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A192967
Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
4
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
0,3
COMMENTS
The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + n(n-1)/2, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232 and A192744.
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
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 *)
PROG
(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
(Magma) [3*Fibonacci(n+1) -n-2: n in [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
(Sage) [3*fibonacci(n+1) -n-2 for n in (0..40)] # G. C. Greubel, Jul 11 2019
(GAP) List([0..40], n-> 3*Fibonacci(n+1) -n-2); # G. C. Greubel, Jul 11 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jul 13 2011
STATUS
approved