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.

A370259 a(n) = (T(n,n+1) - 1)/n^3 for n >= 1, where T(n,x) is the n-th Chebyshev polynomial of the first kind.

Original entry on oeis.org

1, 2, 9, 75, 961, 16900, 380689, 10498005, 343323841, 13007560326, 560789801881, 27125634729375, 1455389462287489, 85805768251305992, 5515372218107327521, 383931652351786775721, 28778117694539885440129, 2311202255914842794592010, 198009919900727928789497641, 18027589454633803742596931571
Offset: 1

Views

Author

Peter Bala, Mar 11 2024

Keywords

Comments

It appears that a(2*n+1) is always a square, while a(2*n) = (n + 1) * a square. See A370260 and A370261.

Links

Crossrefs

Cf. A008310, A342205, A370260, A370261.

Programs

  • Maple
    seq( simplify( (ChebyshevT(n, n+1) - 1)/n^3 ), n = 1..20);
  • Mathematica
    Array[(ChebyshevT[#, #+1]-1)/#^3 &, 20] (* Paolo Xausa, Mar 14 2024 *)
  • Python
    from sympy import chebyshevt
def A370259(n): return (chebyshevt(n,n+1)-1)//n**3 # Chai Wah Wu, Mar 13 2024

Formula

a(n) = Sum_{k = 1..n} (2^k)*n^(k-2)*binomial(n+k, 2*k)/(n + k) (shows that a(n) is an integer).
a(n) = (cos(n*arccos(n+1)) - 1)/n^3.
a(n) = (A342205(n) - 1)/n^3.
a(n) = ( (n + 1 + sqrt(n*(n+2)))^n + (n + 1 - sqrt(n*(n+2)))^n - 2 )/(2*n^3).

A370260 a(n) = sqrt(A370259(2*n+1)).

Original entry on oeis.org

1, 3, 31, 617, 18529, 748859, 38149567, 2348482961, 169641143873, 14071599763379, 1318414335714015, 137720427724123513, 15871136311527376801, 2000355821099358166891, 273735526097742996298111, 40419227378551955037029921, 6405616571975691389276400257
Offset: 0

Views

Author

Peter Bala, Mar 11 2024

Keywords

Comments

The sequence is conjectured to be integral [added 03 Mar 2024: now confirmed - see the Formula section].

Links

Crossrefs

Cf. A008310, A342205, A370259, A370261, A370262.

Programs

  • Maple
    A370259 := n -> simplify( (ChebyshevT(n, n+1) - 1)/n^3 ):
seq(sqrt(A370259(2*n+1)), n = 0..20);
  • Mathematica
    Table[Sqrt[(ChebyshevT[k, k + 1] - 1)/k^3], {k, 1, 40, 2}] (* Paolo Xausa, Jul 24 2024 *)

Formula

a(n) = sqrt( (T(2*n+1, 2*n+2) - 1)/(2*n+1)^3 ), where T(n, x) denotes the n-th Chebyshev polynomial of the first kind.
a(n) = sqrt( Sum_{k = 1..2*n+1} (2^k)*(2*n + 1)^(k-2)*binomial(2*n + k + 1, 2*k)/(2*n + k + 1) ).
a(n) = Sum_{k = 0..n} binomial(n+k, n-k)/(2*k + 1) * (4*n + 2)^k (shows the sequence to be integral) = R(n,2), where R(n, x) is the n-th row polynomial of A370262. - Peter Bala, Apr 03 2024

A343259 a(n) = 2 * T(n,n/2) where T(n,x) is a Chebyshev polynomial of the first kind.

Original entry on oeis.org

2, 1, 2, 18, 194, 2525, 39202, 710647, 14760962, 345946302, 9034502498, 260219353691, 8195978831042, 280256592535933, 10340256951198914, 409468947059131650, 17322711762013765634, 779742677038695037937, 37210469265847998489922, 1876572071974094803391179
Offset: 0

Views

Author

Seiichi Manyama, Apr 09 2021

Keywords

Links

Crossrefs

Main diagonal of A298675.
Cf. A097690, A115066, A342205, A343260, A343261.

Programs

  • Mathematica
    Table[2*ChebyshevT[n, n/2], {n, 1, 20}] (* Amiram Eldar, Apr 09 2021 *)
  • PARI
    a(n) = 2*polchebyshev(n, 1, n/2);
  • PARI
    a(n) = round(2*cos(n*acos(n/2)));
  • PARI
    a(n) = if(n==0, 2, 2*n*sum(k=0, n, (n-2)^k*binomial(n+k, 2*k)/(n+k)));

Formula

a(n) = 2 * cos(n*arccos(n/2)).
a(n) = 2 * n * Sum_{k=0..n} (n-2)^k * binomial(n+k,2*k)/(n+k) for n > 0.
a(n) ~ n^n. - Vaclav Kotesovec, Apr 09 2021

A370261 a(n) = sqrt(A370259(2*n)/(n+1)) for n >= 1.

Original entry on oeis.org

1, 5, 65, 1449, 46561, 1968525, 103565057, 6531391313, 480749649601, 40482981221781, 3840053099665729, 405275779792031225, 47113209228513626017, 5982545638922153790749, 823992221632687352744961, 122360935410018418223907489, 19489013519781051891806113153
Offset: 1

Views

Author

Peter Bala, Mar 11 2024

Keywords

Comments

The sequence is conjectured to be integral.

