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A192953
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Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
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3
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0, 1, 2, 6, 13, 26, 48, 85, 146, 246, 409, 674, 1104, 1801, 2930, 4758, 7717, 12506, 20256, 32797, 53090, 85926, 139057, 225026, 364128, 589201, 953378, 1542630, 2496061, 4038746, 6534864, 10573669, 17108594, 27682326, 44790985, 72473378
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OFFSET
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0,3
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COMMENTS
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The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + 2n - 1, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232 and A192744.
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).
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FORMULA
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a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
G.f.: x*(1 -x +2*x^2)/((1-x-x^2)*(1-x)^2). - R. J. Mathar, Aug 01 2011
a(n) = -2*n - 3 + 3*A000045(n+2). - R. J. Mathar, Aug 01 2011
a(n) = A131300(n) - 1. - R. J. Mathar, Mar 24 2018
a(n) = 3*Fibonacci(n+2) - (2*n+3). - G. C. Greubel, Jul 12 2019
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MATHEMATICA
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(* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + 2n - 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}] (* A111314 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192953 *)
(* Second program *)
With[{F=Fibonacci}, Table[3*F[n+2]-(2*n+3), {n, 0, 40}]] (* G. C. Greubel, Jul 12 2019 *)
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PROG
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(PARI) vector(40, n, n--; f=fibonacci; 3*f(n+2)-(2*n+3)) \\ G. C. Greubel, Jul 12 2019
(Magma) F:=Fibonacci; [3*F(n+2)-(2*n+3): n in [0..40]]; // G. C. Greubel, Jul 12 2019
(Sage) f=fibonacci; [3*f(n+2)-(2*n+3) for n in (0..40)] # G. C. Greubel, Jul 12 2019
(GAP) F:=Fibonacci;; List([0..40], n-> 3*F(n+2)-(2*n+3)); # G. C. Greubel, Jul 12 2019
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CROSSREFS
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Cf. A000045, A192232, A192744, A192951.
Sequence in context: A259577 A172348 A254821 * A353232 A275970 A124677
Adjacent sequences: A192950 A192951 A192952 * A192954 A192955 A192956
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KEYWORD
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nonn,easy
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AUTHOR
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Clark Kimberling, Jul 13 2011
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STATUS
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approved
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