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 1-7 of 7 results.

A266803 Coefficient of x^0 in the minimal polynomial of the continued fraction [1^n,sqrt(2)+sqrt(3),1,1,...], where 1^n means n ones.

Original entry on oeis.org

49, 49, 25281, 606409, 37676521, 1596669889, 78061422609, 3612062087761, 170677159358209, 8000461380881641, 376169445225673929, 17666248458032362369, 830040053693500377841, 38992376127586237335409, 1831844657768331755159361, 86057114020320867143580169
Offset: 0

Views

Author

Clark Kimberling, Jan 10 2016

Keywords

Comments

See A265762 for a guide to related sequences.

Examples

			Let u = sqrt(2) and v = sqrt(3), and let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[u+v,1,1,1,...] has p(0,x) = 49 - 168 x - 50 x^2 + 212 x^3 + 47 x^4 - 68 x^5 - 18 x^6 + 4 x^7 + x^8, so that a(0) = 49.
[1,u+v,1,1,1,...] has p(1,x) = 49 - 560 x + 2498 x^2 - 5760 x^3 + 7547 x^4 - 5760 x^5 + 2498 x^6 - 560 x^7 + 49 x^8, so that a(1) = 49;
[1,1,u+v,1,1,1...] has p(2,x) = 25281 - 101124 x + 173262 x^2 - 165852 x^3 + 96847 x^4 - 35252 x^5 + 7790 x^6 - 952 x^7 + 49 x^8, so that a(2) = 25281.

Links

Crossrefs

Cf. A265762, A266808, A267061, A267062, A267063, A267064, A267065, A267066.

Programs

  • Mathematica
    u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[2] + Sqrt[3]}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}];
Coefficient[t, x, 0];  (* A266803 *)
Coefficient[t, x, 1];  (* A266808 *)
Coefficient[t, x, 2];  (* A267061 *)
Coefficient[t, x, 3];  (* A267062 *)
Coefficient[t, x, 4];  (* A267063 *)
Coefficient[t, x, 5];  (* A267064 *)
Coefficient[t, x, 6];  (* A267065 *)
Coefficient[t, x, 7];  (* A267066 *)
Coefficient[t, x, 8];  (* A266803 *)

Formula

a(n) = 34*a(n-1) + 714*a(n-2) - 4641*a(n-3) - 12376*a(n-4) + 12376*a(n-5) + 4641*a(n-6) - 714*a(n-7) - 34*a(n-8) + a(n-9).
G.f.: (-49 + 1617 x + 11371 x^2 + 60722 x^3 + 158186 x^4 - 21270 x^5 + 1619 x^6 + 25 x^7 - x^8)/(-1 + 34 x + 714 x^2 - 4641 x^3 - 12376 x^4 + 12376 x^5 + 4641 x^6 - 714 x^7 - 34 x^8 + x^9).

A266808 Coefficient of x in the minimal polynomial of the continued fraction [1^n,sqrt(2)+sqrt(3),1,1,...], where 1^n means n ones.

Original entry on oeis.org

-168, -560, -101124, -3288624, -180132168, -7998247028, -384048485640, -17892957477264, -843263161727364, -39567408316416848, -1859687400468342888, -87350263553726629620, -4103880417768964672104, -192790045902230868971504, -9057117701582885083841028
Offset: 0

Views

Author

Clark Kimberling, Jan 10 2016

Keywords

Comments

See A265762 for a guide to related sequences.

Examples

			Let u = sqrt(2) and v = sqrt(3), and let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[u+v,1,1,1,...] has p(0,x) = 49 - 168 x - 50 x^2 + 212 x^3 + 47 x^4 - 68 x^5 - 18 x^6 + 4 x^7 + x^8, so that a(0) = -168.
[1,u+v,1,1,1,...] has p(1,x) = 49 - 560 x + 2498 x^2 - 5760 x^3 + 7547 x^4 - 5760 x^5 + 2498 x^6 - 560 x^7 + 49 x^8, so that a(1) = -560;
[1,1,u+v,1,1,1...] has p(2,x) = 25281 - 101124 x + 173262 x^2 - 165852 x^3 + 96847 x^4 - 35252 x^5 + 7790 x^6 - 952 x^7 + 49 x^8, so that a(2) = -101124.

