A327320 -id:A327320 - cp's OEIS frontend

A329005 a(n) = p(0,n), where p(x,n) is the strong divisibility sequence of polynomials based on sqrt(2) as in A327320.

1, 1, 1, 5, 11, 7, 43, 85, 19, 341, 683, 455, 2731, 5461, 3641, 21845, 43691, 9709, 174763, 349525, 233017, 1398101, 2796203, 1864135, 11184811, 22369621, 1657009, 89478485, 178956971, 119304647, 715827883, 1431655765, 954437177, 5726623061, 11453246123

Offset: 1

Clark Kimberling, Nov 08 2019

nonn

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

			See Example in A327320.

Cf. A327320, A329006, A329007.

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[2]; 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}]];  (* A327320 *)
Table[f[x, n] /. x -> 0, {n, 1, 30}]   (* A329005 *)
Table[f[x, n] /. x -> 1, {n, 1, 30}]   (* A329006 *)
Table[f[x, n] /. x -> 2, {n, 1, 30}]   (* A329007 *)
(* Peter J. C. Moses, Nov 01 2019 *)

A329006 a(n) = p(1,n), where p(x,n) is the strong divisibility sequence of polynomials based on sqrt(2) as in A327320.

Original entry on oeis.org

1, 5, 7, 85, 341, 455, 5461, 21845, 9709, 349525, 1398101, 1864135, 22369621, 89478485, 119304647, 1431655765, 5726623061, 2545165805, 91625968981, 366503875925, 488671834567, 5864062014805, 23456248059221, 31274997412295, 375299968947541, 15011998757901653

Offset: 1

Clark Kimberling, Nov 08 2019

nonn

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

			See Example in A327320.

Cf. A327320, A329005, A329007.

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[2]; 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}]];  (* A327320 *)
Table[f[x, n] /. x -> 0, {n, 1, 30}]   (* A329005 *)
Table[f[x, n] /. x -> 1, {n, 1, 30}]   (* A329006 *)
Table[f[x, n] /. x -> 2, {n, 1, 30}]   (* A329007 *)
(* Peter J. C. Moses, Nov 01 2019 *)

A329007 a(n) = p(2,n), where p(x,n) is the strong divisibility sequence of polynomials based on sqrt(2) as in A327320.

Original entry on oeis.org

1, 9, 21, 405, 2511, 5103, 92583, 557685, 372519, 20135709, 120873303, 241805655, 4353033231, 26119793709, 52241181741, 940355620245, 5642176768191, 3761465527701, 203119525916343, 1218718317759525, 2437437797780517, 43873890820402509, 263243376303474663

Offset: 1

Clark Kimberling, Nov 08 2019

nonn

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

			See Example in A327320.

Cf. A327320, A329005, A329006.

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[2]; 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}]];  (* A327320 *)
Table[f[x, n] /. x -> 0, {n, 1, 30}]   (* A329005 *)
Table[f[x, n] /. x -> 1, {n, 1, 30}]   (* A329006 *)
Table[f[x, n] /. x -> 2, {n, 1, 30}]   (* A329007 *)
(* Peter J. C. Moses, Nov 01 2019 *)

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

Clark Kimberling, Nov 08 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(3). If x is an integer, then p(x,n) is a strong divisibility sequence of integers.

			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).

Cf. A327320, A329008, A329000, A031364.

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 *)

A327322 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, 2, 5, 7, 20, 25, 26, 105, 150, 125, 521, 2600, 5250, 5000, 3125, 434, 2605, 6500, 8750, 6250, 3125, 13021, 91140, 273525, 455000, 459375, 262500, 109375, 8138, 65105, 227850, 455875, 568750, 459375, 218750, 78125, 36169, 325520, 1302100, 3038000, 4558750

Offset: 1

Clark Kimberling, Nov 08 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(5). If x is an integer, then p(x,n) is a strong divisibility sequence of integers.

			p(x,3) = (1/k)((18 (7 + 20 x + 25 x^2))/(5 sqrt(5))), where k = 18/(5 sqrt(5)).
First six rows:
    1;
    2,    5;
    7,   20,   25;
   26,  105,  150,  125;
  521, 2600, 5250, 5000, 3125;
  434, 2605, 6500, 8750, 6250, 3125;
The first six polynomials, not factored:
1, 2 + 5 x, 7 + 20 x + 25 x^2, 26 + 105 x + 150 x^2 + 125 x^3, 521 + 2600 x + 5250 x^2 + 5000 x^3 + 3125 x^4, 434 + 2605 x + 6500 x^2 + 8750 x^3 + 6250 x^4 + 3125 x^5.
The first six polynomials, factored:
1, 2 + 5 x, 7 + 20 x + 25 x^2, (2 + 5 x) (13 + 20 x + 25 x^2), 521 + 2600 x + 5250 x^2 + 5000 x^3 + 3125 x^4, (2 + 5 x) (7 + 20 x + 25 x^2) (31 + 20 x + 25 x^2).

Cf. A327320, A327321.

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]]];
Table[f[x, n], {n, 1, 6}]
Flatten[Table[CoefficientList[f[x, n], x], {n, 1, 12}]]  (* A327322 *)
(* 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 *)

A328644 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, 6, 7, 9, 27, 13, 84, 54, 108, 11, 39, 126, 54, 81, 133, 990, 1755, 3780, 1215, 1458, 463, 2793, 10395, 12285, 19845, 5103, 5103, 1261, 11112, 33516, 83160, 73710, 95256, 20412, 17496, 4039, 34047, 150012, 301644, 561330, 398034, 428652, 78732, 59049

Offset: 1

Clark Kimberling, Nov 03 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(3/2). If x is an integer, then p(x,n) is a strong divisibility sequence of integers.

			We have p(x,3) = (1/k)((5 (7 + 9 x + 27 x^2))/(6 sqrt(6))), where k = 5/(6 sqrt(6)).
First six rows:
  1;
  1, 6;
  7, 9, 27;
  13, 84, 54, 108;
  11, 39, 126, 54, 81;
  133, 990, 1755, 3780, 1215, 1458;
The first six polynomials, not factored:
1, 1 + 6 x, 7 + 9 x + 27 x^2, 13 + 84 x + 54 x^2 + 108 x^3, 11 + 39 x + 126 x^2 + 54 x^3 + 81 x^4, 133 + 990 x + 1755 x^2 + 3780 x^3 + 1215 x^4 + 1458 x^5.
The first six polynomials, factored:
1, 1 + 6 x, 7 + 9 x + 27 x^2, (1 + 6 x) (13 + 6 x + 18 x^2), 11 + 39 x + 126 x^2 + 54 x^3 + 81 x^4, (1 + 6 x) (19 + 3 x + 9 x^2) (7 + 9 x + 27 x^2).

Cf. A327320, A327321, A329017, A329018, A329019.

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[3/2]; 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}]]  (* A328644 *)
(* Peter J. C. Moses, Nov 01 2019 *)

A329011 a(n) = p(0,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, 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 *)

cp's OEIS Frontend

A329005 a(n) = p(0,n), where p(x,n) is the strong divisibility sequence of polynomials based on sqrt(2) as in A327320.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A329006 a(n) = p(1,n), where p(x,n) is the strong divisibility sequence of polynomials based on sqrt(2) as in A327320.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A329007 a(n) = p(2,n), where p(x,n) is the strong divisibility sequence of polynomials based on sqrt(2) as in A327320.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A327322 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.

Views

Author

Keywords

Comments

Examples

Crossrefs

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A328644 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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs