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

1,2

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).

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

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 *)

Prog

(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
(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

Crossrefs

Cf. A051048. Essentially the same as A045899.

Sequence in context: A134803 A030063 A195568 * A192629 A245458 A036504

Adjacent sequences: A051044 A051045 A051046 * A051048 A051049 A051050

Keyword

nonn,easy

Author

Eric W. Weisstein

Extensions

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

Status

approved