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 1-4 of 4 results.

A209927 Decimal expansion of sqrt(3 + sqrt(3 + sqrt(3 + sqrt(3 + ... )))).

Original entry on oeis.org

2, 3, 0, 2, 7, 7, 5, 6, 3, 7, 7, 3, 1, 9, 9, 4, 6, 4, 6, 5, 5, 9, 6, 1, 0, 6, 3, 3, 7, 3, 5, 2, 4, 7, 9, 7, 3, 1, 2, 5, 6, 4, 8, 2, 8, 6, 9, 2, 2, 6, 2, 3, 1, 0, 6, 3, 5, 5, 2, 2, 6, 5, 2, 8, 1, 1, 3, 5, 8, 3, 4, 7, 4, 1, 4, 6, 5, 0, 5, 2, 2, 2, 6, 0, 2, 3, 0, 9, 5, 4, 1, 0, 0, 9, 2, 4, 5, 3, 5, 8, 8, 3, 6, 7, 5, 7
Offset: 1

Views

Author

Alonso del Arte, Mar 17 2012

Keywords

Comments

The number x given by the infinitely nested radical for n = 3 is such that x^2 = x + 3, bearing some similarity to the golden ratio phi with its property that phi^2 = phi + 1. Also, 3/x = x - 1.
The mentioned polynomial x^2 - x - 3 has the present number as positive root, and the negative one is -A223139. - Wolfdieter Lang, Aug 29 2022
It is the spectral radius of the bull-graph (see Seeger and Sossa, 2023). - Stefano Spezia, Sep 19 2023
c^n = A006130(n) + A006130(n-1) * d, where c = (1 + sqrt(13))/2 and d = (-1 + sqrt(13))/2. - Gary W. Adamson, Nov 25 2023
c^n = A052533(n) + A006130(n-1)*c, with A006130(-1) = 0. This is also valid for powers of 1/c = A356033, with A052533 and A006130 given there in terms of S-Chebyshev polynomials (A049310), used for negative indices. - Wolfdieter Lang, Nov 26 2023

Examples

			2.30277563773...

Links

Crossrefs

Cf. A157532 (continued fraction), A001622, A010470, A085550, A098316, A188943, A223139.
Cf. A006130, A049310, A052533, A209927.

Programs

  • Maple
    Digits:=140:
evalf((sqrt(13)+1)/2);  # Alois P. Heinz, Sep 19 2023
  • Mathematica
    RealDigits[(1 + Sqrt[13])/2, 10, 130][[1]]
RealDigits[ Fold[ Sqrt[#1 + #2] &, 0, Table[3, {n, 168}]], 10, 111][[1]] (* Robert G. Wilson v, Oct 02 2018 *)
  • PARI
    (sqrt(13)+1)/2 \\ Altug Alkan, Oct 03 2018

Formula

Closed form: (sqrt(13) + 1)/2 = A098316-1 = A085550+2 = 3*(A188943-1).

A143621 a(n) = (-1)^binomial(n,4): Periodic sequence 1,1,1,1,-1,-1,-1,-1,... .

Original entry on oeis.org

1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1
Offset: 0

Views

Author

Peter Bala, Aug 30 2008

Keywords

Comments

Periodic sequence with period 8. More generally, it appears that (-1)^binomial(n,2^r) gives a periodic sequence of period 2^(r+1), the period consisting of a block of 2^r plus ones followed by a block of 2^r minus ones. See A033999 (r = 0), A057077 (r = 1) and A143622 (r = 3).
Nonsimple continued fraction expansion of A188943 = 1.767591879243... - R. J. Mathar, Mar 08 2012

Examples

			G.f. = 1 + x + x^2 + x^3 - x^4 - x^5 - x^6 - x^7 + x^8 + x^9 + x^10 + ...

Links

Crossrefs

Cf. A033999, A057077, A130151, A143622.

Programs

  • Maple
    with(combinat):
a := n -> (-1)^binomial(n,4):
seq(a(n),n = 0..103);
  • Mathematica
    Table[(-1)^Binomial[n, 4], {n, 0, 100}] (* Wesley Ivan Hurt, May 20 2014 *)
a[ n_] := (-1)^Quotient[n, 4]; (* Michael Somos, May 05 2015 *)
PadRight[{},120,{1,1,1,1,-1,-1,-1,-1}] (* Harvey P. Dale, Nov 29 2024 *)
  • PARI
    {a(n) = (-1)^(n \ 4)}; /* Michael Somos, Sep 30 2011 */
  • PARI
    x='x+O('x^99); Vec((1-x^4)^2/((1-x)*(1-x^8))) \\ Altug Alkan, Apr 15 2016
  • Python
    def A143621(n): return -1 if n&4 else 1 # Chai Wah Wu, Jan 18 2023