Links

Crossrefs

Cf. A008310, A342205, A370259, A370260.

Programs

  • Maple
    A370259 := n -> simplify( (ChebyshevT(n, n+1) - 1)/n^3 ):
seq(sqrt(A370259(2*n)/(n+1)), n = 1..20);
  • Mathematica
    Table[Sqrt[(ChebyshevT[2*n, 2*n + 1] - 1)/(2*n)^3/(n + 1)], {n, 20}] (* Paolo Xausa, Jul 24 2024 *)
  • Python
    from math import isqrt
from sympy import chebyshevt
def A370261(n): return isqrt((chebyshevt((m:=n<<1),m+1)-1)//((n+1)*m**3)) # Chai Wah Wu, Mar 13 2024

Formula

a(n) = sqrt( (T(2*n, 2*n+1) - 1)/((n+1)*(2*n)^3) ), where T(n, x) is the n-th Chebyshev polynomial of the first kind.

A342206 a(n) = T(n,n+2) where T(n,x) is a Chebyshev polynomial of the first kind.

Original entry on oeis.org

1, 3, 31, 485, 10081, 262087, 8193151, 299537289, 12545596801, 592479412811, 31154649926687, 1805486216133613, 114342125644787041, 7857107443850071695, 582268591681887560191, 46292552162781456490001, 3930448770533424343942657
Offset: 0

Views

Author

Seiichi Manyama, Mar 05 2021

Keywords

Links

Crossrefs

Cf. A107995, A115066, A323117, A342205.

Programs

  • Mathematica
    Table[ChebyshevT[n, n + 2], {n, 0, 16}] (* Amiram Eldar, Mar 05 2021 *)
  • PARI
    a(n) = polchebyshev(n, 1, n+2);
  • PARI
    a(n) = round(cos(n*acos(n+2)));
  • PARI
    a(n) = if(n==0, 1, n*sum(k=0, n, (2*n+2)^k*binomial(n+k, 2*k)/(n+k)));

Formula

a(n) = cos(n*arccos(n+2)).
a(n) = n * Sum_{k=0..n} (2*n+2)^k * binomial(n+k,2*k)/(n+k) for n > 0.
a(n) ~ exp(2) * 2^(n-1) * n^n. - Vaclav Kotesovec, Mar 12 2024

A343260 a(n) = 2 * T(n,(n+1)/2) where T(n,x) is a Chebyshev polynomial of the first kind.

Original entry on oeis.org

2, 2, 7, 52, 527, 6726, 103682, 1874888, 38925119, 912670090, 23855111399, 687808321212, 21687295069442, 742397047217294, 27420344506901023, 1086932029484351248, 46027034321342899967, 2073668380220713167378, 99042070146811639444802
Offset: 0

Views

Author

Seiichi Manyama, Apr 09 2021

Keywords

Links

Crossrefs

Cf. A097691, A115066, A342205, A343259, A343261.

Programs

  • Mathematica
    Table[2*ChebyshevT[n, (n+1)/2], {n, 0, 18}] (* Amiram Eldar, Apr 09 2021 *)
  • PARI
    a(n) = 2*polchebyshev(n, 1, (n+1)/2);
  • PARI
    a(n) = round(2*cos(n*acos((n+1)/2)));
  • PARI
    a(n) = if(n==0, 2, 2*n*sum(k=0, n, (n-1)^k*binomial(n+k, 2*k)/(n+k)));

Formula

a(n) = 2 * cos(n*arccos((n+1)/2)).
a(n) = 2 * n * Sum_{k=0..n} (n-1)^k * binomial(n+k,2*k)/(n+k) for n > 0.
a(n) ~ exp(1) * n^n. - Vaclav Kotesovec, Apr 09 2021
Conjecture: a(p^r) == 1 (mod p^(2*r)) for positive integer r and all primes p >= 5. - Peter Bala, Mar 11 2024

A342207 a(n) = U(n,n+1) where U(n,x) is a Chebyshev polynomial of the second kind.

Original entry on oeis.org

1, 4, 35, 496, 9701, 241956, 7338631, 262184896, 10783446409, 501827040100, 26069206375211, 1495427735314800, 93885489910449901, 6403169506981578436, 471427031236487965199, 37265225545829174607616, 3147895910861898495432209
Offset: 0

Views

Author

Seiichi Manyama, Mar 05 2021

Keywords

Links

Crossrefs

Cf. A107995, A323118, A342168, A342205.

Programs

  • Mathematica
    Table[ChebyshevU[n, n + 1], {n, 0, 16}] (* Amiram Eldar, Mar 05 2021 *)
  • PARI
    a(n) = polchebyshev(n, 2, n+1);
  • PARI
    a(n) = sum(k=0, n, (2*n)^(n-k)*binomial(2*n+1-k, k));
  • PARI
    a(n) = sum(k=0, n, (2*n)^k*binomial(n+1+k, 2*k+1));

Formula

a(n) = Sum_{k=0..n} (2*n)^(n-k) * binomial(2*n+1-k,k) = Sum_{k=0..n} (2*n)^k * binomial(n+1+k,2*k+1).
a(n) ~ exp(1) * 2^n * n^n. - Vaclav Kotesovec, Mar 05 2021
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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