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 1-4 of 4 results.

A265762 Coefficient of x in minimal polynomial of the continued fraction [1^n,2,1,1,1,...], where 1^n means n ones.

Original entry on oeis.org

-3, -5, -15, -37, -99, -257, -675, -1765, -4623, -12101, -31683, -82945, -217155, -568517, -1488399, -3896677, -10201635, -26708225, -69923043, -183060901, -479259663, -1254718085, -3284894595, -8599965697, -22515002499, -58945041797, -154320122895
Offset: 0

Views

Author

Clark Kimberling, Jan 04 2016

Keywords

Comments

In the following guide to related sequences, d(n), e(n), f(n) represent the coefficients in the minimal polynomial written as d(n)*x^2 + e(n)*x + f(n), except, in some cases, for initial terms. All of these sequences (eventually) satisfy the linear recurrence relation a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
continued fractions d(n) e(n) f(n)
[1^n,2,1,1,1,...] A236428 A265762 A236428
[1^n,3,1,1,1,...] A236428 A265762 A236428
[1^n,4,1,1,1,...] A265802 A265803 A265802
[1^n,5,1,1,1,...] A265804 A265805 A265804
[1^n,1/2,1,1,1,...] A266699 A266700 A266699
[1^n,1/3,1,1,1,...] A266701 A266702 A266701
[1^n,2/3,1,1,1,...] A266703 A266704 A266703
[1^n,sqrt(5),1,1,1,...] A266705 A266706 A266705
[1^n,tau,1,1,1,...] A266707 A266708 A266707
[2,1^n,2,1,1,1,...] A236428 A266709 A236428
The following forms of continued fractions have minimal polynomials of degree 4 and, after initial terms, satisfy the following linear recurrence relation:
a(n) = 5*a(n-1) + 15*a(n-2) - 15*a(n-3) - 5*a(n-4) + a(n-5).
[1^n,sqrt(2),1,1,1,...]: A266710, A266711, A266712, A266713, A266710
[1^n,sqrt(3),1,1,1,...]: A266799, A266800, A266801, A266802, A266799
[1^n,sqrt(6),1,1,1,...]: A266804, A266805, A266806, A266807, A277804
Continued fractions [1^n,2^(1/3),1,1,1,...] have minimal polynomials of degree 6. The coefficient sequences are linearly recurrenct with signature {13, 104, -260, -260, 104, 13, -1, 0, 0}; see A267078-A267083.
Continued fractions [1^n,sqrt(2)+sqrt(3),1,1,1,...] have minimal polynomials of degree 8. The coefficient sequences are linearly recurrenct with signature {13, 104, -260, -260, 104, 13, -1}; see A266803, A266808, A267061-A267066.

Examples

			Let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[2,1,1,1,1,...] = (3 + sqrt(5))/2 has p(0,x) = x^2 - 3x + 1, so a(0) = -3;
[1,2,1,1,1,...] = (5 - sqrt(5))/2 has p(1,x) = x^2 - 5x + 5, so a(1) = -5;
[1,1,2,1,1,...] = (15 + sqrt(5))/10 has p(2,x) = 5x^2 - 15x + 11, so a(2) = -15.

Links

Crossrefs

Cf. A236428, A265802.

Programs

  • Magma
    I:=[-3,-5,-15]; [n le 3 select I[n] else 2*Self(n-1)+2*Self(n-2)-Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jan 05 2016
  • Mathematica
    Program 1:
u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {2}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 20}]
Coefficient[t, x, 0] (* A236428 *)
Coefficient[t, x, 1] (* A265762 *)
Coefficient[t, x, 2] (* A236428 *)
Program 2:
LinearRecurrence[{2, 2, -1}, {-3, -5, -15}, 50] (* Vincenzo Librandi, Jan 05 2016 *)
  • PARI
    Vec((-3+x+x^2)/(1-2*x-2*x^2+x^3) + O(x^100)) \\ Altug Alkan, Jan 04 2016

Formula

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
G.f.: (-3 + x + x^2)/(1 - 2 x - 2 x^2 + x^3).
a(n) = (-1)*(2^(-n)*(3*(-2)^n+2*((3-sqrt(5))^(1+n)+(3+sqrt(5))^(1+n))))/5. - Colin Barker, Sep 27 2016

A266711 Coefficient of x in the minimal polynomial of the continued fraction [1^n,sqrt(2),1,1,...], where 1^n means n ones.

