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 1-8 of 8 results.

A097828 Partial sums of Chebyshev sequence S(n,13)= U(n,13/2)=A078362(n).

Original entry on oeis.org

1, 14, 182, 2353, 30408, 392952, 5077969, 65620646, 847990430, 10958254945, 141609323856, 1829962955184, 23647909093537, 305592855260798, 3949059209296838, 51032176865598097, 659469240043478424
Offset: 0

Views

Author

Wolfdieter Lang, Aug 31 2004

Keywords

Links

Crossrefs

Cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).

Formula

a(n) = sum(S(k, 13), k=0..n) with S(k, 13)=U(k, 13/2)=A078362(k) Chebyshev's polynomials of the second kind.
G.f.: 1/((1-x)*(1-13*x+x^2)) = 1/(1-14*x+14*x^2-x^3).
a(n) = 14*a(n-1)-14*a(n-2)+a(n-3) with n>=2, a(-1)=0, a(0)=1, a(1)=14.
a(n) = 13*a(n-1)-a(n-2)+1 with n>=1, a(-1)=0, a(0)=1.
a(n) = (S(n+1, 13) - S(n, 13) -1)/11.

A085260 Ratio-determined insertion sequence I(0.0833344) (see the link below).

Original entry on oeis.org

1, 12, 155, 2003, 25884, 334489, 4322473, 55857660, 721827107, 9327894731, 120540804396, 1557702562417, 20129592507025, 260127000028908, 3361521407868779, 43439651302265219, 561353945521579068
Offset: 1

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Author

John W. Layman, Jun 23 2003

Keywords

Comments

This sequence is the ratio-determined insertion sequence (RDIS) "twin" to A078362 (see the link for an explanation of "twin"). See A082630 or A082981 for recent examples of RDIS sequences.
a(n) = L(n,13), where L is defined as in A108299. - Reinhard Zumkeller, Jun 01 2005
For n >= 2, a(n) equals the permanent of the (2n-2) X (2n-2) tridiagonal matrix with sqrt(11)'s along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011
Seems to be positive values of x (or y) satisfying x^2 - 13xy + y^2 + 11 = 0. - Colin Barker, Feb 10 2014
It appears that the b-file, formulas and programs are based on the conjectured, so far apparently unproved recurrence relation. - M. F. Hasler, Nov 05 2018
Nonnegative y values in solutions to the Diophantine equation 11*x^2 - 15*y^2 = -4. The corresponding x values are in A126866. Note that a(n+1)^2 - a(n)*a(n+2) = -11. - Klaus Purath, Mar 21 2025

Links

Crossrefs

Cf. A078362, A082630, A082981.
Row 13 of array A094954.
Cf. similar sequences listed in A238379.

Programs

  • Magma
    I:=[1,12]; [n le 2 select I[n] else 13*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 18 2018
  • Mathematica
    CoefficientList[Series[(1 - x)/(1 - 13 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2014 *)
LinearRecurrence[{13,-1}, {1,12}, 30] (* G. C. Greubel, Jan 18 2018 *)
  • PARI
    my(x='x+O('x^30)); Vec(x*(1-x)/(1-13*x+x^2)) \\ G. C. Greubel, Jan 18 2018

Formula

It appears that the sequence satisfies a(n+1) = 13*a(n) - a(n-1). [Corrected by M. F. Hasler, Nov 05 2018]
If the recurrence a(n+2) = 13*a(n+1) - a(n) holds then for n > 0, a(n)*a(n+3) = 143 + a(n+1)*a(n+2). - Ralf Stephan, May 29 2004
G.f.: x*(1-x)/(1 - 13*x + x^2). - Philippe Deléham, Nov 17 2008
For n>1, a(n) is the numerator of the continued fraction [1,11,1,11,...,1,11] with (n-1) repetitions of 1,11. - Greg Dresden, Sep 10 2019

A078364 A Chebyshev S-sequence with Diophantine property.

