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.

A222132 Decimal expansion of sqrt(4 + sqrt(4 + sqrt(4 + sqrt(4 + ... )))).

Original entry on oeis.org

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

Views

Author

Jaroslav Krizek, Feb 08 2013

Keywords

Comments

Sequence with a(1) = 1 is decimal expansion of sqrt(4 - sqrt(4 - sqrt(4 - sqrt(4 - ... )))) = A222133.
Because 17 == 1 (mod 4), the basis for integers in the real quadratic number field K(sqrt(17)) is <1, omega(17)>, where omega(17) = (1 + sqrt(17))/2. - Wolfdieter Lang, Feb 10 2020
This is the positive root of the polynomial x^2 - x - 4, with negative root -A222133. - Wolfdieter Lang, Dec 10 2022
It is the spectral radius of the diamond graph (see Seeger and Sossa, 2023). - Stefano Spezia, Sep 19 2023
c^n = A006131(n) + A006131(n-1) * d, where c = (1 + sqrt(17))/2 and d = (-1 + sqrt(17))/2. - Gary W. Adamson, Nov 25 2023
c^n = A052923(n) + A006131(n-1) * c. Also for negative n. - Wolfdieter Lang, Nov 27 2023
The effective degree of maximal entropy random walk on the barred-square graph (see Burda et al.). - Stefano Spezia, Feb 07 2025

Examples

			2.561552812808830274910704...

Links

Crossrefs

Cf. A222133, A178255, A082486.
Cf. A006131, A052923.

Programs

  • Maple
    Digits:=140:
evalf((sqrt(17)+1)/2);  # Alois P. Heinz, Sep 19 2023
  • Mathematica
    RealDigits[(1 + Sqrt[17])/2, 10, 130]

Formula

Closed form: (sqrt(17) + 1)/2 = A178255 - 1 = A082486 - 2.
sqrt(4 + sqrt(4 + sqrt(4 + sqrt(4 + ... )))) - 1 = sqrt(4 - sqrt(4 - sqrt(4 - sqrt(4 - ... )))). See A222133.

A178255 Decimal expansion of (3+sqrt(17))/2.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, May 24 2010

Keywords

Comments

Continued fraction expansion of (3+sqrt(17))/2 is A109007.
a(n) = A082486(n) for n > 1.
The rectangle R whose shape (i.e., length/width) is (3+sqrt(17))/2 can be partitioned into rectangles of shapes 3 and 3/2 in a manner that matches the periodic continued fraction [3, 3/2, 3, 3/2, ...]. R can also be partitioned into squares so as to match the periodic continued fraction [3, 1, 1, 3, 1, 1,...]. For details, see A188635. - Clark Kimberling, May 07 2011
The positive eigenvalue of the real symmetric 2 X 2 matrix M defined by M(i,j) = max(i,j) = [(1 2), (2 2)] is (3+sqrt(17))/2, while the negative one is (3-sqrt(17))/2. For a generalization, see A085984. - Bernard Schott, Apr 13 2020
A quadratic integer with minimal polynomial x^2 - 3x - 2. - Charles R Greathouse IV, Apr 14 2020
The positive root of x^2 - 3^x - 2. The negative root is -(-3 + sqrt(17))/2 = -0.56155... - Wolfdieter Lang, Dec 10 2022

Examples

			(3+sqrt(17))/2 = 3.56155281280883027491...

Links

Crossrefs

Cf. A082486 (decimal expansion of (5+sqrt(17))/2), A010473 (decimal expansion of sqrt(17)), A109007 (repeat 3, 1, 1), A085984.

Programs

  • Mathematica
    FromContinuedFraction[{3, 3/2, {3, 3/2}}]
ContinuedFraction[%, 100] (* [3,1,1,3,1,1,...] *)
RealDigits[N[%%, 120]]    (* A178255 *)
N[%%%, 40]
(* Clark Kimberling, May 07 2011 *)
  • PARI
    (3+sqrt(17))/2 \\ Charles R Greathouse IV, Apr 14 2020

A222133 Decimal expansion of sqrt(4 - sqrt(4 - sqrt(4 - sqrt(4 - ... )))).

