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A046727 Related to Pythagorean triples: alternate terms of A001652 and A046090. 4
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.

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[0] == 3,

  a[1] == 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

N. J. A. Sloane

EXTENSIONS

More terms from Sascha Kurz, Jan 23 2003

STATUS

approved

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Last modified September 26 01:25 EDT 2017. Contains 292500 sequences.