login
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

%I #36 Aug 06 2024 09:05:57

%S 1,3,8,120,1680,23408,326040,4541160,63250208,880961760,12270214440,

%T 170902040408,2380358351280,33154114877520,461777249934008,

%U 6431727384198600,89582406128846400,1247721958419651008,17378525011746267720,242051628206028097080

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

%C 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.

%H Colin Barker, <a href="/A051047/b051047.txt">Table of n, a(n) for n = 1..850</a>

%H Andrej Dujella and Attila Petho, <a href="https://citeseerx.ist.psu.edu/pdf/9e894b39c9b7662910b3fa48407ec6d7ecce921e">Generalization of a theorem of Baker and Davenport</a>

%H B. W. Jones, <a href="http://qjmath.oxfordjournals.org/content/27/3/349.extract">A Variation of a Problem of Davenport and Diophantus</a>, Quart. J. Math. (Oxford) Ser. (2) 27, 349-353, 1976.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (15,-15,1).

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

%F For n>4, a(n) = 14*a(n-1)-a(n-2)+8. - _Vincenzo Librandi_, Mar 05 2016

%t With[{x = 1},

%t Join[{x, x + 2},

%t RecurrenceTable[{c[-1] == c[0] == 0,

%t c[k] == (4 x^2 + 8 x + 2) c[k - 1] - c[k - 2] + 4 (x + 1)}, c, {k, 1, 12}]]]

%t LinearRecurrence[{15, -15, 1}, {1, 3, 8, 120, 1680}, 22] (* _Charles R Greathouse IV_, Oct 31 2011 *)

%t 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 *)

%o (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

%o (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

%Y Cf. A051048. Essentially the same as A045899.

%K nonn,easy

%O 1,2

%A _Eric W. Weisstein_

%E Entry revised by _N. J. A. Sloane_, Oct 25 2009, following correspondence with Eric Weisstein