A195034 Vertex number of a square spiral in which the length of the first two edges are the legs of the primitive Pythagorean triple [21, 20, 29]. The edges of the spiral have length A195033.

0, 21, 41, 83, 123, 186, 246, 330, 410, 515, 615, 741, 861, 1008, 1148, 1316, 1476, 1665, 1845, 2055, 2255, 2486, 2706, 2958, 3198, 3471, 3731, 4025, 4305, 4620, 4920, 5256, 5576, 5933, 6273, 6651, 7011, 7410, 7790, 8210, 8610, 9051, 9471

Offset: 0

Author

Omar E. Pol, Sep 12 2011

Keywords

• nonn
• easy

Comments

Zero together with partial sums of A195033.

The only primes in the sequence are 41 and 83 since a(n) = (1/2)*((2*n+(-1)^n+3)/4)*((82*n-43*(-1)^n+43)/4). - Bruno Berselli, Oct 12 2011

The spiral contains infinitely many Pythagorean triples in which the hypotenuses on the main diagonal are the positives multiples of 29 (Cf. A195819). The vertices on the main diagonal are the numbers A195038 = (21+20)*A000217 = 41*A000217, where both 21 and 20 are the first two edges in the spiral. The distance "a" between nearest edges that are perpendicular to the initial edge of the spiral is 21, while the distance "b" between nearest edges that are parallel to the initial edge is 20, so the distance "c" between nearest vertices on the same axis is 29 because from the Pythagorean theorem we can write c = (a^2+b^2)^(1/2) = sqrt(21^2+20^2) = sqrt(441+400) = sqrt(841) = 29. - Omar E. Pol, Oct 12 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Ron Knott, Pythagorean triangles and Triples
• Eric Weisstein's World of Mathematics, Pythagorean Triple
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A195020, A195032, A195033, A195036, A195038.

Programs

Magma

[(2*n*(41*n+83)-(2*n+43)*(-1)^n+43)/16: n in [0..50]]; // Vincenzo Librandi, Oct 14 2011

Mathematica

LinearRecurrence[{1,2,-2,-1,1},{0,21,41,83,123},50] (* Harvey P. Dale, May 02 2012 *)

PARI

concat(0, Vec(x*(21+20*x)/((1+x)^2*(1-x)^3) + O(x^60))) \\ Michel Marcus, Mar 08 2016

Formula

From Bruno Berselli, Oct 12 2011: (Start)

G.f.: x*(21+20*x)/((1+x)^2*(1-x)^3).

a(n) = (2*n*(41*n+83)-(2*n+43)*(-1)^n+43)/16.

a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5).

a(n)-a(-n-2) = A142150(n+1). (End)