A085482 Product of three solutions of the Diophantine equation x^2 - y^2 = z^3.

54, 13824, 354294, 3538944, 21093750, 90699264, 311299254, 905969664, 2324522934, 5400000000, 11575379574, 23219011584, 44049458934, 79692609024, 138396093750, 231928233984, 376690901814, 595077871104, 917112404214

Offset

1,1

Comments

Parametric representation of the solution is (x,y,z) = (6*n^3, 3*n^3, 3*n^2), thus getting a(n) = 54*n^8.

Links

Table of n, a(n) for n=1..19.

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Formula

a(n) = 54*n^8.

a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9); a(1)=54, a(2)=13824, a(3)=354294, a(4)=3538944, a(5)=21093750, a(6)=90699264, a(7)=311299254, a(8)=905969664, a(9)=2324522934. - Harvey P. Dale, Jul 10 2013

From Elmo R. Oliveira, May 27 2026: (Start)

a(n) = 54*A001016(n).

G.f.: 54*x*(1 + 247*x + 4293*x^2 + 15619*x^3 + 15619*x^4 + 4293*x^5 + 247*x^6 + x^7)/(1 - x)^9.

E.g.f.: 54*exp(x)*x*(1 + 127*x + 966*x^2 + 1701*x^3 + 1050*x^4 + 266*x^5 + 28*x^6 + x^7). (End)

Maple

A085482:=n->54*n^8; seq(A085482(n), n=1..50); # Wesley Ivan Hurt, Nov 26 2013

Mathematica

54*Range[20]^8 (* Harvey P. Dale, Jul 10 2013 *)
(* Alternative: *)
LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {54, 13824, 354294, 3538944, 21093750, 90699264, 311299254, 905969664, 2324522934}, 20] (* Harvey P. Dale, Jul 10 2013 *)

Crossrefs

Cf. A001016, A085409.

Sequence in context: A228607 A364304 A030254 * A084226 A071800 A393844

Adjacent sequences: A085479 A085480 A085481 * A085483 A085484 A085485

Keyword

nonn,easy

Author

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Aug 15 2003

Extensions

More terms from Ray Chandler, Nov 06 2003

Status

approved