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A192754
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Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.
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3
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1, 6, 12, 23, 40, 68, 113, 186, 304, 495, 804, 1304, 2113, 3422, 5540, 8967, 14512, 23484, 38001, 61490, 99496, 160991, 260492, 421488, 681985, 1103478, 1785468, 2888951, 4674424, 7563380, 12237809, 19801194, 32039008, 51840207, 83879220, 135719432
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OFFSET
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0,2
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COMMENTS
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The titular polynomial is defined recursively by p(n,x)=x*p(n-1,x)+5*n+1 for n>0, where p(0,x)=1. For discussions of polynomial reduction, see A192232 and A192744.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).
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FORMULA
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Conjecture: G.f.: ( 1+4*x ) / ( (x-1)*(x^2+x-1) ), partial sums of A022095. a(n) = A000071(n+3)+4*A000071(n+2). - R. J. Mathar, May 04 2014
a(n) = 8*Fibonacci(n) + 3*Lucas(n) - 5. - Greg Dresden, Oct 10 2020
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MATHEMATICA
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p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + 5 n + 1;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}] :=
FixedPoint[(s PolynomialQuotient @@ #1 +
PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
(* A192754 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
(* A192755 *)
LinearRecurrence[{2, 0, -1}, {1, 6, 12}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2012 *)
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CROSSREFS
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Cf. A192744, A192232, A192755.
Sequence in context: A144568 A222001 A078472 * A005694 A309359 A172079
Adjacent sequences: A192751 A192752 A192753 * A192755 A192756 A192757
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KEYWORD
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nonn,easy
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AUTHOR
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Clark Kimberling, Jul 09 2011
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STATUS
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approved
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