A192799
Coefficient of x in the reduction of the n-th Fibonacci polynomial by x^3->x^2+2.
Original entry on oeis.org
0, 1, 0, 2, 2, 5, 12, 22, 54, 109, 242, 520, 1118, 2427, 5218, 11290, 24352, 52579, 113526, 245038, 529068, 1142087, 2465644, 5322896, 11491188, 24807721, 53555508, 115617714, 249599214, 538843277, 1163273304, 2511313222, 5421508714
Offset: 1
The first five polynomials p(n,x) and their reductions:
F1(x)=1 -> 1
F2(x)=x -> x
F3(x)=x^2+1 -> x^2+1
F4(x)=x^3+2x -> x^2+2x+2
F5(x)=x^4+3x^2+1 -> 4x^2+2*x+3, so that
A192798=(1,0,1,2,3,...), A192799=(0,1,0,2,2,...), A192800=(0,0,1,1,4,...)
A192800
Coefficient of x^2 in the reduction of the n-th Fibonacci polynomial by x^3->x^2+2.
Original entry on oeis.org
0, 0, 1, 1, 4, 7, 16, 35, 73, 162, 344, 748, 1612, 3478, 7517, 16213, 35020, 75585, 163184, 352295, 760517, 1641880, 3544484, 7652008, 16519388, 35662584, 76989693, 166207785, 358815192, 774622191, 1672280660, 3610176155, 7793770037
Offset: 1
The first five polynomials p(n,x) and their reductions:
F1(x)=1 -> 1
F2(x)=x -> x
F3(x)=x^2+1 -> x^2+1
F4(x)=x^3+2x -> x^2+2x+2
F5(x)=x^4+3x^2+1 -> 4x^2+2*x+3, so that
A192798=(1,0,1,2,3,...), A192799=(0,1,0,2,2,...), A192800=(0,0,1,1,4,...)
A192801
Constant term in the reduction of the polynomial (x+2)^n by x^3->x^2+x+1. See Comments.
Original entry on oeis.org
1, 2, 4, 9, 25, 84, 312, 1199, 4637, 17906, 68976, 265249, 1019069, 3913484, 15026092, 57690143, 221487945, 850350482, 3264725772, 12534190569, 48122302705, 184755243892, 709328262928, 2723314511871, 10455585321989, 40141990468066
Offset: 0
The first five polynomials p(n,x) and their reductions:
p(1,x)=1 -> 1
p(2,x)=x+2 -> x+2
p(3,x)=x^2+4x+4 -> x^2+1
p(4,x)=x^3+6x^2+12x+8 -> x^2+4x+4
p(5,x)=x^4+8x^3+24x^2+32x+16 -> 7x^2+13*x+9, so that
A192798=(1,2,4,9,25,...), A192799=(0,1,4,13,42,...), A192800=(0,0,1,7,34,...).
-
q = x^3; s = x^2 + x + 1; z = 40;
p[n_, x_] := (x + 2)^n;
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}] (* A192801 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
(* A192802 *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
(* A192803 *)
A192802
Coefficient of x in the reduction of the polynomial (x+2)^n by x^3->x^2+x+1.
Original entry on oeis.org
0, 1, 4, 13, 42, 143, 514, 1915, 7268, 27805, 106680, 409633, 1573086, 6040587, 23193782, 89051615, 341901032, 1312664601, 5039704492, 19348873781, 74285859698, 285204660583, 1094982340202, 4203950929347, 16140172668812
Offset: 0
The first five polynomials p(n,x) and their reductions:
p(1,x)=1 -> 1
p(2,x)=x+2 -> x+2
p(3,x)=x^2+4x+4 -> x^2+1
p(4,x)=x^3+6x^2+12x+8 -> x^2+4x+4
p(5,x)=x^4+8x^3+24x^2+32x+16 -> 7x^2+13*x+9, so that
A192798=(1,2,4,9,25,...), A192799=(0,1,4,13,42,...), A192800=(0,0,1,7,34,...).
A192803
Coefficient of x^2 in the reduction of the polynomial (x+2)^n by x^3->x^2+x+1.
Original entry on oeis.org
0, 0, 1, 7, 34, 144, 575, 2239, 8632, 33164, 127297, 488571, 1875346, 7199124, 27637959, 106107659, 407374592, 1564024808, 6004739025, 23053921567, 88510638482, 339817775144, 1304657986015, 5008956298247, 19230819824088, 73832632141076
Offset: 0
The first five polynomials p(n,x) and their reductions:
p(1,x)=1 -> 1
p(2,x)=x+2 -> x+2
p(3,x)=x^2+4x+4 -> x^2+1
p(4,x)=x^3+6x^2+12x+8 -> x^2+4x+4
p(5,x)=x^4+8x^3+24x^2+32x+16 -> 7x^2+13*x+9, so that
A192798=(1,2,4,9,25,...), A192799=(0,1,4,13,42,...), A192800=(0,0,1,7,34,...).
Showing 1-5 of 5 results.
Comments