A195032 Vertex number of a square spiral in which the length of the first two edges are the legs of the primitive Pythagorean triple [5, 12, 13]. The edges of the spiral have length A195031.

0, 5, 17, 27, 51, 66, 102, 122, 170, 195, 255, 285, 357, 392, 476, 516, 612, 657, 765, 815, 935, 990, 1122, 1182, 1326, 1391, 1547, 1617, 1785, 1860, 2040, 2120, 2312, 2397, 2601, 2691, 2907, 3002, 3230, 3330, 3570, 3675, 3927, 4037, 4301, 4416, 4692

Offset

0,2

Comments

Zero together with partial sums of A195031.

The spiral contains infinitely many Pythagorean triples in which the hypotenuses on the main diagonal are the positives multiples of 13 (cf. A008595). The vertices on the main diagonal are the numbers A195037 = (5+12)*A000217 = 17*A000217, where both 5 and 12 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 5, while the distance "b" between nearest edges that are parallel to the initial edge is 12, so the distance "c" between nearest vertices on the same axis is 13 because from the Pythagorean theorem we can write c = (a^2 + b^2)^(1/2) = sqrt(5^2 + 12^2) = sqrt(25 + 144) = sqrt(169) = 13. - 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).

Formula

From Bruno Berselli, Oct 13 2011: (Start)

G.f.: x*(5 + 12*x)/((1 + x)^2*(1 - x)^3).

a(n) = (1/2)*((2*n + (-1)^n + 3)/4)*((34*n - 3*(-1)^n+3)/4) = (2*n*(17*n + 27) + (14*n - 3)*(-1)^n + 3)/16.

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

E.g.f.: (1/16)*((3 + 88*x + 34*x^2)*exp(x) - (3 + 14*x)*exp(-x)). - Franck Maminirina Ramaharo, Nov 23 2018

Mathematica

a[n_] := (2 n (17 n + 27) + (14 n - 3)*(-1)^n + 3)/16; Array[a, 50, 0] (* Amiram Eldar, Nov 23 2018 *)

Prog

(Magma) [(2*n*(17*n+27)+(14*n-3)*(-1)^n+3)/16: n in [0..50]]; // Vincenzo Librandi, Oct 14 2011
(PARI) vector(50, n, n--; (2*n*(17*n+27)+(14*n-3)*(-1)^n+3)/16) \\ G. C. Greubel, Nov 23 2018
(Sage) [(2*n*(17*n+27)+(14*n-3)*(-1)^n+3)/16 for n in range(50)] # G. C. Greubel, Nov 23 2018

Crossrefs

Cf. A195020, A195031, A195034, A195036, A195037.

Sequence in context: A268526 A018447 A196142 * A264618 A271471 A217512

Adjacent sequences: A195029 A195030 A195031 * A195033 A195034 A195035

Keyword

nonn,easy

Author

Omar E. Pol, Sep 12 2011

Status

approved