Original entry on oeis.org

-6, 2, 18, -102, -714, -4826, -33222, -227298, -1558962, -10682534, -73226346, -501882042, -3439999878, -23577981122, -161606223954, -1107664654566, -7592048797962, -52036670543258, -356664661728582, -2444615917773474, -16755646877311986, -114844911923314982
Offset: 0

Views

Author

Clark Kimberling, Jan 09 2016

Keywords

Comments

See A265762 for a guide to related sequences.

Examples

			Let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[sqrt(2),1,1,1,...] has p(0,x) = -1 - 6 x - 5 x^2 + 2 x^3 + x^4, so a(0) = -6;
[1,sqrt(2),1,1,1,...] has p(1,x) = 1 + 2 x - 7 x^2 + 2 x^3 + x^4, so a(1) = 2;
[1,1,sqrt(2),1,1,1...] has p(2,x) = -9 + 18 x - 7 x^2 - 2 x^3 + x^4, so a(2) = 18.

Links

Crossrefs

Cf. A265762, A266710, A266712, A266713.

Programs

  • Magma
    I:=[-102, -714, -4826, -33222, -227298]; [-6, 2, 18] cat [n le 5 select I[n] else 5*Self(n-1) + 15*Self(n-2) - 15*Self(n-3) - 5*Self(n-4) + Self(n-5): n in [1..30]]; // G. C. Greubel, Jan 26 2018
  • Mathematica
    u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[2]}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}];
Coefficient[t, x, 0] ; (* A266710 *)
Coefficient[t, x, 1];  (* A266711 *)
Coefficient[t, x, 2];  (* A266712 *)
Coefficient[t, x, 3];  (* A266713 *)
Coefficient[t, x, 4];  (* A266710 *)
LinearRecurrence[{5,15,-15,-5,1}, {-6, 2, 18, -102, -714, -4826, -33222, -227298}, 30] (* G. C. Greubel, Jan 26 2018 *)
  • PARI
    x='x+O('x^30); Vec(2*(3 -16*x -49*x^2 +156*x^3 +237*x^4 -280*x^5 -88*x^6 +18*x^7)/(-1 +5*x +15*x^2 -15*x^3 -5*x^4 +x^5)) \\ G. C. Greubel, Jan 26 2018

Formula

a(n) = 5*a(n-1) + 15*a(n-2) - 15*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: 2*(3 -16*x -49*x^2 +156*x^3 +237*x^4 -280*x^5 -88*x^6 +18*x^7)/(-1 +5*x +15*x^2 -15*x^3 -5*x^4 +x^5).

A266712 Coefficient of x^2 in the minimal polynomial of the continued fraction [1^n,sqrt(2),1,1,...], where 1^n means n ones.

Original entry on oeis.org

-5, -7, -7, 115, 607, 4615, 30427, 211687, 1442695, 9909907, 67867135, 465315847, 3188935867, 21858303175, 149816390407, 1026863749555, 7038210692767, 48240661271047, 330646286854555, 2266283690589607, 15533338646986375, 106467089195295187
Offset: 0

Views

Author

Clark Kimberling, Jan 09 2016

Keywords

Comments

See A265762 for a guide to related sequences.

Examples

			Let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[sqrt(2),1,1,1,...] has p(0,x) = -1 - 6 x - 5 x^2 + 2 x^3 + x^4, so a(0) = -5;
[1,sqrt(2),1,1,1,...] has p(1,x) = 1 + 2 x - 7 x^2 + 2 x^3 + x^4, so a(1) = -7;
[1,1,sqrt(2),1,1,1...] has p(2,x) = -9 + 18 x - 7 x^2 - 2 x^3 + x^4, so a(2) = -7.

Links

Crossrefs

Cf. A265762, A266710, A266711, A266713.

