A192966 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
0, 1, 2, 6, 14, 30, 59, 110, 197, 343, 585, 983, 1634, 2695, 4420, 7220, 11760, 19116, 31029, 50316, 81535, 132061, 213827, 346141, 560244, 906685, 1467254, 2374290, 3841922, 6216618, 10058975, 16276058, 26335529, 42612115, 68948205, 111560915
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..400
- Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).
Programs
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GAP
F:=Fibonacci;; List([0..40], n-> F(n+4) +2*F(n+2) -(n^2+5*n+10)/2); # G. C. Greubel, Jul 11 2019
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Magma
I:=[0, 1, 2, 6, 14]; [n le 5 select I[n] else 4*Self(n-1)-5*Self(n-2)+Self(n-3)+2*Self(n-4)-Self(n-5): n in [1..40]]; // Vincenzo Librandi, Nov 16 2011
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Magma
F:=Fibonacci; [F(n+4) +2*F(n+2) -(n^2+5*n+10)/2: n in [0..40]]; // G. C. Greubel, Jul 11 2019
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Mathematica
q = x^2; s = x + 1; z = 40; p[0, x]:= 1; p[n_, x_]:= x*p[n-1, x] + n(n+1)/2; 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}] (* A030119 *) u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192966 *) LinearRecurrence[{4,-5,1,2,-1},{0,1,2,6,14},40] (* Vincenzo Librandi, Nov 16 2011 *) Table[Fibonacci[n+4] +2*Fibonacci[n+2] -(n^2+5*n+10)/2, {n,0,40}] (* G. C. Greubel, Jul 11 2019 *) -
PARI
vector(40, n, n--; f=fibonacci; f(n+4)+2*f(n+2)-(n^2+5*n+10)/2) \\ G. C. Greubel, Jul 11 2019
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Sage
f=fibonacci; [f(n+4) +2*f(n+2) -(n^2+5*n+10)/2 for n in (0..40)] # G. C. Greubel, Jul 11 2019
Formula
a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).
G.f.: x*(1 - 2*x + 3*x^2 - x^3)/((1-x-x^2)*(1-x)^3). - R. J. Mathar, May 11 2014
a(n) = Fibonacci(n+4) + 2*Fibonacci(n+2) - (n^2 + 5*n + 10)/2. - G. C. Greubel, Jul 11 2019
Comments