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 1-10 of 13 results. Next

A115066 Chebyshev polynomial of the first kind T(n,x), evaluated at x=n.

Original entry on oeis.org

1, 1, 7, 99, 1921, 47525, 1431431, 50843527, 2081028097, 96450076809, 4993116004999, 285573847759211, 17882714781360001, 1216895030905226413, 89415396036432386311, 7055673735003659189775, 595077930963909484707841, 53421565080956452077519377
Offset: 0

Views

Author

Roger L. Bagula, Mar 01 2006

Keywords

Examples

			a(3) = 99 because T[3, x] = 4x^3 - 3x and T[3, 3] = 4*3^3 - 3*3 = 99.

References

  • G. Freud, Orthogonal Polynomials, Pergamon Press, Oxford, 1966, p. 35.
  • M. Rosenblum and J. Rovnyak, Hardy Classes and Operator Theory, Dover, New York, 1985, page 18.
  • G. Szego, Orthogonal polynomials, Amer. Math. Soc., Providence, 1939, p. 29.

Links

Crossrefs

Main diagonal of A322836.
Cf. A053120, A107995, A173129, A323118, A349070, A349071, A349072.

Programs

  • Maple
    with(orthopoly): seq(T(n,n),n=0..17);
  • Mathematica
    Table[ChebyshevT[n, n], {n, 0, 17}] (* Arkadiusz Wesolowski, Nov 17 2012 *)
  • PARI
    A115066(n)=cos(n*acos(n))  \\ M. F. Hasler, Apr 06 2012
  • PARI
    a(n) = polchebyshev(n, 1, n); \\ Seiichi Manyama, Dec 28 2018
  • PARI
    a(n) = if(n==0, 1, n*sum(k=0, n, (2*n-2)^k*binomial(n+k, 2*k)/(n+k))); \\ Seiichi Manyama, Mar 05 2021

Formula

a(n) = cos(n*arccos(n)).
a(n) ~ 2^(n-1) * n^n. - Vaclav Kotesovec, Jan 19 2019
a(n) = (A323118(n) - A107995(n-2))/2 for n > 1. - Seiichi Manyama, Mar 05 2021
a(n) = n * Sum_{k=0..n} (2*n-2)^k * binomial(n+k,2*k)/(n+k) for n > 0. - Seiichi Manyama, Mar 05 2021
It appears that a(2*n+1) == 0 (mod (2*n+1)^2) and 2*a(4*n+2) == -2 (mod (4*n+2)^4), while for k > 1, 2*a(2^k*(2*n+1)) == 2 (mod (2^k*(2*n+1))^4). - Peter Bala, Feb 01 2022

Extensions

Edited by N. J. A. Sloane, Apr 05 2006

A323182 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Chebyshev polynomial of the second kind U_{n}(x), evaluated at x=k.

Original entry on oeis.org

1, 1, 0, 1, 2, -1, 1, 4, 3, 0, 1, 6, 15, 4, 1, 1, 8, 35, 56, 5, 0, 1, 10, 63, 204, 209, 6, -1, 1, 12, 99, 496, 1189, 780, 7, 0, 1, 14, 143, 980, 3905, 6930, 2911, 8, 1, 1, 16, 195, 1704, 9701, 30744, 40391, 10864, 9, 0, 1, 18, 255, 2716, 20305, 96030, 242047, 235416, 40545, 10, -1
Offset: 0

Views

Author

Seiichi Manyama, Jan 06 2019

Keywords

Examples

			Square array begins:
   1, 1,    1,     1,      1,      1,       1, ...
   0, 2,    4,     6,      8,     10,      12, ...
  -1, 3,   15,    35,     63,     99,     143, ...
   0, 4,   56,   204,    496,    980,    1704, ...
   1, 5,  209,  1189,   3905,   9701,   20305, ...
   0, 6,  780,  6930,  30744,  96030,  241956, ...
  -1, 7, 2911, 40391, 242047, 950599, 2883167, ...

Links

Crossrefs

Mirror of A228161.
Columns 0-19 give A056594, A000027(n+1), A001353(n+1), A001109(n+1), A001090(n+1), A004189(n+1), A004191, A007655(n+2), A077412, A049660(n+1), A075843(n+1), A077421, A077423, A097309, A097311, A097313, A029548, A029547, A144128(n+1), A078987.
Rows 0-10 give A000012, A005843, A000466, A144138, A144139, A242850, A242851, A242852, A242853, A242854, A243130.
Main diagonal gives A323118.
Cf. A179943, A322836 (Chebyshev polynomial of the first kind).

