A327321
Triangular array read by rows: row n shows the coefficients of the polynomial p(x,n) constructed as in Comments; these polynomials form a strong divisibility sequence.
Original entry on oeis.org
1, 1, 3, 7, 18, 27, 5, 21, 27, 27, 61, 300, 630, 540, 405, 91, 549, 1350, 1890, 1215, 729, 547, 3822, 11529, 18900, 19845, 10206, 5103, 205, 1641, 5733, 11529, 14175, 11907, 5103, 2187, 4921, 44280, 177228, 412776, 622566, 612360, 428652, 157464, 59049, 7381
Offset: 1
p(x,3) = (1/k)((4 (7 + 18 x + 27 x^2))/(3 sqrt(3))), where k = 4/(3 sqrt(3)).
First six rows:
1;
1, 3;
7, 18, 27;
5, 21, 27, 27;
61, 300, 630, 540, 405;
91, 549, 1350, 1890, 1215, 729;
The first six polynomials, not factored:
1, 1 + 3 x, 7 + 18 x + 27 x^2, 5 + 21 x + 27 x^2 + 27 x^3, 61 + 300 x + 630 x^2 + 540 x^3 + 405 x^4, 91 + 549 x + 1350 x^2 + 1890 x^3 + 1215 x^4 + 729 x^5.
The first six polynomials, factored:
1, 1 + 3 x, 7 + 18 x + 27 x^2, (1 + 3 x) (5 + 6 x + 9 x^2), 61 + 300 x + 630 x^2 + 540 x^3 + 405 x^4, (1 + 3 x) (13 + 6 x + 9 x^2) (7 + 18 x + 27 x^2).
-
c[poly_] := If[Head[poly] === Times, Times @@ DeleteCases[(#1 (Boole[
MemberQ[#1, x] || MemberQ[#1, y] || MemberQ[#1, z]] &) /@
Variables /@ #1 &)[List @@ poly], 0], poly];
r = Sqrt[3]; f[x_, n_] := c[Factor[Expand[(r x + r)^n - (r x - 1/r)^n]]];
Table[f[x, n], {n, 1, 6}]
Flatten[Table[CoefficientList[f[x, n], x], {n, 1, 12}]] (* A327321 *)
(* Peter J. C. Moses, Nov 01 2019 *)
A329009
a(n) = p(1,n), where p(x,n) is the strong divisibility sequence of polynomials based on sqrt(3) as in A327321.
Original entry on oeis.org
1, 4, 52, 80, 1936, 5824, 69952, 52480, 2519296, 7558144, 90698752, 136048640, 3265171456, 9795518464, 117546237952, 44079841280, 4231664828416, 12694994550784, 152339934871552, 228509902438400, 5484237659570176, 16452712979759104, 197432555761303552
Offset: 1
-
A329009 := n -> 2^(n - 1 - padic[ordp](2*n, 2))*(3^n - 1):
seq(A329009(n), n = 1..22); # Peter Luschny, Mar 05 2022
-
c[poly_] := If[Head[poly] === Times, Times @@ DeleteCases[(#1 (Boole[MemberQ[#1, x] || MemberQ[#1, y] || MemberQ[#1, z]] &) /@Variables /@ #1 &)[List @@ poly], 0], poly];
r = Sqrt[3]; f[x_, n_] := c[Factor[Expand[(r x + r)^n - (r x - 1/r)^n]]];
Flatten[Table[CoefficientList[f[x, n], x], {n, 1, 12}]]; (* A327321 *)
Table[f[x, n] /. x -> 0, {n, 1, 30}] (* A329008 *)
Table[f[x, n] /. x -> 1, {n, 1, 30}] (* A329009 *)
Table[f[x, n] /. x -> 2, {n, 1, 30}] (* A329010 *)
(* Peter J. C. Moses, Nov 01 2019 *)
A329010
a(n) = p(2,n), where p(x,n) is the strong divisibility sequence of polynomials based on sqrt(3) as in A327321.
Original entry on oeis.org
1, 7, 151, 371, 13981, 64477, 1176211, 1333003, 96366841, 434627347, 7833057871, 17636587241, 635161281301, 2858836117417, 51465153629131, 28951056265019, 4169104690053361, 18761352574966687, 337708161046665991, 759848130726580511, 27354628073588539021
Offset: 1
-
c[poly_] := If[Head[poly] === Times, Times @@ DeleteCases[(#1 (Boole[MemberQ[#1, x] || MemberQ[#1, y] || MemberQ[#1, z]] &) /@Variables /@ #1 &)[List @@ poly], 0], poly];
r = Sqrt[3]; f[x_, n_] := c[Factor[Expand[(r x + r)^n - (r x - 1/r)^n]]];
Flatten[Table[CoefficientList[f[x, n], x], {n, 1, 12}]]; (* A327321 *)
Table[f[x, n] /. x -> 0, {n, 1, 30}] (* A329008 *)
Table[f[x, n] /. x -> 1, {n, 1, 30}] (* A329009 *)
Table[f[x, n] /. x -> 2, {n, 1, 30}] (* A329010 *)
(* Peter J. C. Moses, Nov 01 2019 *)
Showing 1-3 of 3 results.
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