cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A267078 Coefficient of x^0 in the minimal polynomial of the continued fraction [1^n,2^(1/3),1,1,...], where 1^n means n ones.

Original entry on oeis.org

-5, -11, 131, 3421, 56209, 1049105, 18561659, 334918459, 5997328339, 107703879581, 1932077585345, 34673771913121, 622167861459451, 11164539354582251, 200338227165577379, 3594932551574173405, 64508386001097153649, 1157556438367284595889
Offset: 0

Views

Author

Clark Kimberling, Jan 10 2016

Keywords

Comments

See A265762 for a guide to related sequences.

Examples

			Let u = 2^(1/3), and let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[u,1,1,1,...] has p(0,x)  = -5 - 15 x - 6 x^2 - 9 x^3 + 3 x^5 + x^6, so that a(0) = -5.
[1,u,1,1,1,...] has p(1,x) = -11 + 45 x - 66 x^2 + 35 x^3 + 6 x^4 - 15 x^5 + 5 x^6, so that a(1) = -11;
[1,1,u,1,1,1...] has p(2,x) = 131 - 633 x + 1110 x^2 - 969 x^3 + 456 x^4 - 111 x^5 + 11 x^6, so that a(2) = 131.

Links

Crossrefs

Cf. A265762, A267079, A267080, A267081, A267082, A267083, A267084.

Programs

  • Mathematica
    u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {2^(1/3)}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 30}]
Coefficient[t, x, 0]; (* A267078 *)
Coefficient[t, x, 1]; (* A267079 *)
Coefficient[t, x, 2]; (* A267080 *)
Coefficient[t, x, 3]; (* A267081 *)
Coefficient[t, x, 4]; (* A267082 *)
Coefficient[t, x, 5]; (* A267083 *)
Coefficient[t, x, 6]; (* A267084 *)
  • PARI
    Vec(-(5 - 54*x - 794*x^2 - 1562*x^3 + 6048*x^4 + 5676*x^5 - 2287*x^6 - 286*x^7 + 22*x^8)/((1 + x)*(1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2)) + O(x^30)) \\ Andrew Howroyd, Mar 07 2018

Formula

a(n) = 13*a(n-1) + 104*a(n-2) - 260*a(n-3) - 260*a(n-4) + 104*a(n-5) + 13*a(n-6) - a(n-7) for n > 8.
G.f.: (-5 + 54 x + 794 x^2 + 1562 x^3 - 6048 x^4 - 5676 x^5 + 2287 x^6 + 286 x^7 - 22 x^8)/(1 - 13 x - 104 x^2 + 260 x^3 + 260 x^4 - 104 x^5 - 13 x^6 + x^7).
G.f.: -(5 - 54*x - 794*x^2 - 1562*x^3 + 6048*x^4 + 5676*x^5 - 2287*x^6 - 286*x^7 + 22*x^8)/((1 + x)*(1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2)). - Andrew Howroyd, Mar 07 2018

A267083 Coefficient of x^5 in the minimal polynomial of the continued fraction [1^n,2^(1/3),1,1,...], where 1^n means n ones.

Original entry on oeis.org

3, -15, -111, -1419, -32847, -549633, -10161885, -180368529, -3250331151, -58231071723, -1045558051887, -18757368165345, -336617548680381, -6040149618970929, -108387507398297007, -1944925169961946443, -34900332815651650575, -626260604480315081409
Offset: 0

Views

Author

Clark Kimberling, Jan 11 2016

Keywords

Comments

See A265762 for a guide to related sequences.

Examples

			Let u = 2^(1/3), and let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[u,1,1,1,...] has p(0,x)  = -5 - 15 x - 6 x^2 - 9 x^3 + 3 x^5 + x^6, so that a(0) = 3.
[1,u,1,1,1,...] has p(1,x) = -11 + 45 x - 66 x^2 + 35 x^3 + 6 x^4 - 15 x^5 + 5 x^6, so that a(1) = -15;
[1,1,u,1,1,1...] has p(2,x) = 131 - 633 x + 1110 x^2 - 969 x^3 + 456 x^4 - 111 x^5 + 11 x^6, so that a(2) = -111.

Links

Crossrefs

Cf. A265762, A267078, A267079, A267080, A267081, A267082, A266527.

