtint(2*j*n - j*j))}; /* Michael Somos, May 24 2015 */

Crossrefs

Cf. A046080, A002623, A224921, A247586, A247587, A247589.

Sequence in context: A050834 A294911 A179799 * A275341 A191884 A291693

Adjacent sequences: A247585 A247586 A247587 * A247589 A247590 A247591

Keyword

nonn

Author

Vladimir Letsko, Sep 20 2014

Status

approved