Links

Crossrefs

Cf. A265762, A266803, A267061, A267062, A267063, A267064, A267065, A267066.

Programs

  • Mathematica
    u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[2] + Sqrt[3]}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}];
Coefficient[t, x, 0];  (* A266803 *)
Coefficient[t, x, 1];  (* A266808 *)
Coefficient[t, x, 2];  (* A267061 *)
Coefficient[t, x, 3];  (* A267062 *)
Coefficient[t, x, 4];  (* A267063 *)
Coefficient[t, x, 5];  (* A267064 *)
Coefficient[t, x, 6];  (* A267065 *)
Coefficient[t, x, 7];  (* A267066 *)
Coefficient[t, x, 8];  (* A266803 *)

Formula

a(n) = 34*a(n-1) + 714*a(n-2) - 4641*a(n-3) - 12376*a(n-4) + 12376*a(n-5) + 4641*a(n-6) - 714*a(n-7) - 34*a(n-8) + a(n-9).
G.f.: -((4 (-42 + 1288 x + 9467 x^2 - 57564 x^3 - 198636 x^4 + 39086 x^5 - 5774 x^6 - 48 x^7 + 3 x^8))/(-1 + 34 x + 714 x^2 - 4641 x^3 - 2376 x^4 + 12376 x^5 + 4641 x^6 - 714 x^7 - 34 x^8 + x^9)).

A267061 Coefficient of x^2 in the minimal polynomial of the continued fraction [1^n,sqrt(2)+sqrt(3),1,1,...], where 1^n means n ones.

Original entry on oeis.org

-50, 2498, 173262, 7783550, 376636138, 17527857350, 826628182158, 38778106729442, 1822757247598510, 85612705715717438, 4022299792573538250, 188956642021519970918, 8877044611408850508622, 417030260830076184423170, 19591578937460413027671438
Offset: 0

Views

Author

Clark Kimberling, Jan 10 2016

Keywords

Comments

See A265762 for a guide to related sequences.

Examples

			Let u = sqrt(2) and v = sqrt(3), and let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[u+v,1,1,1,...] has p(0,x) = 49 - 168 x - 50 x^2 + 212 x^3 + 47 x^4 - 68 x^5 - 18 x^6 + 4 x^7 + x^8, so that a(0) = -168.
[1,u+v,1,1,1,...] has p(1,x) = 49 - 560 x + 2498 x^2 - 5760 x^3 + 7547 x^4 - 5760 x^5 + 2498 x^6 - 560 x^7 + 49 x^8, so that a(1) = -560;
[1,1,u+v,1,1,1...] has p(2,x) = 25281 - 101124 x + 173262 x^2 - 165852 x^3 + 96847 x^4 - 35252 x^5 + 7790 x^6 - 952 x^7 + 49 x^8, so that a(2) = -101124.

Links

Crossrefs

Cf. A265762, A266803, A266808, A267062, A267063, A267064, A267065, A267066.

Programs

  • Mathematica
    u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[2] + Sqrt[3]}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}];
Coefficient[t, x, 0];  (* A266803 *)
Coefficient[t, x, 1];  (* A266808 *)
Coefficient[t, x, 2];  (* A267061 *)
Coefficient[t, x, 3];  (* A267062 *)
Coefficient[t, x, 4];  (* A267063 *)
Coefficient[t, x, 5];  (* A267064 *)
Coefficient[t, x, 6];  (* A267065 *)
Coefficient[t, x, 7];  (* A267066 *)
Coefficient[t, x, 8];  (* A266803 *)

Formula

a(n) = 34*a(n-1) + 714*a(n-2) - 4641*a(n-3) - 12376*a(n-4) + 12376*a(n-5) + 4641*a(n-6) - 714*a(n-7) - 34*a(n-8) + a(n-9).
G.f.: -((2 (-25 + 2099 x + 62015 x^2 - 61490 x^3 - 369606 x^4 + 208474 x^5 - 53705 x^6 - x^7 + 19 x^8))/(-1 + 34 x + 714 x^2 - 4641 x^3 - 12376 x^4 + 12376 x^5 + 4641 x^6 - 714 x^7 - 34 x^8 + x^9)).