Programs

  • Mathematica
    u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {2^(1/3)}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 30}]
Coefficient[t, x, 0]; (* A267078 *)
Coefficient[t, x, 1]; (* A267079 *)
Coefficient[t, x, 2]; (* A267080 *)
Coefficient[t, x, 3]; (* A267081 *)
Coefficient[t, x, 4]; (* A267082 *)
Coefficient[t, x, 5]; (* A267083 *)
Coefficient[t, x, 6]; (* A266527 *)
  • PARI
    Vec(3*(1 - 18*x - 76*x^2 + 788*x^3 - 1992*x^4 - 2706*x^5 + 1051*x^6 + 130*x^7 - 10*x^8)/((1 + x)*(1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2)) + O(x^30)) \\ Andrew Howroyd, Mar 07 2018

Formula

a(n) = 13*a(n-1) + 104*a(n-2) - 260*a(n-3) - 260*a(n-4) + 104*a(n-5) + 13*a(n-6) - a(n-7) for n > 8.
G.f.: -((3 (-1 + 18 x + 76 x^2 - 788 x^3 + 1992 x^4 + 2706 x^5 - 1051 x^6 - 130 x^7 + 10 x^8))/(1 - 13 x - 104 x^2 + 260 x^3 + 260 x^4 - 104 x^5 - 13 x^6 + x^7)).
G.f.: 3*(1 - 18*x - 76*x^2 + 788*x^3 - 1992*x^4 - 2706*x^5 + 1051*x^6 + 130*x^7 - 10*x^8)/((1 + x)*(1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2)). - Andrew Howroyd, Mar 07 2018

A267079 Coefficient of x in the minimal polynomial of the continued fraction [1^n,2^(1/3),1,1,...], where 1^n means n ones.

Original entry on oeis.org

-15, 45, -633, -12321, -212379, -3867255, -68998575, -1240820397, -22247101689, -399334774401, -7164902653275, -128574917201655, -2307142450214223, -41400271270803501, -742895806968482169, -13330737506206610145, -239210288473732159515
Offset: 0

Views

Author

Clark Kimberling, Jan 11 2016

Keywords

Comments

See A265762 for a guide to related sequences.

Examples

			Let u = 2^(1/3), and let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[u,1,1,1,...] has p(0,x)  = -5 - 15 x - 6 x^2 - 9 x^3 + 3 x^5 + x^6, so that a(0) = -15.
[1,u,1,1,1,...] has p(1,x) = -11 + 45 x - 66 x^2 + 35 x^3 + 6 x^4 - 15 x^5 + 5 x^6, so that a(1) = 45;
[1,1,u,1,1,1...] has p(2,x) = 131 - 633 x + 1110 x^2 - 969 x^3 + 456 x^4 - 111 x^5 + 11 x^6, so that a(2) = -633.

Links

Crossrefs

Cf. A265762, A267078, A267080, A267081, A267082, A267083, A266527.

Programs

  • Mathematica
    u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {2^(1/3)}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 30}]
Coefficient[t, x, 0]; (* A267078 *)
Coefficient[t, x, 1]; (* A267079 *)
Coefficient[t, x, 2]; (* A267080 *)
Coefficient[t, x, 3]; (* A267081 *)
Coefficient[t, x, 4]; (* A267082 *)
Coefficient[t, x, 5]; (* A267083 *)
Coefficient[t, x, 6]; (* A266527 *)
  • PARI
    Vec(-3*(5 - 80*x - 114*x^2 + 4224*x^3 - 7142*x^4 - 7912*x^5 + 3123*x^6 + 390*x^7 - 30*x^8)/((1 + x)*(1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2)) + O(x^30)) \\ Andrew Howroyd, Mar 07 2018

Formula

a(n) = 13*a(n-1) + 104*a(n-2) - 260*a(n-3) - 260*a(n-4) + 104*a(n-5) + 13*a(n-6) - a(n-7) for n > 8.
G.f.: (3 (-5 + 80 x + 114 x^2 - 4224 x^3 + 7142 x^4 + 7912 x^5 - 3123 x^6 - 390 x^7 + 30 x^8))/(1 - 13 x - 104 x^2 + 260 x^3 + 260 x^4 - 104 x^5 - 13 x^6 + x^7).
G.f.: -3*(5 - 80*x - 114*x^2 + 4224*x^3 - 7142*x^4 - 7912*x^5 + 3123*x^6 + 390*x^7 - 30*x^8)/((1 + x)*(1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2)). - Andrew Howroyd, Mar 07 2018

A267081 Coefficient of x^3 in the minimal polynomial of the continued fraction [1^n,2^(1/3),1,1,...], where 1^n means n ones.

