|
%I #62 Feb 12 2023 10:17:27
%S 0,3,21,119,697,4059,23661,137903,803761,4684659,27304197,159140519,
%T 927538921,5406093003,31509019101,183648021599,1070379110497,
%U 6238626641379,36361380737781,211929657785303,1235216565974041,7199369738058939,41961001862379597
%N Related to Pythagorean triples: alternate terms of A001652 and A046090.
%C 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
%C 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
%D A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, pp. 122-125, 1964.
%H Reinhard Zumkeller, <a href="/A046727/b046727.txt">Table of n, a(n) for n = 0..1000</a>
%H A. Bogomolny, <a href="http://www.cut-the-knot.org/pythagoras/PT_matrix.shtml">The Trinary Tree(s) underlying Primitive Pythagorean Triples</a>.
%H Dan Romik, <a href="https://doi.org/10.1090/S0002-9947-08-04467-X">The dynamics of Pythagorean Triples</a>, Trans. Amer. Math. Soc. 360 (2008), 6045-6064.
%H P. E. Trier, <a href="https://archim.org.uk/eureka/archive/Eureka-4.pdf">"Almost Isosceles" Right-Angled Triangles</a>, Eureka, No. 4, May 1940, pp. 9 - 11.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,5,-1).
%F 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
%F For n > 0, a(n) = A001333(n)*A001333(n+1). - Lambert Klasen (Lambert.Klasen(AT)gmx.de), Nov 10 2004
%F G.f.: x*(3+6*x-x^2)/((1+x)*(1-6*x+x^2)). - _R. J. Mathar_, Jul 08 2009
%F a(n) + a(n+1) = A005319(n+1), n > 0. - _R. J. Mathar_, Jul 13 2009
%F a(n) = 6*a(n-1) - a(n-2) - 4*(-1)^n. - _Ron Knott_, Jul 01 2013
%F From _Colin Barker_, Nov 03 2016: (Start)
%F a(n) = (2*(-1)^n + (1+sqrt(2))^(2*n+1) + (1-sqrt(2))^(2*n+1))/4 for n > 0.
%F a(n) = 5*a(n-1) + 5*a(n-2) - a(n-3) for n > 3. (End)
%F From _G. C. Greubel_, Feb 11 2023: (Start)
%F a(n) = (1/2)*(A001109(n+1) + A001109(n) + (-1)^n) - [n=0].
%F a(n) = (A001333(2*n+1) + (-1)^n)/2 - [n=0]. (End)
%t 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 *)
%t LinearRecurrence[{5,5,-1}, {0,3,21,119}, 30] (* _Vincenzo Librandi_, Nov 04 2016 *)
%o (Haskell)
%o a046727 n = a046727_list !! n
%o a046727_list = 0 : f (tail a001652_list) (tail a046090_list) where
%o f (x:_:xs) (_:y:ys) = x : y : f xs ys
%o -- _Reinhard Zumkeller_, Jan 10 2012
%o (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
%o (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
%o (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
%Y Cf. A001109, A001333, A001652, A005319, A046090, A046729.
%Y Essentially the same as A084159.
%K easy,nonn
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Sascha Kurz_, Jan 23 2003
| 