A267066 Coefficient of x^6 in the minimal polynomial of the continued fraction [1^n,sqrt(2)+sqrt(3),1,1,...], where 1^n means n ones.

Original entry on oeis.org

4, -560, -952, -303372, -8139896, -481544336, -20771606140, -1008539866512, -46789454179352, -2208680436593036, -103571099363469976, -4869042962273734320, -228680251217985528572, -10744200847316967694832, -504729054922920767654776
Offset: 0

Views

Author

Clark Kimberling, Jan 10 2016

Keywords

Comments

See A265762 for a guide to related sequences.

Examples

			Let u = sqrt(2) and v = sqrt(3), and let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[u+v,1,1,1,...] has p(0,x)  = 49 - 168 x - 50 x^2 + 212 x^3 + 47 x^4 - 68 x^5 - 18 x^6 + 4 x^7 + x^8, so that a(0) = 4.
[1,u+v,1,1,1,...] has p(1,x) = 49 - 560 x + 2498 x^2 - 5760 x^3 + 7547 x^4 - 5760 x^5 + 2498 x^6 - 560 x^7 + 49 x^8, so that a(1) = -560;
[1,1,u+v,1,1,1...] has p(2,x) = 25281 - 101124 x + 173262 x^2 - 165852 x^3 + 96847 x^4 - 35252 x^5 + 7790 x^6 - 952 x^7 + 49 x^8, so that a(2) = -952.

Links

Crossrefs

Cf. A265762, A266803, A266808, A267061, A267062, A267063, A267064, A267065.

Programs

  • Mathematica
    u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[2] + Sqrt[3]}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}];
Coefficient[t, x, 0];  (* A266803 *)
Coefficient[t, x, 1];  (* A266808 *)
Coefficient[t, x, 2];  (* A267061 *)
Coefficient[t, x, 3];  (* A267062 *)
Coefficient[t, x, 4];  (* A267063 *)
Coefficient[t, x, 5];  (* A267064 *)
Coefficient[t, x, 6];  (* A267065 *)
Coefficient[t, x, 7];  (* A267066 *)
Coefficient[t, x, 8];  (* A266803 *)

Formula

a(n) = 34*a(n-1) + 714*a(n-2) - 4641*a(n-3) - 12376*a(n-4) + 12376*a(n-5) + 4641*a(n-6) - 714*a(n-7) - 34*a(n-8) + a(n-9).
G.f.: -((4 (1 - 174 x + 3808 x^2 + 36850 x^3 + 76256 x^4 + 105360 x^5 - 8095 x^6 - 1822 x^7 + 36 x^8))/(-1 + 34 x + 714 x^2 - 4641 x^3 - 12376 x^4 + 12376 x^5 + 4641 x^6 - 714 x^7 - 34 x^8 + x^9)).

A267062 Coefficient of x^3 in the minimal polynomial of the continued fraction [1^n,sqrt(2)+sqrt(3),1,1,...], where 1^n means n ones.

Original entry on oeis.org

212, -5760, -165852, -10501476, -449827456, -21948311748, -1016699956620, -48023357086272, -2251419462422716, -105852417560435076, -4971310326775823808, -233572686675369390180, -10972461323000994899692, -515480788238950647507456, -24216468853316695676874396
Offset: 0

Views

Author

Clark Kimberling, Jan 10 2016

Keywords

Comments

See A265762 for a guide to related sequences.

Examples

			Let u = sqrt(2) and v = sqrt(3), and let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[u+v,1,1,1,...] has p(0,x)  = 49 - 168 x - 50 x^2 + 212 x^3 + 47 x^4 - 68 x^5 - 18 x^6 + 4 x^7 + x^8, so that a(0) = 212.
[1,u+v,1,1,1,...] has p(1,x) = 49 - 560 x + 2498 x^2 - 5760 x^3 + 7547 x^4 - 5760 x^5 + 2498 x^6 - 560 x^7 + 49 x^8, so that a(1) = -5760;
[1,1,u+v,1,1,1...] has p(2,x) = 25281 - 101124 x + 173262 x^2 - 165852 x^3 + 96847 x^4 - 35252 x^5 + 7790 x^6 - 952 x^7 + 49 x^8, so that a(2) = -165852.

Links

Crossrefs

Cf. A265762, A266803, A266808, A267061, A267063, A267064, A267065, A267066.

Programs

  • Mathematica
    u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[2] + Sqrt[3]}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}];
Coefficient[t, x, 0];  (* A266803 *)
Coefficient[t, x, 1];  (* A266808 *)
Coefficient[t, x, 2];  (* A267061 *)
Coefficient[t, x, 3];  (* A267062 *)
Coefficient[t, x, 4];  (* A267063 *)
Coefficient[t, x, 5];  (* A267064 *)
Coefficient[t, x, 6];  (* A267065 *)
Coefficient[t, x, 7];  (* A267066 *)
Coefficient[t, x, 8];  (* A266803 *)

Formula

a(n) = 34*a(n-1) + 714*a(n-2) - 4641*a(n-3) - 12376*a(n-4) + 12376*a(n-5) + 4641*a(n-6) - 714*a(n-7) - 34*a(n-8) + a(n-9).
G.f.: (4 (-53 + 3242 x + 30345 x^2 - 58506 x^3 - 383152 x^4 - 61754 x^5 + 46551 x^6 - 1122 x^7 + 9 x^8))/(-1 + 34 x + 714 x^2 - 4641 x^3 - 12376 x^4 + 12376 x^5 + 4641 x^6 - 714 x^7 - 34 x^8 + x^9).

A267064 Coefficient of x^5 in the minimal polynomial of the continued fraction [1^n,sqrt(2)+sqrt(3),1,1,...], where 1^n means n ones.

Original entry on oeis.org

-68, -5760, -35252, -4744764, -160222784, -8602304988, -384492157220, -18412926914112, -858719581400084, -40454410268348124, -1898470063828865408, -89224033424689993980, -4190977987082560730372, -196898460771438377224704, -9249826380311085293230964
Offset: 0

Views

Author

Clark Kimberling, Jan 10 2016

Keywords

Comments

See A265762 for a guide to related sequences.

Examples

			Let u = sqrt(2) and v = sqrt(3), and let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[u+v,1,1,1,...] has p(0,x)  = 49 - 168 x - 50 x^2 + 212 x^3 + 47 x^4 - 68 x^5 - 18 x^6 + 4 x^7 + x^8, so that a(0) = -68.
[1,u+v,1,1,1,...] has p(1,x) = 49 - 560 x + 2498 x^2 - 5760 x^3 + 7547 x^4 - 5760 x^5 + 2498 x^6 - 560 x^7 + 49 x^8, so that a(1) = -5760;
[1,1,u+v,1,1,1...] has p(2,x) = 25281 - 101124 x + 173262 x^2 - 165852 x^3 + 96847 x^4 - 35252 x^5 + 7790 x^6 - 952 x^7 + 49 x^8, so that a(2) = -35252.

Links

Crossrefs

Cf. A265762, A266803, A266808, A267061, A267062, A267063, A267065, A267066.