Original entry on oeis.org

1, 15, 224, 3345, 49951, 745920, 11138849, 166336815, 2483913376, 37092363825, 553901543999, 8271430796160, 123517560398401, 1844491975179855, 27543862067299424, 411313439034311505
Offset: 0

Views

Author

Wolfdieter Lang, Nov 29 2002

Keywords

Comments

a(n) gives the general (positive integer) solution of the Pell equation b^2 - 221*a^2 = +4 with companion sequence b(n)=A078365(n+1), n>=0.
This is the m=17 member of the m-family of sequences S(n,m-2) = S(2*n+1,sqrt(m))/sqrt(m). The m=4..16 (nonnegative) sequences are: A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913, A004189, A004190, A004191, A078362 and A007655. The m=1..3 (signed) sequences are A049347, A056594, A010892.
For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 15's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011
For n>=2, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,14}. - Milan Janjic, Jan 23 2015

Links

Crossrefs

a(n) = sqrt((A078365(n+1)^2 - 4)/221), n>=0, (Pell equation d=221, +4).
Cf. A077428, A078355 (Pell +4 equations).

Programs

  • Mathematica
    LinearRecurrence[{15,-1},{1,15},30] (* Harvey P. Dale, Oct 16 2011 *)
  • Sage
    [lucas_number1(n,15,1) for n in range(1,20)] # Zerinvary Lajos, Jun 25 2008

Formula

a(n) = 15*a(n-1) - a(n-2), n>= 1; a(-1)=0, a(0)=1.
a(n) = S(2*n+1, sqrt(17))/sqrt(17) = S(n, 15); S(n, x) := U(n, x/2), Chebyshev polynomials of the 2nd kind, A049310.
a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap = (15+sqrt(221))/2 and am = (15-sqrt(221))/2.
G.f.: 1/(1 - 15*x + x^2). - Philippe Deléham, Nov 17 2008
a(n) = Sum_{k=0..n} A101950(n,k)*14^k. - Philippe Deléham, Feb 10 2012
Product {n >= 0} (1 + 1/a(n)) = 1/13*(13 + sqrt(221)). - Peter Bala, Dec 23 2012
Product {n >= 1} (1 - 1/a(n)) = 1/30*(13 + sqrt(221)). - Peter Bala, Dec 23 2012
For n>=1, a(n) = U(n-1,15/2), where U(k,x) is Chebyshev polynomial of the second kind. - Milan Janjic, Jan 23 2015

A078366 A Chebyshev S-sequence with Diophantine property.

Original entry on oeis.org

1, 17, 288, 4879, 82655, 1400256, 23721697, 401868593, 6808044384, 115334885935, 1953885016511, 33100710394752, 560758191694273, 9499788548407889, 160935647131239840, 2726406212682669391, 46187969968474139807, 782469083251377707328, 13255786445304946884769
Offset: 0

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Author

Wolfdieter Lang, Nov 29 2002

Keywords

Comments

a(n) gives the general (positive integer) solution of the Pell equation b^2 - 285*a^2 = +4 with companion sequence b(n)=A078367(n+1), n >= 0.
This is the m=19 member of the m-family of sequences S(n,m-2) = S(2*n+1,sqrt(m))/sqrt(m). The m=4..18 (nonnegative) sequences are: A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913, A004189, A004190, A004191, A078362, A007655, A078364 and A077412. The m=1..3 (signed) sequences are A049347, A056594, A010892.
For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 17's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011
For n >= 2, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,16}. - Milan Janjic, Jan 23 2015

Links

Crossrefs

Cf. A077428, A078355 (Pell +4 equations). Cf. A078367.

