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 A046727 Related to Pythagorean triples: alternate terms of A001652 and A046090. 5
 0, 3, 21, 119, 697, 4059, 23661, 137903, 803761, 4684659, 27304197, 159140519, 927538921, 5406093003, 31509019101, 183648021599, 1070379110497, 6238626641379, 36361380737781, 211929657785303, 1235216565974041, 7199369738058939, 41961001862379597 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS For a triple (a,b,c) there exist k,m such that (a,b,c) = (k^2-m^2, 2km, 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. 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 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 - (3-2*sqrt(2))^n*(-1+sqrt(2)) + (1+sqrt(2))*(3+2*sqrt(2))^n)/4 for n > 0. a(n) = 5*a(n-1) + 5*a(n-2) - a(n-3) for n > 3. (End) MATHEMATICA RecurrenceTable[{a[n + 2] == 6 a[n + 1] - a[n] - (-1)^n 4, a == 3,   a == 21}, a, {n, 30}] (* Ron Knott, Jul 01 2013 *) Join[{0}, LinearRecurrence[{5, 5, -1}, {3, 21, 119}, 25]] (* 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 CROSSREFS Cf. A046729. Essentially the same as A084159. Sequence in context: A092634 A178537 A084159 * A283421 A117512 A068127 Adjacent sequences:  A046724 A046725 A046726 * A046728 A046729 A046730 KEYWORD easy,nonn AUTHOR EXTENSIONS More terms from Sascha Kurz, Jan 23 2003 STATUS approved

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Last modified July 2 02:41 EDT 2020. Contains 335389 sequences. (Running on oeis4.)