Programs

  • PARI
    T(n,k)  = polchebyshev(n, 2, k);
matrix(7, 7, n, k, T(n-1,k-1)) \\ Michel Marcus, Jan 07 2019
  • PARI
    T(n, k) = sum(j=0, n, (2*k-2)^j*binomial(n+1+j, 2*j+1)); \\ Seiichi Manyama, Mar 03 2021

Formula

T(0,k) = 1, T(1,k) = 2 * k and T(n,k) = 2 * k * T(n-1,k) - T(n-2,k) for n > 1.
T(n, k) = Sum_{j=0..n} (2*k-2)^j * binomial(n+1+j,2*j+1). - Seiichi Manyama, Mar 03 2021

A097690 Numerators of the continued fraction n-1/(n-1/...) [n times].

Original entry on oeis.org

1, 3, 21, 209, 2640, 40391, 726103, 15003009, 350382231, 9127651499, 262424759520, 8254109243953, 281944946167261, 10393834843080975, 411313439034311505, 17391182043967249409, 782469083251377707328
Offset: 1

Views

Author

T. D. Noe, Aug 19 2004

Keywords

Comments

The n-th term of the Lucas sequence U(n,1). The denominator is the (n-1)-th term. Adjacent terms of the sequence U(n,1) are relatively prime.

Examples

			a(4) = 209 because 4-1/(4-1/(4-1/4)) = 209/56.

Links

Crossrefs

Cf. A084844, A084845, A097691 (denominators), A179943, A323118.

Programs

  • Mathematica
    Table[s=n; Do[s=n-1/s, {n-1}]; Numerator[s], {n, 20}]
Table[DifferenceRoot[Function[{y, m}, {y[1 + m] == n*y[m] - y[m - 1], y[0] == 1, y[1] == n}]][n], {n, 1, 20}] (* Benedict W. J. Irwin, Nov 05 2016 *)
  • PARI
    {a(n)=polcoeff(1/(1-n*x+x^2+x*O(x^n)), n)} \\ Paul D. Hanna, Dec 27 2012
  • PARI
    a(n) = polchebyshev(n, 2, n/2); \\ Seiichi Manyama, Mar 03 2021
  • PARI
    a(n) = sum(k=0, n, (n-2)^k*binomial(n+1+k, 2*k+1)); \\ Seiichi Manyama, Mar 03 2021
  • Sage
    [lucas_number1(n,n-1,1) for n in range(19)] # Zerinvary Lajos, Jun 25 2008

Formula

a(n) = [x^n] 1/(1 - n*x + x^2). - Paul D. Hanna, Dec 27 2012
a(n) = y(n,n), where y(m+1,n) = n*y(m,n) - y(m-1,n) with y(0,n)=1, y(1,n)=n. - Benedict W. J. Irwin, Nov 05 2016
From Seiichi Manyama, Mar 03 2021: (Start)
a(n) = U(n,n/2) where U(n,x) is a Chebyshev polynomial of the second kind.
a(n) = Sum_{k=0..n} (n-2)^(n-k) * binomial(2*n+1-k,k) = Sum_{k=0..n} (n-2)^k * binomial(n+1+k,2*k+1). (End)

A349073 a(n) = U(2*n, n), where U(n, x) is the Chebyshev polynomial of the second kind.

Original entry on oeis.org

1, 3, 209, 40391, 15003009, 9127651499, 8254109243953, 10393834843080975, 17391182043967249409, 37326390852372133364819, 99976027392046047055178001, 326887883645157139828711692503, 1281398359905415379814555044995201, 5932135472283024519893762690145006075
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 07 2021

Keywords

Links

Crossrefs

Cf. A173129, A323118, A349071, A349074, A349075.

Programs

  • Mathematica
    Table[ChebyshevU[2*n, n], {n, 0, 15}]
  • PARI
    a(n) = polchebyshev(2*n, 2, n); \\ Michel Marcus, Nov 07 2021
  • Python
    from sympy import chebyshevu
def A349073(n): return chebyshevu(n<<1,n) # Chai Wah Wu, Nov 08 2023

Formula

For n>1, a(n) = ((n + sqrt(n^2-1))^(2*n+1) - (n - sqrt(n^2-1))^(2*n+1)) / (2*sqrt(n^2-1)).
a(n) ~ 2^(2*n) * n^(2*n).