Programs

  • Magma
    I:=[115, 607, 4615, 30427, 211687]; [-5, -7, -7] cat [n le 5 select I[n] else 5*Self(n-1) + 15*Self(n-2) - 15*Self(n-3) - 5*Self(n-4) + Self(n-5): n in [1..30]]; // G. C. Greubel, Jan 26 2018
  • Mathematica
    u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[2]}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}];
Coefficient[t, x, 0] ; (* A266710 *)
Coefficient[t, x, 1];  (* A266711 *)
Coefficient[t, x, 2];  (* A266712 *)
Coefficient[t, x, 3];  (* A266713 *)
Coefficient[t, x, 4];  (* A266710 *)
LinearRecurrence[{5,15,-15,-5,1}, {-5, -7, -7, 115, 607, 4615, 30427, 211687}, 30] (* G. C. Greubel, Jan 26 2018 *)
  • PARI
    x='x+O('x^30); Vec((5 -18*x -103*x^2 -180*x^3 -7*x^4 +280*x^5 +56*x^6 -14*x^7)/(-1 + 5*x +15*x^2 -15*x^3 -5*x^4 +x^5)) \\ G. C. Greubel, Jan 26 2018

Formula

a(n) = 5*a(n-1) + 15*a(n-2) - 15*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: (5 -18*x -103*x^2 -180*x^3 -7*x^4 +280*x^5 +56*x^6 -14*x^7)/(-1 + 5*x +15*x^2 -15*x^3 -5*x^4 +x^5).

A266713 Coefficient of x^3 in the minimal polynomial of the continued fraction [1^n,sqrt(2),1,1,...], where 1^n means n ones.

Original entry on oeis.org

2, 2, -2, -54, -226, -1958, -12382, -87618, -593374, -4085846, -27955618, -191739462, -1313864638, -9006244994, -61727410366, -423092015478, -2899899974242, -19876251587558, -136233746512414, -933760274094786, -6400087386491038, -43866853488227222
Offset: 0

Views

Author

Clark Kimberling, Jan 09 2016

Keywords

Comments

See A265762 for a guide to related sequences.

Examples

			Let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[sqrt(2),1,1,1,...] has p(0,x) = -1 - 6 x - 5 x^2 + 2 x^3 + x^4, so a(0) = 2;
[1,sqrt(2),1,1,1,...] has p(1,x) = 1 + 2 x - 7 x^2 + 2 x^3 + x^4, so a(1) = 2;
[1,1,sqrt(2),1,1,1...] has p(2,x) = -9 + 18 x - 7 x^2 - 2 x^3 + x^4, so a(2) = -2.

Links

Crossrefs

Cf. A265762, A266710, A266711, A266712.

Programs

  • Magma
    I:=[-54, -226, -1958, -12382, -87618]; [2, 2, -2] cat [n le 5 select I[n] else 5*Self(n-1) + 15*Self(n-2) - 15*Self(n-3) - 5*Self(n-4) + Self(n-5): n in [1..30]]; // G. C. Greubel, Jan 26 2018
  • Mathematica
    u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[2]}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}];
Coefficient[t, x, 0] ; (* A266710 *)
Coefficient[t, x, 1];  (* A266711 *)
Coefficient[t, x, 2];  (* A266712 *)
Coefficient[t, x, 3];  (* A266713 *)
Coefficient[t, x, 4];  (* A266710 *)
LinearRecurrence[{5,15,-15,-5,1}, {2, 2, -2, -54, -226, -1958, -12382, -87618}, 30] (* G. C. Greubel, Jan 26 2018 *)
  • PARI
    my(x='x+O('x^30)); Vec(-2*(1 -4*x -21*x^2 -22*x^3 +57*x^4 -20*x^5 -12*x^6 +2*x^7)/(-1 +5*x +15*x^2 -15*x^3 -5*x^4 +x^5)) \\ G. C. Greubel, Jan 26 2018

Formula

a(n) = 5*a(n-1) + 15*a(n-2) - 15*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: -2*(1 -4*x -21*x^2 -22*x^3 +57*x^4 -20*x^5 -12*x^6 +2*x^7)/(-1 +5*x +15*x^2 -15*x^3 -5*x^4 +x^5).
Showing 1-4 of 4 results.

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