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 21 results. Next

A090248 a(n) = 27a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 27.

Original entry on oeis.org

2, 27, 727, 19602, 528527, 14250627, 384238402, 10360186227, 279340789727, 7531841136402, 203080369893127, 5475638145978027, 147639149571513602, 3980781400284889227, 107333458658120495527, 2894022602368968490002, 78031276805304028734527
Offset: 0

Views

Author

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 24 2004

Keywords

Comments

a(n+1)/a(n) converges to ((27+sqrt(725))/2) = 26.96291201...
Lim a(n)/a(n+1) as n approaches infinity = 0.03708798... = 2/(27+sqrt(725)) = (27-sqrt(725))/2.
Lim a(n+1)/a(n) as n approaches infinity = 26.96291201... = (27+sqrt(725))/2 = 2/(27-sqrt(725)).
Lim a(n)/a(n+1) = 27 - Lim a(n+1)/a(n).
A Chebyshev T-sequence with Diophantine property.
a(n) gives the general (nonnegative integer) solution of the Pell equation a^2 - 29*(5*b)^2 =+4 with companion sequence b(n)=A097781(n-1), n>=0.

Examples

			a(4) = 528527 = 27a(3) - a(2) = 27*19602 - 727 = ((27+sqrt(725))/2)^4 + ((27-sqrt(725))/2)^4 = 528526.999998107 + 0.000001892 = 528527.
(x;y) = (2;0), (27;1), (727;27), (19602;728), ... give the nonnegative integer solutions to x^2 - 29*(5*y)^2 = +4.

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. A046213, A078046.
a(n)=sqrt(4 + 29*(5*A097781(n-1))^2), n>=1.
Cf. A077428, A078355 (Pell +4 equations).
Cf. A090733 for 2*T(n, 25/2).
Cf. A087130.

Programs

  • Mathematica
    a[0] = 2; a[1] = 27; a[n_] := 27a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *)
RecurrenceTable[{a[0]==2,a[1]==27,a[n]==27a[n-1]-a[n-2]},a,{n,20}] (* or *) LinearRecurrence[{27,-1},{2,27},20] (* Harvey P. Dale, Jan 03 2018 *)
  • PARI
    {a(n) = (-1)^n * subst(2 * poltchebi(2*n), 'x, -5/2 * I)}; /* Michael Somos, Nov 04 2008 */
  • Python
    def aupton(idx):
  alst = [2, 27]
  for n in range(2, idx+1): alst.append(27*alst[-1] - alst[-2])
  return alst
print(aupton(16)) # Michael S. Branicky, Feb 27 2021
  • Sage
    [lucas_number2(n,27,1) for n in range(0,16)] # Zerinvary Lajos, Jun 27 2008

Formula

a(n) = 27a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 27. a(n) = ((27+sqrt(725))/2)^n + ((27-sqrt(725))/2)^n, (a(n))^2 = a(2n)+2.
a(n) = S(n, 27) - S(n-2, 27) = 2*T(n, 27/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 27)=A097781(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, kind. See A049310 and A053120.
a(n) = ap^n + am^n, with ap := (27+5*sqrt(29))/2 and am := (27-5*sqrt(29))/2.
G.f.: (2-27*x)/(1-27*x+x^2).
a(-n) = a(n). - Michael Somos, Nov 01 2008
A087130(2*n) = a(n). - Michael Somos, Nov 01 2008

Extensions

More terms from Robert G. Wilson v, Jan 30 2004
Chebyshev and Pell comments from Wolfdieter Lang, Aug 31 2004

A046224 Distinct numbers seen when writing first numerator and then denominator of central elements of 1/2-Pascal triangle.

Original entry on oeis.org

1, 2, 3, 11, 40, 147, 546, 2046, 7722, 29315, 111826, 428298, 1646008, 6344366, 24515700, 94942620, 368404110, 1431985635, 5574725970, 21732560850, 84828633120, 331488081210, 1296712152060, 5077282282020, 19897457591700, 78039200913102, 306302623291476
Offset: 1

Views

Author

Mohammad K. Azarian

Keywords

Examples

			1/1; <-- hence 1;
1/1 1/1;
1/1 1/2 1/1; <-- hence 2
1/1 3/2 3/2 1/1;
1/1 5/2 3/1 5/2 1/1; <-- hence 3
1/1 7/2 11/2 11/2 7/2 1/1;
1/1 9/2 9/1 11/1 9/1 9/2 1/1; <-- hence 11
1/1 11/2 27/2 20/1 20/1 27/2 11/2 1/1;
...

