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A192970
Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
3
0, 1, 3, 9, 21, 44, 85, 156, 276, 476, 806, 1347, 2230, 3667, 6001, 9787, 15923, 25862, 41955, 68006, 110170, 178406, 288828, 467509, 756636, 1224469, 1981455, 3206301, 5188161, 8394896, 13583521, 21978912, 35562960, 57542432, 93105986
OFFSET
0,3
COMMENTS
The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + n(n+3)/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(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).
G.f.: x*(1-x+2*x^2-x^3)/((1-x-x^2)*(1-x)^3). - R. J. Mathar, May 11 2014
a(n) = Fibonacci(n+4) + Lucas(n+3) - (n^2 + 7*n + 14)/2. - Ehren Metcalfe, Jul 13 2019
MATHEMATICA
(* First progream *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + n*(n+3)/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}] (* A192969 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192970 *)
(* Additional programs *)
CoefficientList[Series[x*(1-x+2*x^2-x^3)/((1-x-x^2)*(1-x)^3), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 13 2019 *)
Table[LucasL[n+3]+Fibonacci[n+4]-(n^2+7*n+14)/2, {n, 0, 40}] (* G. C. Greubel, Jul 24 2019 *)
PROG
(Magma) [Fibonacci(n+4)+Lucas(n+3)-(n^2+7*n+14)/2: n in [0..40]]; // Vincenzo Librandi, Jul 13 2019
(PARI) vector(40, n, n--; f=fibonacci; 2*f(n+4)+f(n+2)-(n^2+7*n+14)/2) \\ G. C. Greubel, Jul 24 2019
(Sage) f=fibonacci; [2*f(n+4)+f(n+2)-(n^2+7*n+14)/2 for n in (0..40)] # G. C. Greubel, Jul 24 2019
(GAP) F:=Fibonacci;; List([0..40], n-> 2*F(n+4)+F(n+2)-(n^2+7*n+14)/2); # G. C. Greubel, Jul 24 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jul 13 2011
STATUS
approved