A342167 a(n) = U(n, (n+2)/2) where U(n, x) is a Chebyshev polynomial of the 2nd kind.

Original entry on oeis.org

1, 3, 15, 115, 1189, 15456, 242047, 4435929, 93149001, 2205405829, 58130412911, 1688353631328, 53577891882061, 1844491975179855, 68470281953483775, 2726406212682669391, 115921586524134874897, 5241862216131004082160, 251197634537351883217999
Offset: 0

Views

Author

Seiichi Manyama, Mar 03 2021

Keywords

Links

Crossrefs

Cf. A097690, A097691, A323118, A342168.

Programs

  • Mathematica
    Table[ChebyshevU[n, (n + 2)/2], {n, 0, 18}] (* Amiram Eldar, Apr 27 2021 *)
  • PARI
    a(n) = polchebyshev(n, 2, (n+2)/2);
  • PARI
    a(n) = sum(k=0, n, n^(n-k)*binomial(2*n+1-k, k));
  • PARI
    a(n) = sum(k=0, n, n^k*binomial(n+1+k, 2*k+1));

Formula

a(n) = Sum_{k=0..n} n^(n-k) * binomial(2*n+1-k,k) = Sum_{k=0..n} n^k * binomial(n+1+k,2*k+1).
a(n) ~ exp(2) * n^n. - Vaclav Kotesovec, May 06 2021

A342168 a(n) = U(n, (n+3)/2) where U(n, x) is a Chebyshev polynomial of the 2nd kind.

Original entry on oeis.org

1, 4, 24, 204, 2255, 30744, 499121, 9409960, 202176360, 4878316860, 130651068911, 3846719565780, 123517560398401, 4296240885694576, 160935647131239840, 6460088606857290384, 276655979838719058119, 12591439417867717440180, 606947064800948702246681
Offset: 0

Views

Author

Seiichi Manyama, Mar 03 2021

Keywords

Links

Crossrefs

Cf. A097690, A097691, A323118, A342167.

Programs

  • Mathematica
    Table[ChebyshevU[n, (n + 3)/2], {n, 0, 18}] (* Amiram Eldar, Apr 27 2021 *)
  • PARI
    a(n) = polchebyshev(n, 2, (n+3)/2);
  • PARI
    a(n) = sum(k=0, n, (n+1)^(n-k)*binomial(2*n+1-k, k));
  • PARI
    a(n) = sum(k=0, n, (n+1)^k*binomial(n+1+k, 2*k+1));

Formula

a(n) = Sum_{k=0..n} (n+1)^(n-k) * binomial(2*n+1-k,k) = Sum_{k=0..n} (n+1)^k * binomial(n+1+k,2*k+1).
a(n) ~ exp(3) * n^n. - Vaclav Kotesovec, May 06 2021

A349074 a(n) = U(3*n, n), where U(n, x) is the Chebyshev polynomial of the second kind.

Original entry on oeis.org

1, 4, 2911, 7997214, 57641556673, 867583274883920, 23630375698358890319, 1056918444955456528983706, 72383076947075470731692782081, 7200266529428094485775774835670652, 998383804974887102441600687728515247999, 186701261436825568741051032736345268517903734
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 07 2021

Keywords

Comments

In general, for k>=1, U(k*n, n) ~ 2^(k*n) * n^(k*n).

Links

Crossrefs

Cf. A323118, A349070, A349073, A349076.

Programs

  • Mathematica
    Table[ChebyshevU[3*n, n], {n, 0, 13}]
  • PARI
    a(n) = polchebyshev(3*n, 2, n); \\ Michel Marcus, Nov 07 2021
  • Python
    from sympy import chebyshevu
def A349074(n): return chebyshevu(3*n,n) # Chai Wah Wu, Nov 08 2023

Formula

For n>1, a(n) = ((n + sqrt(n^2-1))^(3*n+1) - (n - sqrt(n^2-1))^(3*n+1)) / (2*sqrt(n^2-1)).
a(n) ~ 2^(3*n) * n^(3*n).

A349076 a(n) = U(n, 3*n), where U(n, x) is the Chebyshev polynomial of the second kind.

Original entry on oeis.org

1, 6, 143, 5796, 330049, 24192090, 2168392031, 229755926568, 28093745899009, 3893604149949966, 603151411514453999, 103272803655639197580, 19367259480582106560193, 3947962681769909551857186, 869179946261864224288867775, 205535515565731164929726435280
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 07 2021

Keywords

Comments

In general, for k>=1, U(n, k*n) ~ 2^n * k^n * n^n.

