OFFSET
0,2
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).
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
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) = (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
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 *)
PROG
(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
(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
(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
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
EXTENSIONS
More terms from Sascha Kurz, Jan 23 2003
STATUS
approved