Programs

  • GAP
    a:=[1,17,288];; for n in [4..20] do a[n]:=17*a[n-1]-a[n-2]; od; a; # G. C. Greubel, May 25 2019
  • Magma
    I:=[1, 17, 288]; [n le 3 select I[n] else 17*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 24 2012
  • Mathematica
    CoefficientList[Series[1/(1 - 17 x + x^2), {x, 0, 20}], x] (* Vincenzo Librandi, Dec 24 2012 *)
LinearRecurrence[{17,-1},{1,17},20] (* Harvey P. Dale, Aug 02 2018 *)
  • PARI
    my(x='x+O('x^20)); Vec(1/(1-17*x+x^2)) \\ G. C. Greubel, May 25 2019
  • Sage
    [lucas_number1(n,17,1) for n in range(1,20)] # Zerinvary Lajos, Jun 25 2008

Formula

a(n) = 17*a(n-1) - a(n-2), n >= 1; a(-1)=0, a(0)=1.
a(n) = S(2*n+1, sqrt(19))/sqrt(19) = S(n, 17), where S(n, x) = U(n, x/2), Chebyshev polynomials of the 2nd kind, A049310.
a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap = (17+sqrt(285))/2 and am = (17-sqrt(285))/2.
G.f.: 1/(1-17*x+x^2).
a(n) = Sum_{k=0..n} A101950(n,k)*16^k. - Philippe Deléham, Feb 10 2012
Product {n >= 0} (1 + 1/a(n)) = (1/15)*(15 + sqrt(285)). - Peter Bala, Dec 23 2012
Product {n >= 1} (1 - 1/a(n)) = (1/34)*(15 + sqrt(285)). - Peter Bala, Dec 23 2012
For n >= 1, a(n) = U(n-1,13/2), where U(k,x) represents Chebyshev polynomial of the second order. - Milan Janjic, Jan 23 2015
a(n) = sqrt((A078367(n+1)^2 - 4)/285), n>=0, (Pell equation d=285, +4).
E.g.f.: exp(17*x/2)*(sqrt(285)*cosh(sqrt(285)*x/2) + 17*sinh(sqrt(285)*x/2))/sqrt(285). - Stefano Spezia, Aug 19 2023

A097781 Chebyshev polynomials S(n,27) with Diophantine property.

Original entry on oeis.org

1, 27, 728, 19629, 529255, 14270256, 384767657, 10374456483, 279725557384, 7542215592885, 203360095450511, 5483180361570912, 147842509666964113, 3986264580646460139, 107481301167787459640, 2898008866949614950141
Offset: 0

Views

Author

Wolfdieter Lang, Aug 31 2004

Keywords

Comments

All positive integer solutions of Pell equation b(n)^2 - 725*a(n)^2 = +4 together with b(n)=A090248(n+1), n>=0. Note that D=725=29*5^2 is not squarefree.
For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 27's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011
For n>=1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,26}. - Milan Janjic, Jan 26 2015

Examples

			(x,y) = (27;1), (727;27), (19602;728), ... give the positive integer solutions to x^2 - 29*(5*y)^2 =+4.

Links

Crossrefs

Cf. A078362, A078366.

Programs

  • Magma
    I:=[1, 27, 728]; [n le 3 select I[n] else 27*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 24 2012
  • Maple
    with (combinat):seq(fibonacci(2*n, 5)/5, n=1..16); # Zerinvary Lajos, Apr 20 2008
  • Mathematica
    Join[{a=1,b=27},Table[c=27*b-a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2011 *)
CoefficientList[Series[1/(1 - 27 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 24 2012 *)
  • Sage
    [lucas_number1(n,27,1) for n in range(1,20)] # Zerinvary Lajos, Jun 25 2008

Formula

a(n) = S(n, 27) = U(n, 27/2) = S(2*n+1, sqrt(29))/sqrt(29) with S(n, x)=U(n, x/2) Chebyshev's polynomials of the 2nd kind, A049310. S(-1, x)= 0 = U(-1, x).
a(n) = 27*a(n-1)-a(n-2), n >= 1; a(0)=1, a(1)=27; a(-1)=0.
a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap = (27+5*sqrt(29))/2 and am = (27-5*sqrt(29))/2.
G.f.: 1/(1-27*x+x^2).
a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*26^k. - Philippe Deléham, Feb 10 2012
Product {n >= 0} (1 + 1/a(n)) = 1/5*(5 + sqrt(29)). - Peter Bala, Dec 23 2012
Product {n >= 1} (1 - 1/a(n)) = 5/54*(5 + sqrt(29)). - Peter Bala, Dec 23 2012

A078363 A Chebyshev T-sequence with Diophantine property.