Original entry on oeis.org

-9, 35, -969, -14359, -279261, -4862231, -88270665, -1576950691, -28345226121, -508305487319, -9123426587229, -163697793422935, -2937543639603849, -52711355807057699, -945871877489577801, -16972948054702729111, -304567428780675699165, -5465239154667149397911
Offset: 0

Views

Author

Clark Kimberling, Jan 11 2016

Keywords

Comments

See A265762 for a guide to related sequences.

Examples

			Let u = 2^(1/3), and let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[u,1,1,1,...] has p(0,x)  = -5 - 15 x - 6 x^2 - 9 x^3 + 3 x^5 + x^6, so that a(0) = -9.
[1,u,1,1,1,...] has p(1,x) = -11 + 45 x - 66 x^2 + 35 x^3 + 6 x^4 - 15 x^5 + 5 x^6, so that a(1) = 35;
[1,1,u,1,1,1...] has p(2,x) = 131 - 633 x + 1110 x^2 - 969 x^3 + 456 x^4 - 111 x^5 + 11 x^6, so that a(2) = -969.

Links

Crossrefs

Cf. A265762, A267078, A267079, A267080, A267082, A267083, A266527.

Programs

  • Mathematica
    u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {2^(1/3)}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 30}]
Coefficient[t, x, 0]; (* A267078 *)
Coefficient[t, x, 1]; (* A267079 *)
Coefficient[t, x, 2]; (* A267080 *)
Coefficient[t, x, 3]; (* A267081 *)
Coefficient[t, x, 4]; (* A267082 *)
Coefficient[t, x, 5]; (* A267083 *)
Coefficient[t, x, 6]; (* A266527 *)
  • PARI
    Vec((-9 + 125*x - 104*x^2 - 8179*x^3 - 9491*x^4 - 700*x^5 + 70*x^6)/((1 + x)*(1 + 7*x + x^2)*(1 - 18*x + x^2)) + O(x^30)) \\ Andrew Howroyd, Mar 07 2018

Formula

a(n) = 13*a(n-1) + 104*a(n-2) - 260*a(n-3) - 260*a(n-4) + 104*a(n-5) + 13*a(n-6) - a(n-7) for n > 8.
G.f.: (-9 + 125 x - 104 x^2 - 8179 x^3 - 9491 x^4 - 700 x^5 + 70 x^6)/(1 - 10 x - 135 x^2 - 135 x^3 - 10 x^4 + x^5).
From Andrew Howroyd, Mar 07 2018: (Start)
a(n) = 10*a(n-1) + 135*a(n-2) + 135*a(n-3) + 10*a(n-4) - a(n-5) for n > 6.
G.f.: (-9 + 125*x - 104*x^2 - 8179*x^3 - 9491*x^4 - 700*x^5 + 70*x^6)/((1 + x)*(1 + 7*x + x^2)*(1 - 18*x + x^2)).
(End)

A267082 Coefficient of x^4 in the minimal polynomial of the continued fraction [1^n,2^(1/3),1,1,...], where 1^n means n ones.

Original entry on oeis.org

0, 6, 456, 6240, 131238, 2238780, 41011296, 730283034, 13143304440, 235581102912, 4229156006790, 75876624195564, 1361636473680576, 24432987781993530, 438436202143461288, 7867390833380267040, 141174789462751501926, 2533277512666920359964
Offset: 0

Views

Author

Clark Kimberling, Jan 11 2016

Keywords

Comments

See A265762 for a guide to related sequences.

Examples

			Let u = 2^(1/3), and let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[u,1,1,1,...] has p(0,x)  = -5 - 15 x - 6 x^2 - 9 x^3 + 3 x^5 + x^6, so that a(0) = 0.
[1,u,1,1,1,...] has p(1,x) = -11 + 45 x - 66 x^2 + 35 x^3 + 6 x^4 - 15 x^5 + 5 x^6, so that a(1) = 6;
[1,1,u,1,1,1...] has p(2,x) = 131 - 633 x + 1110 x^2 - 969 x^3 + 456 x^4 - 111 x^5 + 11 x^6, so that a(2) = 456.

Links

Crossrefs

Cf. A265762, A267078, A267079, A267080, A267081, A267083, A266527.

