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A192762 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments. 5

%I #11 Jun 13 2015 00:53:53

%S 0,1,6,13,26,47,82,139,232,383,628,1025,1668,2709,4394,7121,11534,

%T 18675,30230,48927,79180,128131,207336,335493,542856,878377,1421262,

%U 2299669,3720962,6020663,9741658,15762355,25504048,41266439,66770524

%N Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

%C The titular polynomial is defined recursively by p(n,x)=x*(n-1,x)+n+4 for n>0, where p(0,x)=1. For discussions of polynomial reduction, see A192232 and A192744.

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).

%F a(n) = 3*a(n-1)-2*a(n-2)-a(n-3)+a(n-4). G.f.: x*(3*x^2-3*x-1) / ((x-1)^2*(x^2+x-1)). [_Colin Barker_, Dec 08 2012]

%t p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + n + 4;

%t Table[Expand[p[n, x]], {n, 0, 7}]

%t reduce[{p1_, q_, s_, x_}] :=

%t FixedPoint[(s PolynomialQuotient @@ #1 +

%t PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]

%t t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];

%t u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]

%t (* A022319 *)

%t u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]

%t (* A192762 *)

%Y Cf. A192744, A192232, A022319 (first differences).

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, Jul 09 2011

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Last modified March 28 13:42 EDT 2024. Contains 371254 sequences. (Running on oeis4.)