A192967 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
1, 0, 2, 4, 9, 17, 31, 54, 92, 154, 255, 419, 685, 1116, 1814, 2944, 4773, 7733, 12523, 20274, 32816, 53110, 85947, 139079, 225049, 364152, 589226, 953404, 1542657, 2496089, 4038775, 6534894, 10573700, 17108626, 27682359, 44791019, 72473413, 117264468
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).
Programs
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GAP
List([0..40], n-> 3*Fibonacci(n+1) -n-2); # G. C. Greubel, Jul 11 2019
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Magma
I:=[1, 0, 2, 4]; [n le 4 select I[n] else 3*Self(n-1)-2*Self(n-2)-Self(n-3)+Self(n-4): n in [1..40]]; // Vincenzo Librandi, Feb 20 2012
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Magma
[3*Fibonacci(n+1) -n-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}] (* A192967 *) u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192968 *) LinearRecurrence[{3,-2,-1,1}, {1,0,2,4}, 41] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2012 *) Table[3*Fibonacci[n+1] -n-2, {n,0,40}] (* G. C. Greubel, Jul 11 2019 *) -
PARI
vector(40, n, n--; f=fibonacci; 3*f(n+1)-n-2) \\ G. C. Greubel, Jul 11 2019
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Sage
[3*fibonacci(n+1) -n-2 for n in (0..40)] # G. C. Greubel, Jul 11 2019
Formula
a(0)=1, a(1)=0, for n > 1, a(n) = a(n-1) + a(n-2) + n - 1. - Alex Ratushnyak, May 10 2012
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
G.f.: (1 -3*x +4*x^2 -x^3)/((1-x-x^2)*(1-x)^2). - R. J. Mathar, May 11 2014
a(n) = 3*Fibonacci(n+1) - n - 2. - G. C. Greubel, Jul 11 2019
Comments