A046727 Related to Pythagorean triples: alternate terms of A001652 and A046090.

0, 3, 21, 119, 697, 4059, 23661, 137903, 803761, 4684659, 27304197, 159140519, 927538921, 5406093003, 31509019101, 183648021599, 1070379110497, 6238626641379, 36361380737781, 211929657785303, 1235216565974041, 7199369738058939, 41961001862379597, 244566641436218639

Offset: 0

Author

N. J. A. Sloane

Keywords

• easy
• nonn

Comments

For a triple (a,b,c) there exist k,m such that (a,b,c) = (k^2 - m^2, 2*k*m, k^2 + m^2). Here k = A001333(n) and m = A001333(n+1), so this sequence is identical to the Pell oblongs A084159 for n > 0. - Lambert Klasen (Lambert.Klasen(AT)gmx.de), Nov 10 2004

a(n), for n >= 1, gives the odd length (in some unit) catheti (legs) of the (primitive) Pythagorean triples which have absolute length difference of the catheti equal to one. See a W. Lang comment on A001653 on how to generate all such Pythagorean triples. - Wolfdieter Lang, Mar 08 2012

References

• A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, pp. 122-125, 1964.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
• A. Bogomolny, The Trinary Tree(s) underlying Primitive Pythagorean Triples.
• Dan Romik, The dynamics of Pythagorean Triples, Trans. Amer. Math. Soc. 360 (2008), 6045-6064.
• P. E. Trier, "Almost Isosceles" Right-Angled Triangles, Eureka, No. 4, May 1940, pp. 9 - 11.
• Index entries for linear recurrences with constant coefficients, signature (5,5,-1).

Crossrefs

Cf. A001109, A001333, A001652, A005319, A046090, A046729.

Essentially the same as A084159.

Programs

Haskell

a046727 n = a046727_list !! n
a046727_list = 0 : f (tail a001652_list) (tail a046090_list) where
   f (x::xs) (:y:ys) = x : y : f xs ys
-- Reinhard Zumkeller, Jan 10 2012

Magma

I:=[0,3,21,119]; [n le 4 select I[n] else 5*Self(n-1)+5*Self(n-2)-Self(n-3): n in [1..30]]; // Vincenzo Librandi, Nov 04 2016

Mathematica

RecurrenceTable[{a[n+2]==6a[n+1] -a[n] -4*(-1)^n, a[0]==3, a[1]==21}, a, {n, 30}] (* Ron Knott, Jul 01 2013 *)
LinearRecurrence[{5,5,-1}, {0,3,21,119}, 30] (* Vincenzo Librandi, Nov 04 2016 *)

PARI

concat(0, Vec(x*(3+6*x-x^2)/((1+x)*(1-6*x+x^2)) + O(x^30))) \\ Colin Barker, Nov 03 2016

SageMath

[(lucas_number2(2*n+1,2,-1) +2*(-1)^n)/4 -int(n==0) for n in range(41)] # G. C. Greubel, Feb 11 2023

Formula

Values of x obtained by repeatedly multiplying the triple (x, y, z) = (3, 4, 5) by the matrix A = ([1 2 2], [2 1 2], [2 2 3]), the Across matrix of "The Trinary Tree(s) underlying Primitive Pythagorean Triples" generating matrices. - Vim Wenders, Jan 14 2004

For n > 0, a(n) = A001333(n)*A001333(n+1). - Lambert Klasen (Lambert.Klasen(AT)gmx.de), Nov 10 2004

G.f.: x*(3+6*x-x^2)/((1+x)*(1-6*x+x^2)). - R. J. Mathar, Jul 08 2009

a(n) + a(n+1) = A005319(n+1), n > 0. - R. J. Mathar, Jul 13 2009

a(n) = 6*a(n-1) - a(n-2) - 4*(-1)^n. - Ron Knott, Jul 01 2013

From Colin Barker, Nov 03 2016: (Start)

a(n) = (2*(-1)^n + (1+sqrt(2))^(2*n+1) + (1-sqrt(2))^(2*n+1))/4 for n > 0.

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

From G. C. Greubel, Feb 11 2023: (Start)

a(n) = (1/2)*(A001109(n+1) + A001109(n) + (-1)^n) - [n=0].

a(n) = (A001333(2*n+1) + (-1)^n)/2 - [n=0]. (End)

E.g.f.: exp(-x)*(1 + exp(4*x)*(cosh(2*sqrt(2)*x) + sqrt(2)*sinh(2*sqrt(2)*x)))/2 - 1. - Stefano Spezia, Aug 03 2024

Extensions

More terms from Sascha Kurz, Jan 23 2003