A327322 -id:A327322 - cp's OEIS frontend

A329011 a(n) = p(0,n), where p(x,n) is the strong divisibility sequence of polynomials based on sqrt(5) as in A327322.

1, 2, 7, 26, 521, 434, 13021, 8138, 36169, 813802, 8138021, 3390842, 203450521, 508626302, 1695421007, 1589457194, 127156575521, 35321270978, 3178914388021, 3973642985026, 26490953233507, 198682149251302, 1986821492513021, 413921144273546, 49670537312825521

Offset: 1

Clark Kimberling, Nov 23 2019

a(n) is a strong divisibility sequence; i.e., gcd(a(h),a(k)) = a(gcd(h,k)).

			See Example in A327322.

Cf. A327320, A327321, A329012, A329013.

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[5]; 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}]];  (* A327322 *)
Table[f[x, n] /. x -> 0, {n, 1, 30}]   (* A329011 *)
Table[f[x, n] /. x -> 1, {n, 1, 30}]   (* A329012 *)
Table[f[x, n] /. x -> 2, {n, 1, 30}]   (* A329013 *)
(* Peter J. C. Moses, Nov 01 2019 *)

A329012 a(n) = p(1,n), where p(x,n) is the strong divisibility sequence of polynomials based on sqrt(5) as in A327322.

Original entry on oeis.org

1, 7, 52, 406, 16496, 27664, 1663936, 2081968, 18513664, 833245952, 16665967616, 13888655872, 1666655481856, 8333310963712, 55555495903232, 104166621927424, 16666663803355136, 9259258622967808, 1666666620853682176, 4166666620853682176, 55555555311219638272

Offset: 1

Clark Kimberling, Nov 23 2019

a(n) is a strong divisibility sequence; i.e., gcd(a(h),a(k)) = a(gcd(h,k)). Conjecture: there is no upper bound for the number of consecutive equal digits among numbers in this sequence, as suggested, for example, by 34 straight 1's in a(96) and 38 straight 6's in a(97).

			See Example in A327322.

Cf. A327320, A327321, A329011, A329013.

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[5]; 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}]];  (* A327322 *)
Table[f[x, n] /. x -> 0, {n, 1, 30}]   (* A329011 *)
Table[f[x, n] /. x -> 1, {n, 1, 30}]   (* A329012 *)
Table[f[x, n] /. x -> 2, {n, 1, 30}]   (* A329013 *)
(* Peter J. C. Moses, Nov 01 2019 *)

A329013 a(n) = p(2,n), where p(x,n) is the strong divisibility sequence of polynomials based on sqrt(5) as in A327322.

Original entry on oeis.org

1, 12, 147, 1836, 116721, 301644, 27679401, 52496748, 704739609, 47763633852, 1436395799961, 1798109838252, 323942200421841, 2430837436077972, 24315999958264707, 68401618078375404, 16418241358998948801, 13682794309260216588, 3694504558135555477881

Offset: 1

Clark Kimberling, Nov 23 2019

a(n) is a strong divisibility sequence; i.e., gcd(a(h),a(k)) = a(gcd(h,k)).

			See Example in A327322.

Cf. A327320, A327321, A329011, A329012.

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[5]; 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}]];  (* A327322 *)
Table[f[x, n] /. x -> 0, {n, 1, 30}]   (* A329011 *)
Table[f[x, n] /. x -> 1, {n, 1, 30}]   (* A329012 *)
Table[f[x, n] /. x -> 2, {n, 1, 30}]   (* A329013 *)
(* Peter J. C. Moses, Nov 01 2019 *)

A327323 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, 5, 12, 31, 90, 108, 185, 744, 1080, 864, 1111, 5550, 11160, 10800, 6480, 6665, 39996, 99900, 133920, 97200, 46656, 5713, 39990, 119988, 199800, 200880, 116640, 46656, 239945, 1919568, 6718320, 13438656, 16783200, 13499136, 6531840, 2239488, 1439671

Offset: 1

Clark Kimberling, Nov 09 2019

Suppose q is a rational number such that the number r = sqrt(q) is irrational. The function (r x + r)^n - (r x - 1/r)^n of x can be represented as k*p(x,n), where k is a constant and p(x,n) is a product of nonconstant polynomials having GCD = 1; the sequence p(x,n) is a strong divisibility sequence of polynomials; i.e., gcd(p(x,h),p(x,k)) = p(x,gcd(h,k)). For A327320, r = sqrt(6). If x is an integer, then p(x,n) is a strong divisibility sequence of integers.

			p(x,3) = (1/k)((7 (31 + 90 x + 108 x^2))/(6 sqrt(6))), where k = 7/(6 sqrt(6)).
First six rows:
     1;
     5,    12;
    31,    90,    108;
   185,   744,   1080,    864;
  1111,  5550,  11160,  10800,   6480;
  6665, 39996,  99900, 133920,  97200,  46656;
  5713, 39990, 119988, 199800, 200880, 116640, 46656;
The first six polynomials, not factored:
1, 5 + 12 x, 31 + 90 x + 108 x^2, 185 + 744 x + 1080 x^2 + 864 x^3, 1111 + 5550 x + 11160 x^2 + 10800 x^3 + 6480 x^4, 6665 + 39996 x + 99900 x^2 + 133920 x^3 + 97200 x^4 + 46656 x^5.
The first six polynomials, factored:
1, 5 + 12 x, 31 + 90 x + 108 x^2, (5 + 12 x) (37 + 60 x + 72 x^2), 1111 + 5550 x +  11160 x^2 + 10800 x^3 + 6480 x^4, (5 + 12 x) (43 + 30 x + 36 x^2) (31 + 90 x + 108 x^2).

Cf. A327320, A327321, A327322, A329014, A329015, A329016.

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[6]; 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}]]  (* A327323 *)
(* Peter J. C. Moses, Nov 01 2019 *)

cp's OEIS Frontend

A329011 a(n) = p(0,n), where p(x,n) is the strong divisibility sequence of polynomials based on sqrt(5) as in A327322.

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A329012 a(n) = p(1,n), where p(x,n) is the strong divisibility sequence of polynomials based on sqrt(5) as in A327322.

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Mathematica

A329013 a(n) = p(2,n), where p(x,n) is the strong divisibility sequence of polynomials based on sqrt(5) as in A327322.

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Author

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Programs

Mathematica

A327323 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.

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Mathematica