Links

Crossrefs

Cf. A046213.

Programs

  • Magma
    [1,2] cat [(5*n-9)/(8*n-12)*Binomial(2*n-2,n-1): n in [3..40]]; // Vincenzo Librandi, Sep 24 2015
  • Mathematica
    Join[{1, 2}, Table[(5 n - 9)/(8 n - 12) Binomial[2 n - 2, n - 1], {n, 3, 40}]] (* Vincenzo Librandi, Sep 24 2015 *)
  • PARI
    a(n) = if (n<3, n, (5*n-9)/(8*n-12)*binomial(2*n-2,n-1));
vector(40, n, a(n)) \\ Altug Alkan, Oct 01 2015

Formula

a(n) = Sum_{k=1..n-2} (2*k+1)*binomial(2*n-k-5,n-3), n>2; a(1)=1, a(2)=2. - Vladimir Kruchinin, Sep 27 2011
a(n) = (5*n-9)/(8*n-12)*binomial(2*n-2,n-1), n>2; a(1)=1, a(2)=2. - Eric Werley, Sep 16 2015
G.f.: (3/2)*x^2 + (2*x - 3*x^2)/(2*sqrt(1-4*x)). - G. C. Greubel, Sep 24 2015

Extensions

More terms from James Sellers, Dec 13 1999
a(26)-a(27) from Vincenzo Librandi, Sep 24 2015

A046222 First numerator and then denominator of central elements of 1/2-Pascal triangle.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 11, 1, 40, 1, 147, 1, 546, 1, 2046, 1, 7722, 1, 29315, 1, 111826, 1, 428298, 1, 1646008, 1, 6344366, 1, 24515700, 1, 94942620, 1, 368404110, 1, 1431985635, 1, 5574725970, 1, 21732560850, 1, 84828633120, 1, 331488081210, 1
Offset: 1

Views

Author

Mohammad K. Azarian

Keywords

Examples

			1/1; 1/1 1/1; 1/1 1/2 1/1; 1/1 3/2 3/2 1/1; 1/1 5/2 3/1 5/2 1/1; 1/1 7/2 11/2 11/2 7/2 1/1; 1/1 9/2 9/1 11/1 9/1 9/2 1/1; 1/1 11/2 27/2 20/1 20/1 27/2 11/2 1/1; ...

Crossrefs

Cf. A046213.

Extensions

More terms from James Sellers, Dec 13 1999

A046223 First denominator and then numerator of central elements of 1/2-Pascal triangle.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 11, 1, 40, 1, 147, 1, 546, 1, 2046, 1, 7722, 1, 29315, 1, 111826, 1, 428298, 1, 1646008, 1, 6344366, 1, 24515700, 1, 94942620, 1, 368404110, 1, 1431985635, 1, 5574725970, 1, 21732560850, 1, 84828633120, 1, 331488081210, 1
Offset: 1

Views

Author

Mohammad K. Azarian

Keywords

Comments

a(n) = A046222(n-1) for n > 5. - Georg Fischer, Oct 17 2018

Examples

			1/1; 1/1 1/1; 1/1 1/2 1/1; 1/1 3/2 3/2 1/1; 1/1 5/2 3/1 5/2 1/1; 1/1 7/2 11/2 11/2 7/2 1/1; 1/1 9/2 9/1 11/1 9/1 9/2 1/1; 1/1 11/2 27/2 20/1 20/1 27/2 11/2 1/1; ...

Links

Crossrefs

Cf. A046213, A046222.