Programs

  • Mathematica
    u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {2^(1/3)}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 30}]
Coefficient[t, x, 0]; (* A267078 *)
Coefficient[t, x, 1]; (* A267079 *)
Coefficient[t, x, 2]; (* A267080 *)
Coefficient[t, x, 3]; (* A267081 *)
Coefficient[t, x, 4]; (* A267082 *)
Coefficient[t, x, 5]; (* A267083 *)
Coefficient[t, x, 6]; (* A266527 *)
  • PARI
    concat([0], 6*Vec((1 + 62*x - 114*x^2 + 823*x^3 - 182*x^4 - 28*x^5 + 2*x^6)/((1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2)) + O(x^30))) \\ Andrew Howroyd, Mar 07 2018

Formula

a(n) = 13*a(n-1) + 104*a(n-2) - 260*a(n-3) - 260*a(n-4) + 104*a(n-5) + 13*a(n-6) - a(n-7) for n > 8.
G.f.: (6 (x + 62 x^2 - 114 x^3 + 823 x^4 - 182 x^5 - 28 x^6 + 2 x^7))/(1 - 14 x - 90 x^2 + 350 x^3 - 90 x^4 - 14 x^5 + x^6).
From Andrew Howroyd, Mar 07 2018: (Start)
a(n) = 14*a(n-1) + 90*a(n-2) - 350*a(n-3) + 90*a(n-4) + 14*a(n-5) - a(n-6) for n > 7.
G.f.: 6*x*(1 + 62*x - 114*x^2 + 823*x^3 - 182*x^4 - 28*x^5 + 2*x^6)/((1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2)).
(End)

A266527 Coefficient of x^6 in the minimal polynomial of the continued fraction [1^n,2^(1/3),1,1,...], where 1^n means n ones.

Original entry on oeis.org

1, 5, 11, 131, 3421, 56209, 1049105, 18561659, 334918459, 5997328339, 107703879581, 1932077585345, 34673771913121, 622167861459451, 11164539354582251, 200338227165577379, 3594932551574173405, 64508386001097153649, 1157556438367284595889
Offset: 0

Views

Author

Clark Kimberling, Jan 11 2016

Keywords

Comments

See A265762 for a guide to related sequences.

Examples

			Let u = 2^(1/3), and let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[u,1,1,1,...] has p(0,x)  = -5 - 15 x - 6 x^2 - 9 x^3 + 3 x^5 + x^6, so that a(0) = 1.
[1,u,1,1,1,...] has p(1,x) = -11 + 45 x - 66 x^2 + 35 x^3 + 6 x^4 - 15 x^5 + 5 x^6, so that a(1) = 5;
[1,1,u,1,1,1...] has p(2,x) = 131 - 633 x + 1110 x^2 - 969 x^3 + 456 x^4 - 111 x^5 + 11 x^6, so that a(2) = 11.

Links

Crossrefs

Cf. A265762, A267078, A267079, A267080, A267081, A267082, A267083.

Programs

  • Mathematica
    u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {2^(1/3)}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 30}]
Coefficient[t, x, 0]; (* A267078 *)
Coefficient[t, x, 1]; (* A267079 *)
Coefficient[t, x, 2]; (* A267080 *)
Coefficient[t, x, 3]; (* A267081 *)
Coefficient[t, x, 4]; (* A267082 *)
Coefficient[t, x, 5]; (* A267083 *)
Coefficient[t, x, 6]; (* A266527 *)
  • PARI
    Vec((1 - 8*x - 158*x^2 - 272*x^3 + 2134*x^4 + 2168*x^5 - 1009*x^6 - 130*x^7 + 10*x^8)/((1 + x)*(1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2)) + O(x^30)) \\ Andrew Howroyd, Mar 07 2018

Formula

a(n) = 13*a(n-1) + 104*a(n-2) - 260*a(n-3) - 260*a(n-4) + 104*a(n-5) + 13*a(n-6) - a(n-7) for n > 8.
G.f.: (1 - 8 x - 158 x^2 - 272 x^3 + 2134 x^4 + 2168 x^5 - 1009 x^6 - 130 x^7 + 10 x^8)/(1 - 13 x - 104 x^2 + 260 x^3 + 260 x^4 - 104 x^5 - 13 x^6 + x^7).
G.f.: (1 - 8*x - 158*x^2 - 272*x^3 + 2134*x^4 + 2168*x^5 - 1009*x^6 - 130*x^7 + 10*x^8)/((1 + x)*(1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2)). - Andrew Howroyd, Mar 07 2018
Showing 1-6 of 6 results.

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