Original entry on oeis.org

2, 13, 167, 2158, 27887, 360373, 4656962, 60180133, 777684767, 10049721838, 129868699127, 1678243366813, 21687295069442, 280256592535933, 3621648407897687, 46801172710133998, 604793596823844287, 7815515585999841733, 100996909021174098242
Offset: 0

Views

Author

Wolfdieter Lang, Nov 29 2002

Keywords

Comments

a(n) gives the general (positive integer) solution of the Pell equation a^2 - 165*b^2 = +4 with companion sequence b(n)=A078362(n-1), n>=1.
Except for the first term, positive values of x (or y) satisfying x^2 - 13xy + y^2 + 165 = 0. - Colin Barker, Feb 26 2014

References

  • O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

Links

Crossrefs

Cf. A078362.
Cf. A077428, A078355 (Pell +4 equations).
Cf. A049310, A053120.

Programs

  • Mathematica
    a[0] = 2; a[1] = 13; a[n_] := 13a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 16}] (* Robert G. Wilson v, Jan 30 2004 *)
LinearRecurrence[{13,-1},{2,13},20] (* Harvey P. Dale, Oct 28 2016 *)
  • PARI
    a(n)=if(n<0,0,2*subst(poltchebi(n),x,13/2))
  • PARI
    a(n)=if(n<0,0,polsym(1-13*x+x^2,n)[n+1])
  • PARI
    Vec((2-13*x)/(1-13*x+x^2) + O(x^100)) \\ Colin Barker, Feb 26 2014
  • Sage
    [lucas_number2(n,13,1) for n in range(0,20)] # Zerinvary Lajos, Jun 25 2008

Formula

a(n) = 13*a(n-1)-a(n-2), n >= 1; a(-1)=13, a(0)=2.
a(n) = S(n, 13) - S(n-2, 13) = 2*T(n, 13/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 13)=A078362(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120.
G.f.: (2-13*x)/(1-13*x+x^2).
a(n) = ap^n + am^n, with ap := (13+sqrt(165))/2 and am := (13-sqrt(165))/2.
a(n) = sqrt(4 + 165*A078362(n-1)^2), n>=1, (Pell equation d=165, +4).
E.g.f.: 2*exp(13*x/2)*cosh(sqrt(165)*x/2). - Stefano Spezia, Sep 24 2022

Extensions

More terms from Colin Barker, Feb 26 2014

A078368 A Chebyshev S-sequence with Diophantine property.

Original entry on oeis.org

1, 19, 360, 6821, 129239, 2448720, 46396441, 879083659, 16656193080, 315588584861, 5979526919279, 113295422881440, 2146633507828081, 40672741225852099, 770635449783361800, 14601400804658022101
Offset: 0

Views

Author

Wolfdieter Lang, Nov 29 2002

Keywords

Comments

a(n) gives the general (positive integer) solution of the Pell equation b^2 - 357*a^2 =+4 with companion sequence b(n)=A078369(n+1), n>=0.
This is the m=21 member of the m-family of sequences S(n,m-2) = S(2*n+1,sqrt(m))/sqrt(m). The m=4..20 (nonnegative) sequences are: A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913, A004189, A004190, A004191, A078362, A007655, A078364, A077412, A078366 and A049660. The m=1..3 (signed) sequences are A049347, A056594, A010892.
For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 19's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011
For n>=2, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,18}. Milan Janjic, Jan 25 2015

Links

Crossrefs

a(n) = sqrt((A078369(n+1)^2 - 4)/357), n>=0, (Pell equation d=357, +4).
Cf. A077428, A078355 (Pell +4 equations).

