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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. 4
1, 3, 8, 120, 1680, 23408, 326040, 4541160, 63250208, 880961760, 12270214440, 170902040408, 2380358351280, 33154114877520, 461777249934008, 6431727384198600, 89582406128846400, 1247721958419651008, 17378525011746267720, 242051628206028097080 (list; graph; refs; listen; history; text; internal format)
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

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Last modified February 23 00:18 EST 2019. Contains 320411 sequences. (Running on oeis4.)