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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, 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 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) 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 Cf. A001109, A001333, A001652, A005319, A046090, 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 N. J. A. Sloane EXTENSIONS More terms from Sascha Kurz, Jan 23 2003 STATUS approved

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Last modified June 5 17:29 EDT 2023. Contains 363138 sequences. (Running on oeis4.)