Programs

  • Mathematica
    u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[2] + Sqrt[3]}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}];
Coefficient[t, x, 0];  (* A266803 *)
Coefficient[t, x, 1];  (* A266808 *)
Coefficient[t, x, 2];  (* A267061 *)
Coefficient[t, x, 3];  (* A267062 *)
Coefficient[t, x, 4];  (* A267063 *)
Coefficient[t, x, 5];  (* A267064 *)
Coefficient[t, x, 6];  (* A267065 *)
Coefficient[t, x, 7];  (* A267066 *)
Coefficient[t, x, 8];  (* A266803 *)

Formula

a(n) = 34*a(n-1) + 714*a(n-2) - 4641*a(n-3) - 12376*a(n-4) + 12376*a(n-5) + 4641*a(n-6) - 714*a(n-7) - 34*a(n-8) + a(n-9).
G.f.: (4 (17 + 862 x - 52285 x^2 - 62714 x^3 + 326152 x^4 + 254390 x^5 - 38255 x^6 - 3838 x^7 + 111 x^8))/(-1 + 34 x + 714 x^2 - 4641 x^3 - 12376 x^4 + 12376 x^5 + 4641 x^6 - 714 x^7 - 34 x^8 + x^9).

A267065 Coefficient of x^6 in the minimal polynomial of the continued fraction [1^n,sqrt(2)+sqrt(3),1,1,...], where 1^n means n ones.

Original entry on oeis.org

-18, 2498, 7790, 1588998, 47783370, 2692503902, 118222343438, 5700687414690, 265166547527598, 12504559841719910, 586597482595321322, 27572856062170808478, 1295063443722512524590, 60845329039209613792898, 2858347113718106912615150, 134283053266053897759020742
Offset: 0

Views

Author

Clark Kimberling, Jan 10 2016

Keywords

Comments

See A265762 for a guide to related sequences.

Examples

			Let u = sqrt(2) and v = sqrt(3), and let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[u+v,1,1,1,...] has p(0,x)  = 49 - 168 x - 50 x^2 + 212 x^3 + 47 x^4 - 68 x^5 - 18 x^6 + 4 x^7 + x^8, so that a(0) = -18.
[1,u+v,1,1,1,...] has p(1,x) = 49 - 560 x + 2498 x^2 - 5760 x^3 + 7547 x^4 - 5760 x^5 + 2498 x^6 - 560 x^7 + 49 x^8, so that a(1) = 2498;
[1,1,u+v,1,1,1...] has p(2,x) = 25281 - 101124 x + 173262 x^2 - 165852 x^3 + 96847 x^4 - 35252 x^5 + 7790 x^6 - 952 x^7 + 49 x^8, so that a(2) = 7790.

Links

Crossrefs

Cf. A265762, A266803, A266808, A267061, A267062, A267063, A267064, A267066.

Programs

  • Mathematica
    u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[2] + Sqrt[3]}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}];
Coefficient[t, x, 0];  (* A266803 *)
Coefficient[t, x, 1];  (* A266808 *)
Coefficient[t, x, 2];  (* A267061 *)
Coefficient[t, x, 3];  (* A267062 *)
Coefficient[t, x, 4];  (* A267063 *)
Coefficient[t, x, 5];  (* A267064 *)
Coefficient[t, x, 6];  (* A267065 *)
Coefficient[t, x, 7];  (* A267066 *)
Coefficient[t, x, 8];  (* A266803 *)

Formula

a(n) = 34*a(n-1) + 714*a(n-2) - 4641*a(n-3) - 12376*a(n-4) + 12376*a(n-5) + 4641*a(n-6) - 714*a(n-7) - 34*a(n-8) + a(n-9).
G.f.: -((2 (-9 + 1555 x - 32145 x^2 - 271486 x^3 - 217086 x^4 + 308078 x^5 + 819 x^6 - 1961 x^7 + 15 x^8))/(-1 + 34 x + 714 x^2 - 4641 x^3 - 12376 x^4 + 12376 x^5 + 4641 x^6 - 714 x^7 - 34 x^8 + x^9)).
Showing 1-7 of 7 results.

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