Programs

Formula

a(n) = 19*a(n-1)-a(n-2), n >= 1; a(-1)=0, a(0)=1.
a(n) = (ap^(n+1)-am^(n+1))/(ap-am) with ap = (19+sqrt(357))/2 and am = (19-sqrt(357))/2.
a(n) = S(2*n+1, sqrt(21))/sqrt(21) = S(n, 19); S(n, x) := U(n, x/2), Chebyshev polynomials of the 2nd kind, A049310.
G.f.: 1/(1-19*x+x^2).
a(n) = Sum_{k=0..n} A101950(n,k)*18^k. - Philippe Deléham, Feb 10 2012
Product {n >= 0} (1 + 1/a(n)) = 1/17*(17 + sqrt(357)). - Peter Bala, Dec 23 2012
Product {n >= 1} (1 - 1/a(n)) = 1/38*(17 + sqrt(357)). - Peter Bala, Dec 23 2012

A092499 Chebyshev polynomials S(n-1,21) with Diophantine property.

Original entry on oeis.org

0, 1, 21, 440, 9219, 193159, 4047120, 84796361, 1776676461, 37225409320, 779956919259, 16341869895119, 342399310878240, 7174043658547921, 150312517518628101, 3149388824232642200, 65986852791366858099
Offset: 0

Views

Author

Rainer Rosenthal, Apr 05 2004

Keywords

Comments

Sequence R_21: Starts with 0,1,21 and satisfies A*C=B^2-1 for successive A,B,C.
The natural numbers a(n)=n satisfy the recurrence a(n-1)*a(n+1)=a(n)^2-1. Let R_r denote the sequence starting with 0,1,r and with this recurrence. We see that R_2 = "the natural numbers" and we find R_3 = A001906. These R_r form a "family" of sequences, which coincides with the m-family (r=m-2, n -> n+1) provided by Wolfdieter Lang (see A078368). This sequence R_21 is strongly related to A041833, which gives the denominators in the continued fraction of sqrt(437).
All positive integer solutions of Pell equation b(n)^2 - 437*a(n)^2 = +4 together with b(n)=A097777(n), n>=0.
For n>=2, a(n) equals the permanent of the (n-1)X(n-1) tridiagonal matrix with 21's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011
For n>=1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,20}. - Milan Janjic, Jan 25 2015

Examples

			a(3)=440 because a(1)*440 = a(2)^2-1.

Links

Crossrefs

Cf. R_3=A001906, R_4=A001353, R_5=A004254, R_6=A001109, R_7=A004187, R_8=A001090, R_9=A018913, R_10=A004189, R_11=A004190, R_12=A004191, R_13=A078362, R_14=A007655, R_15=A078364, R_16=A077412, R_17=A078366, R_18=A049660, R_19=A078368, R_20=A075843, R_21=this, sequence, R_22=A077421. See also A041219 and A041917.

Programs

  • Mathematica
    LinearRecurrence[{21,-1},{0,1},30] (* Harvey P. Dale, Apr 23 2015 *)
  • Sage
    [lucas_number1(n,21,1) for n in range(0,20)] # Zerinvary Lajos, Jun 25 2008

Formula

a(0)=0, a(1)=1, a(2)=21 and a(n-1)*a(n+1) = a(n)^2-1
a(n) = S(n-1, 21)=U(n-1, 21/2) with S(n, x)=U(n, x/2) Chebyshev's polynomials of the 2nd kind, A049310. S(-1, x)= 0 = U(-1, x).
a(n) = S(2*n-1, sqrt(23))/sqrt(23), n>=1.
a(n) = 21*a(n-1)-a(n-2), n >= 1; a(0)=0, a(1)=1.
a(n) = (ap^n-am^n)/(ap-am) with ap := (21+sqrt(437))/2 and am := (21-sqrt(437))/2.
G.f.: x/(1-21*x+x^2).
a(n+1) = Sum_{k, 0<=k<=n} A101950(n,k)*20^k. - Philippe Deléham, Feb 10 2012
Product {n >= 1} (1 + 1/a(n)) = 1/19*(19 + sqrt(437)). - Peter Bala, Dec 23 2012
Product {n >= 2} (1 - 1/a(n)) = 1/42*(19 + sqrt(437)). - Peter Bala, Dec 23 2012

Extensions

Extension, Chebyshev and Pell comments from Wolfdieter Lang, Aug 31 2004
Corrected by T. D. Noe, Nov 07 2006
Showing 1-8 of 8 results.

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