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A192804
Constant term in the reduction of the polynomial 1+x+x^2+...+x^n by x^3->x^2+x+1. See Comments.
3
1, 1, 1, 2, 3, 5, 9, 16, 29, 53, 97, 178, 327, 601, 1105, 2032, 3737, 6873, 12641, 23250, 42763, 78653, 144665, 266080, 489397, 900141, 1655617, 3045154, 5600911, 10301681, 18947745, 34850336, 64099761, 117897841, 216847937, 398845538
OFFSET
0,4
COMMENTS
For discussions of polynomial reduction, see A192232 and A192744.
This sequence provides the most-significant place-values in the construction of a tribonacci code. - James Dow Allen, Jul 12 2021
FORMULA
a(n) = 2*a(n-1) - a(n-4).
a(n) = a(n-1) + a(n-2) + a(n-3) - 1. - Alzhekeyev Ascar M, Feb 05 2012
G.f.: ( 1-x-x^2 ) / ( (x-1)*(x^3+x^2+x-1) ). - R. J. Mathar, May 06 2014
a(n) - a(n-1) = A000073(n-1). - R. J. Mathar, May 06 2014
EXAMPLE
The first five polynomials p(n,x) and their reductions:
p(1,x)=1 -> 1,
p(2,x)=x+1 -> x+1,
p(3,x)=x^2+x+1 -> x^2+x+1,
p(4,x)=x^3+x^2+x+1 -> 2x^2+2x+2,
p(5,x)=x^4+x^3+x^2+x+1 -> 4x^2+4*x+3, so that
A192804=(1,1,1,2,3,...), A000073=(0,1,1,2,4,...), A008937=(0,0,1,2,4,...).
MATHEMATICA
q = x^3; s = x^2 + x + 1; z = 40;
p[0, x_] := 1; p[n_, x_] := x^n + p[n - 1, x];
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}]
(* A192804 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
(* A000073 *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
(* A008937 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jul 10 2011
STATUS
approved