Links

Crossrefs

Cf. A323118, A349071, A349072, A349074, A349075.

Programs

  • Mathematica
    Table[ChebyshevU[n, 3*n], {n, 0, 20}]
  • PARI
    a(n) = polchebyshev(n, 2, 3*n); \\ Michel Marcus, Nov 07 2021

Formula

a(n) = ((3*n + sqrt(9*n^2-1))^(n+1) - (3*n - sqrt(9*n^2-1))^(n+1)) / (2*sqrt(9*n^2-1)).
a(n) ~ 6^n * n^n.

A107995 Chebyshev polynomial of the second kind U[n,x] evaluated at x=n+2.

Original entry on oeis.org

1, 6, 63, 980, 20305, 526890, 16451071, 600940872, 25154396001, 1187422368110, 62418042417599, 3616337930622300, 228977061309711793, 15731733543660288210, 1165677769357309014015, 92665403695822344828176
Offset: 0

Views

Author

Roger L. Bagula, Mar 01 2006

Keywords

Examples

			a(3)=980 because U[3,x]=8x^3-4x and U[3,5]=8*5^3-4*5=980.

References

  • Rosenblum and Rovnyak, Hardy Classes and Operator Theory,Dover, New York,1985, page 18.
  • G. Szego, Orthogonal polynomials, Amer. Math. Soc., Providence, 1939, p. 29.
  • G. Freud, Orthogonal Polynomials, Pergamon Press, Oxford, 1966, p. 35.

Links

Crossrefs

Cf. A097690, A323118, A342167.

Programs

  • Maple
    with(orthopoly): seq(U(n,n+2),n=0..17);
  • Mathematica
    Table[ChebyshevU[n, n + 2], {n, 0, 15}] (* Amiram Eldar, Mar 05 2021 *)
  • PARI
    a(n) = polchebyshev(n, 2, n+2); \\ Seiichi Manyama, Mar 05 2021
  • PARI
    a(n) = sum(k=0, n, (2*n+2)^k*binomial(n+1+k, 2*k+1)); \\ Seiichi Manyama, Mar 05 2021

Formula

a(n) = Sum_{k=0..n} (2*n+2)^(n-k) * binomial(2*n+1-k,k) = Sum_{k=0..n} (2*n+2)^k * binomial(n+1+k,2*k+1). - Seiichi Manyama, Mar 05 2021
a(n) ~ exp(2) * 2^n * n^n. - Vaclav Kotesovec, Mar 05 2021

Extensions

Edited by N. J. A. Sloane, Apr 05 2006

A323117 a(n) = T_{n}(n-1) where T_{n}(x) is a Chebyshev polynomial of the first kind.

Original entry on oeis.org

1, 0, 1, 26, 577, 15124, 470449, 17057046, 708158977, 33165873224, 1730726404001, 99612037019890, 6269617090376641, 428438743526336412, 31592397706723526737, 2500433598371461203374, 211434761022028192051201, 19023879409608991280267536
Offset: 0

Views

Author

Seiichi Manyama, Jan 05 2019

Keywords

Links

Crossrefs

Cf. A115066, A322836, A323118.

Programs

  • Mathematica
    Table[ChebyshevT[n, n - 1], {n, 0, 20}] (* Vaclav Kotesovec, Jan 05 2019 *)
  • PARI
    a(n) = polchebyshev(n, 1, n-1);
  • PARI
    a(n) = round(cos(n*acos(n-1))); \\ Seiichi Manyama, Mar 05 2021
  • PARI
    a(n) = if(n==0, 1, n*sum(k=0, n, (2*n-4)^k*binomial(n+k, 2*k)/(n+k))); \\ Seiichi Manyama, Mar 05 2021

Formula

a(n)^2 - ((n - 1)^2 - 1) * A323118(n-1)^2 = 1 for n > 0.
a(n) = A322836(n,n-1) for n > 0.
a(n) ~ exp(-1) * 2^(n-1) * n^n. - Vaclav Kotesovec, Jan 05 2019
a(n) = cos(n*arccos(n-1)). - Seiichi Manyama, Mar 05 2021
a(n) = n * Sum_{k=0..n} (2*n-4)^k * binomial(n+k,2*k)/(n+k) for n > 0. - Seiichi Manyama, Mar 05 2021
Showing 1-10 of 13 results. Next

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