A051047 For n > 5, a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3); initial terms are 1, 3, 8, 120, 1680.

1, 3, 8, 120, 1680, 23408, 326040, 4541160, 63250208, 880961760, 12270214440, 170902040408, 2380358351280, 33154114877520, 461777249934008, 6431727384198600, 89582406128846400, 1247721958419651008, 17378525011746267720, 242051628206028097080

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• easy

Comments

The recurrence gives an infinite sequence of polynomials S={x,x+2,c_1(x),c_2(x),...} such that the product of any two consecutive polynomials, increased by 1, is the square of a polynomial - see the Jones reference.

Links

• Colin Barker, Table of n, a(n) for n = 1..850
• Andrej Dujella and Attila Petho, Generalization of a theorem of Baker and Davenport
• B. W. Jones, A Variation of a Problem of Davenport and Diophantus, Quart. J. Math. (Oxford) Ser. (2) 27, 349-353, 1976.
• Index entries for linear recurrences with constant coefficients, signature (15,-15,1).

Crossrefs

Cf. A051048. Essentially the same as A045899.

Programs

Magma

I:=[1,3,8,120,1680]; [n le 5 select I[n] else 14*Self(n-1)-Self(n-2)+8: n in [1..20]]; // Vincenzo Librandi, Mar 05 2016

Mathematica

With[{x = 1},
Join[{x, x + 2},
RecurrenceTable[{c[-1] == c[0] == 0,
c[k] == (4 x^2 + 8 x + 2) c[k - 1] - c[k - 2] + 4 (x + 1)}, c, {k, 1, 12}]]]
LinearRecurrence[{15, -15, 1}, {1, 3, 8, 120, 1680}, 22] (* Charles R Greathouse IV, Oct 31 2011 *)
Join[{1, 3}, RecurrenceTable[{a[1] == 8, a[2] == 120, a[n] == 14 a[n-1] - a[n-2] + 8}, a, {n, 20}]] (* Vincenzo Librandi, Mar 05 2016 *)

PARI

Vec((3*x^4-44*x^3+22*x^2+12*x-1)/(x^3-15*x^2+15*x-1)+O(x^99)) \\ Charles R Greathouse IV, Oct 31 2011

Formula

G.f.: x*(3*x^4-44*x^3+22*x^2+12*x-1) / (x^3-15*x^2+15*x-1).

For n>4, a(n) = 14*a(n-1)-a(n-2)+8. - Vincenzo Librandi, Mar 05 2016

Extensions

Entry revised by N. J. A. Sloane, Oct 25 2009, following correspondence with Eric Weisstein