Formula

a(n) = (-1)^binomial(n,4) = (-1)^floor(n/4), since Sum_{k = 1..n-3} k*(k+1)(k+2)/3! = binomial(n,4) == floor(n/4) (mod 2) for n = 0,1,...,7 by calculation and both sides increase by an even number if we substitute n+8 for n.
a(n) = (1/4)*((n+4) mod 8 - n mod 8).
O.g.f.: (1+x+x^2+x^3)/(1+x^4) = (1+x)*(1+x^2)/(1+x^4) = (1-x^4)/((1-x)*(1+x^4)).
Define E(k) = Sum_{n>=0} a(n)*n^k/n! for k = 0,1,2,... . Then E(k) is an integral linear combination of E(0), E(1), E(2) and E(3) (a Dobinski-type relation).
a(n) = (-1)^A180969(2,n), where the first index in A180969(.,.) is the row index. - Adriano Caroli, Nov 18 2010
Euler transform of length 8 sequence [ 1, 0, 0, -2, 0, 0, 0, 1]. - Michael Somos, Sep 30 2011
G.f.: (1 - x^4)^2 / ((1 - x) * (1 - x^8)). a(n) = -a(-1 - n) for all n in Z. - Michael Somos, Sep 30 2011
E.g.f.: sin(x/sqrt(2))*sinh(x/sqrt(2)) + (sqrt(2)*sin(x/sqrt(2)) + cos(x/sqrt(2)))*cosh(x/sqrt(2)). - Ilya Gutkovskiy, Apr 15 2016

A356033 Decimal expansion of (-1 + sqrt(13))/6 = A223139/3.

Original entry on oeis.org

4, 3, 4, 2, 5, 8, 5, 4, 5, 9, 1, 0, 6, 6, 4, 8, 8, 2, 1, 8, 6, 5, 3, 6, 8, 7, 7, 9, 1, 1, 7, 4, 9, 3, 2, 4, 3, 7, 5, 2, 1, 6, 0, 9, 5, 6, 4, 0, 8, 7, 4, 3, 6, 8, 7, 8, 5, 0, 7, 5, 5, 0, 9, 3, 7, 1, 1, 9, 4, 4, 9, 1, 3, 8, 2, 1, 6, 8
Offset: 0

Views

Author

Wolfdieter Lang, Aug 29 2022

Keywords

Comments

This constant r, an algebraic integer of the quadratic number field Q(13), is the positive root of its monic minimal polynomial x^2 + x/3 - 1/3. The negative root is -(1 + sqrt(13))/6 = -A209927/3 = -(A188943 - 1).
r^n = A052533(-n) + A006130(-(n+1))*r, for n >= 0, with A052533(-n) = 3*sqrt(-3)^(-n-2)*Snx(-n-2,1/sqrt(-3)), and A006130(-(n+1)) = sqrt(-3)^(-(n+1))*Snx(-(n+1), 1/sqrt(-3)), with the S-Chebyshev polynomials (see A049310), with S(-n, x) = -S(n-2, x), for n>=2, and S(-1, x) = 0. - Wolfdieter Lang, Nov 27 2023

Examples

			0.4342585459106648821865368779117493243752160956408743687850755...

Crossrefs

Cf. A006130, A049310, A052533, A188943, A209927, A223139.

Programs

  • Mathematica
    First[RealDigits[x/.N[Last[Solve[3x^2+x-1==0,x]],78]]] (* Stefano Spezia, Aug 29 2022 *)

Formula

r = (-1 + sqrt(13))/6 = A223139/3 = 1/A209927.

A188942 Decimal expansion of (7-sqrt(13))/6.

Original entry on oeis.org

5, 6, 5, 7, 4, 1, 4, 5, 4, 0, 8, 9, 3, 3, 5, 1, 1, 7, 8, 1, 3, 4, 6, 3, 1, 2, 2, 0, 8, 8, 2, 5, 0, 6, 7, 5, 6, 2, 4, 7, 8, 3, 9, 0, 4, 3, 5, 9, 1, 2, 5, 6, 3, 1, 2, 1, 4, 9, 2, 4, 4, 9, 0, 6, 2, 8, 8, 0, 5, 5, 0, 8, 6, 1, 7, 8, 3, 1, 5, 9, 2, 4, 6, 5, 8, 9, 6, 8, 1, 9, 6, 6, 3, 5, 8, 4, 8, 8, 0, 3, 8, 7, 7, 4, 7, 6, 3, 6, 3, 2, 9, 3, 2, 2, 7, 4, 3, 2, 7, 0, 6, 6, 1, 5, 5, 3, 6, 2, 2, 5
Offset: 1

Views

Author

Clark Kimberling, Apr 14 2011

Keywords

Comments

Decimal expansion of the shape (= length/width = (7-sqrt(13))/6) of the lesser (7/3)-contraction rectangle.
See A188738 for an introduction to lesser and greater r-contraction rectangles, their shapes, and partitioning these rectangles into a sets of squares in a manner that matches the continued fractions of their shapes.

Examples

			0.56574145408933511781346312208825067562478390...

Crossrefs

Cf. A188738, A188943.

Programs

  • Mathematica
    r = 7/3; t = (r - (-4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]
Showing 1-4 of 4 results.

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