A192777
Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+3x+1. See Comments.
Original entry on oeis.org
1, 0, 1, 1, 2, 8, 14, 55, 121, 392, 989, 2912, 7797, 22104, 60553, 169289, 467622, 1300888, 3603914, 10008543, 27755249, 77034176, 213702153, 593005504, 1645265209, 4565154816, 12666317073, 35144684065, 97512548090, 270561677224
Offset: 1
The first five polynomials p(n,x) and their reductions are as follows:
F1(x)=1 -> 1
F2(x)=x -> x
F3(x)=x^2+1 -> x^2+1
F4(x)=x^3+2x -> x^2+5x+1
F5(x)=x^4+3x^2+1 -> 7x^2+4x+2, so that
A192777=(1,0,1,1,2,...), A192778=(0,1,0,5,4,...), A192779=(0,0,1,1,7,...)
-
q = x^3; s = x^2 + 3 x + 1; z = 40;
p[n_, x_] := Fibonacci[n, x];
Table[Expand[p[n, x]], {n, 1, 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, 1, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
(* A192777 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
(* A192778 *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
(* A192779 *)
A192778
Coefficient of x in the reduction of the n-th Fibonacci polynomial by x^3->x^2+3x+1.
Original entry on oeis.org
0, 1, 0, 5, 4, 28, 48, 183, 424, 1315, 3420, 9864, 26756, 75237, 207128, 577345, 1597624, 4439764, 12307388, 34166643, 94769936, 262998791, 729644824, 2024614928, 5617339496, 15586328073, 43245649904, 119991232893, 332929027020
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+5x+1
F5(x)=x^4+3x^2+1 -> 7x^2+4x+2, so that
A192777=(1,0,1,1,2,...), A192778=(0,1,0,5,4,...), A192779=(0,0,1,1,7,...)
A192779
Coefficient of x^2 in the reduction of the n-th Fibonacci polynomial by x^3->x^2+3x+1.
Original entry on oeis.org
0, 0, 1, 1, 7, 12, 47, 107, 337, 868, 2520, 6808, 19192, 52756, 147185, 407069, 1131599, 3136292, 8707655, 24151335, 67025633, 185946904, 515971328, 1431563056, 3972149312, 11021051864, 30579529249, 84846231017, 235416993159, 653192251196
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+5x+1
F5(x)=x^4+3x^2+1 -> 7x^2+4x+2, so that
A192777=(1,0,1,1,2,...), A192778=(0,1,0,5,4,...), A192779=(0,0,1,1,7,...)
-
(See A192777.)
LinearRecurrence[{1,6,-1,-6,1,1},{0,0,1,1,7,12}, 30] (* Harvey P. Dale, Oct 29 2018 *)
A192773
Coefficient of x in the reduction of the n-th Fibonacci polynomial by x^3->x^2+2x+1.
Original entry on oeis.org
0, 1, 0, 4, 3, 18, 30, 98, 219, 596, 1464, 3783, 9540, 24328, 61740, 156985, 398904, 1013772, 2576475, 6547574, 16640382, 42288806, 107473443, 273129468, 694130016, 1764047839, 4483130424, 11393354512, 28954911624, 73585574049
Offset: 1
The first five polynomials p(n,x) and their reductions are as follows:
F1(x)=1 -> 1
F2(x)=x -> x
F3(x)=x^2+1 -> x^2+1
F4(x)=x^3+2x -> x^2+4x+1
F5(x)=x^4+3x^2+1 -> 6x^2+3x+2, so that
A192772=(1,0,1,1,2,...), A192773=(0,1,0,4,3,...), A192774=(0,0,1,1,6,...)
-
(See A192772.)
LinearRecurrence[{1,5,-1,-5,1,1},{0,1,0,4,3,18},40] (* Harvey P. Dale, Aug 07 2025 *)
A192774
Coefficient of x^2 in the reduction of the n-th Fibonacci polynomial by x^3->x^2+2x+1.
Original entry on oeis.org
0, 0, 1, 1, 6, 10, 34, 74, 206, 499, 1301, 3264, 8348, 21152, 53828, 136720, 347533, 883157, 2244462, 5704094, 14496130, 36840606, 93625542, 237939591, 604694601, 1536764208, 3905506648, 9925401280, 25224262440, 64104575344
Offset: 1
The first five polynomials p(n,x) and their reductions are as follows:
F1(x)=1 -> 1
F2(x)=x -> x
F3(x)=x^2+1 -> x^2+1
F4(x)=x^3+2x -> x^2+4x+1
F5(x)=x^4+3x^2+1 -> 6x^2+3x+2, so that
A192772=(1,0,1,1,2,...), A192773=(0,1,0,4,3,...), A192774=(0,0,1,1,6,...)
-
(See A192772.)
LinearRecurrence[{1,5,-1,-5,1,1},{0,0,1,1,6,10},30] (* Harvey P. Dale, Jun 25 2017 *)
Showing 1-5 of 5 results.
Comments