Programs

  • Mathematica
    Map[{Denominator@ #, Numerator@ #} &@ #[[Ceiling[Length[#]/2] ]] &, Select[Nest[Append[#, Join[{#[[-1, 1]]}, Total /@ Partition[#[[-1]], 2, 1], {#[[-1, -1]]}]] &, {{1}, {1, 1}, {1, 1/2, 1}}, 2 (20) + 1], OddQ@ Length@ # & ]] // Flatten (* or *)
With[{r = Sqrt[1 - 4 x]}, {1, 1, 2, 1}~Join~Riffle[ConstantArray[1, Length@ #], #] &@ CoefficientList[Series[(2 - 2 r - 3 x - r x)/(2 r x^2), {x, 0, 19}], x]] (* Michael De Vlieger, Oct 17 2018 *)

Extensions

More terms from James Sellers, Dec 13 1999

A046226 First denominator and then numerator of elements to right of central elements of 1/2-Pascal triangle.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 1, 2, 5, 1, 1, 2, 11, 2, 7, 1, 1, 1, 9, 2, 9, 1, 1, 1, 20, 2, 27, 2, 11, 1, 1, 2, 67, 1, 19, 2, 13, 1, 1, 2, 147, 2, 105, 2, 51, 2, 15, 1, 1, 1, 126, 1, 78, 1, 33, 2, 17, 1, 1, 1, 273, 1, 204, 1, 111, 2, 83, 2, 19, 1, 1, 1, 477, 1, 315, 2
Offset: 1

Views

Author

Mohammad K. Azarian

Keywords

Examples

			1/1; -->
1/1  1/1; --> 1 1
1/1  1/2  1/1; --> 1 1
1/1  3/2  3/2  1/1; --> 2 3
1/1  5/2  3/1  5/2  1/1; --> ...
1/1  7/2 11/2 11/2  7/2  1/1;
1/1  9/2  9/1 11/1  9/1  9/2  1/1;
1/1 11/2 27/2 20/1 20/1 27/2 11/2 1/1; ...

Crossrefs

Cf. A046213.

Extensions

More terms from Sean A. Irvine, Apr 07 2021

A046214 First denominator and then numerator of 1/2-Pascal triangle (by row).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 3, 2, 3, 1, 1, 1, 1, 2, 5, 1, 3, 2, 5, 1, 1, 1, 1, 2, 7, 2, 11, 2, 11, 2, 7, 1, 1, 1, 1, 2, 9, 1, 9, 1, 11, 1, 9, 2, 9, 1, 1, 1, 1, 2, 11, 2, 27, 1, 20, 1, 20, 2, 27, 2, 11, 1, 1, 1, 1, 2, 13, 1, 19, 2, 67, 1, 40, 2, 67, 1, 19, 2, 13, 1, 1, 1, 1, 2, 15, 2
Offset: 1

Views

Author

Mohammad K. Azarian

Keywords

Examples

			1/1; --> 1 1
1/1 1/1; --> 1 1 1 1
1/1 1/2 1/1; --> 1 1 2 1 1 1
1/1 3/2 3/2 1/1; --> ...
1/1 5/2 3/1 5/2 1/1;
1/1 7/2 11/2 11/2 7/2 1/1;
1/1 9/2 9/1 11/1 9/1 9/2 1/1;
1/1 11/2 27/2 20/1 20/1 27/2 11/2 1/1;...

Crossrefs

Cf. A046213.

A046215 First numerator and then denominator of 1/2-Pascal triangle (by row), excluding 1's.

Original entry on oeis.org

2, 3, 2, 3, 2, 5, 2, 3, 5, 2, 7, 2, 11, 2, 11, 2, 7, 2, 9, 2, 9, 11, 9, 9, 2, 11, 2, 27, 2, 20, 20, 27, 2, 11, 2, 13, 2, 19, 67, 2, 40, 67, 2, 19, 13, 2, 15, 2, 51, 2, 105, 2, 147, 2, 147, 2, 105, 2, 51, 2, 15, 2, 17, 2, 33, 78, 126, 147, 126, 78, 33, 17, 2, 19, 2, 83, 2, 111, 204
Offset: 1

Views

Author

Mohammad K. Azarian

Keywords

Examples

			1/1; -->
1/1 1/1; -->
1/1 1/2 1/1; --> 2
1/1 3/2 3/2 1/1; --> 3 2 3 2
1/1 5/2 3/1 5/2 1/1;  --> ...
1/1 7/2 11/2 11/2 7/2 1/1;
1/1 9/2 9/1 11/1 9/1 9/2 1/1;
1/1 11/2 27/2 20/1 20/1 27/2 11/2 1/1; ...

