A095861 Number of primitive Pythagorean triangles of form (X,Y,Y+1) with hypotenuse Y+1 less than or equal to n.

0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset

1,13

Comments

Indices where records occur are hypotenuse values A001844 (Y+1); corresponding leg values given by A046092 (Y) and A005408 (X).

The number k, for k >= 0, appears exactly 4*(k+1) = A008586(k+1) times. The number k appears for the first time for a(4*T(k) + 1) = a(A001844(k)), where T(k) = A000217(k); see the preceding comment on records. - Wolfdieter Lang, Mar 01 2022

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

S. Crowley, Mellin and Laplace Integral Transforms Related to the Harmonic Sawtooth Map and a Diversion Into The Theory Of Fractal Strings, vixra:1202.0079, 2012.

S. Crowley, Integral Transforms of the Harmonic Sawtooth Map, The Riemann Zeta Function, Fractal Strings, and a Finite Reflection Formula, arXiv preprint arXiv:1210.5652 [math.NT], 2012. - N. J. A. Sloane, Jan 01 2013

Eric Weisstein's World of Mathematics, Pythagorean Triple.

Formula

a(n) = floor((sqrt(2*n-1) - 1)/2).

Maple

A095861 := proc (x) local y; y := 0; while (2*y+2)*(y+2) <= x do y := y+1 end do; return y end proc; map(A095861, [seq(k, k = 0 .. 100)]) # Stephen Crowley, Aug 01 2009

Mathematica

Table[Floor[(Sqrt[2*n-1] - 1)/2], {n, 1, 100}] (* Jean-François Alcover, Apr 01 2018 *)

Prog

(Magma) [Floor((Sqrt(2*n-1)-1)/2): n in [1..105]]; // Vincenzo Librandi, Apr 01 2018

Crossrefs

Cf. A000217, A001844, A005408, A008586, A046092.

Sequence in context: A110591 A105209 A179076 * A111855 A071701 A342696

Adjacent sequences: A095858 A095859 A095860 * A095862 A095863 A095864

Keyword

nonn,easy,changed

Author

Ray Chandler, Jun 20 2004

Status

approved