Crossrefs

Cf. A046213.

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Apr 06 2000

A046216 First denominator and then numerator of 1/2-Pascal triangle (by row), excluding 1's.

Original entry on oeis.org

2, 2, 3, 2, 3, 2, 5, 3, 2, 5, 2, 7, 2, 11, 2, 11, 2, 7, 2, 9, 9, 11, 9, 2, 9, 2, 11, 2, 27, 20, 20, 2, 27, 2, 11, 2, 13, 19, 2, 67, 40, 2, 67, 19, 2, 13, 2, 15, 2, 51, 2, 105, 2, 147, 2, 147, 2, 105, 2, 51, 2, 15, 2, 17, 33, 78, 126, 147, 126, 78, 33, 2, 17, 2
Offset: 1

Views

Author

Mohammad K. Azarian

Keywords

Examples

			1/1; -->
1/1 1/1; -->
1/1 1/2 1/1; --> 2
1/1 3/2 3/2 1/1; --> 2 3 2 3
1/1 5/2 3/1 5/2 1/1;  --> ...
1/1 7/2 11/2 11/2 7/2 1/1;
1/1 9/2 9/1 11/1 9/1 9/2 1/1;
1/1 11/2 27/2 20/1 20/1 27/2 11/2 1/1; ...

Crossrefs

Cf. A046213.

Extensions

More terms from Sean A. Irvine, Apr 07 2021

A046217 First numerator and then denominator of 1/2-Pascal triangle (by row) excluding 1's and 2's.

Original entry on oeis.org

3, 3, 5, 3, 5, 7, 11, 11, 7, 9, 9, 11, 9, 9, 11, 27, 20, 20, 27, 11, 13, 19, 67, 40, 67, 19, 13, 15, 51, 105, 147, 147, 105, 51, 15, 17, 33, 78, 126, 147, 126, 78, 33, 17, 19, 83, 111, 204, 273, 273, 204, 111, 83, 19, 21, 51, 305, 315, 477, 546, 477, 315, 305, 51, 21, 23
Offset: 1

Views

Author

Mohammad K. Azarian

Keywords

Examples

			1/1; 1/1 1/1; 1/1 1/2 1/1; 1/1 3/2 3/2 1/1; 1/1 5/2 3/1 5/2 1/1; 1/1 7/2 11/2 11/2 7/2 1/1; 1/1 9/2 9/1 11/1 9/1 9/2 1/1; 1/1 11/2 27/2 20/1 20/1 27/2 11/2 1/1; ...

Crossrefs

Cf. A046213.

Extensions

More terms from James Sellers, Dec 13 1999

A046218 Numerators of elements of 1/2-Pascal triangle (by row).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 5, 3, 5, 1, 1, 7, 11, 11, 7, 1, 1, 9, 9, 11, 9, 9, 1, 1, 11, 27, 20, 20, 27, 11, 1, 1, 13, 19, 67, 40, 67, 19, 13, 1, 1, 15, 51, 105, 147, 147, 105, 51, 15, 1, 1, 17, 33, 78, 126, 147, 126, 78, 33, 17, 1, 1, 19, 83, 111, 204
Offset: 1

Views

Author

Mohammad K. Azarian

Keywords

Examples

			1/1; --> 1
1/1  1/1; --> 1 1
1/1  1/2  1/1; --> 1 1 1
1/1  3/2  3/2  1/1; --> 1 3 3 1
1/1  5/2  3/1  5/2  1/1; --> ...
1/1  7/2 11/2 11/2  7/2  1/1;
1/1  9/2  9/1 11/1  9/1  9/2  1/1;
1/1 11/2 27/2 20/1 20/1 27/2 11/2 1/1;
...

Crossrefs

Cf. A046213.

Extensions

More terms from Sean A. Irvine, Apr 07 2021
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