Original entry on oeis.org

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

Views

Author

Jaroslav Krizek, Feb 08 2013

Keywords

Comments

Sequence with a(1) = 2 is the decimal expansion of sqrt(4 + sqrt(4 + sqrt(4 + sqrt(4 + ... )))) = - A222132.
This is the positive root of the minimal polynomial x^2 + x - 4, with negative root -A222132. - Wolfdieter Lang, Dec 10 2022

Examples

			1.561552812808830274910704...

Crossrefs

Cf. A178255, A082486, A222132.

Programs

  • Mathematica
    RealDigits[(Sqrt[17] - 1)/2, 10, 130]

Formula

Closed form: (sqrt(17) - 1)/2 = A178255-2 = A082486-3 = A222132-1.
sqrt(4 - sqrt(4 - sqrt(4 - sqrt(4 - ... )))) + 1 = sqrt(4 + sqrt(4 + sqrt(4 + sqrt(4 + ... )))). See A222132.

A340309 Number of ordered pairs of vertices which have two different shortest paths between them in the n-Hanoi graph (3 pegs, n discs).

Original entry on oeis.org

0, 6, 48, 282, 1476, 7302, 35016, 164850, 767340, 3546366, 16315248, 74837802, 342621396, 1566620022, 7157423256, 32682574050, 149184117180, 680813718126, 3106475197248, 14173073072922, 64659388538916, 294971717255142, 1345602571317096, 6138257708432850
Offset: 1

Views

Author

Kevin Ryde, Jan 04 2021

Keywords

Comments

Vertices of the Hanoi graph are configurations of discs on pegs in the Towers of Hanoi puzzle. Edges are a move of a disc from one peg to another.
The shortest path between a pair of vertices u,v may be unique, or there may be 2 different paths. a(n) is the number of vertex pairs with 2 shortest paths. Pairs are ordered, so both u,v and v,u are counted.
For a given vertex u, Hinz et al. characterize and count the destinations v which have 2 shortest paths. Their total x_n is the number of vertex pairs in the graph of n+1 discs. The present sequence is for n discs so a(n) = x_{n-1}.

Examples

			For n=3 discs, the Hanoi graph is
                *           \
               / \          | top
              A---*         | subgraph,
             /     \        | of n-1 = 2
            B       *       | discs
           / \     / \      |
          C---D---E---*     /
         /             \          two shortest
        *               *           paths for
       / \             / \           A to S
      *---*           *---*          B to T
     /     \         /     \         C to R
    *       *       R       *        C to U
   / \     / \     / \     / \       D to S
  *---*---*---*---S---T---U---*
Going from the top subgraph down to the bottom right subgraph, there are 5 vertex pairs with two shortest paths.  C to R goes around the middle 12-cycle either right or left, and likewise D to S.  The other pairs also go each way around the middle.  There are 6 ordered pairs of n-1 subgraphs repeating these 5 pairs.
Within the n-1 = 2 disc top subgraph, A and E are in separate n-2 subgraphs (unit triangles) and they are the only pair with two shortest paths.  Again 6 combinations of these, and in 3 subgraphs.  Total a(3) = 6*5 + 6*3*1 = 48.

Links

Crossrefs

Cf. A052984, A107839.

Programs

  • PARI
    my(p=Mod('x, 'x^2-5*'x+2)); a(n) = (vecsum(Vec(lift(p^(n+1)))) - 3^n)*3/2;

Formula

With P = (5 + sqrt(17))/2 = A082486, and M = (5 - sqrt(17))/2:
a(n) = (3/(4*sqrt(17)))*( (sqrt(17)+1)*P^n - 2*sqrt(17)*3^n + (sqrt(17)-1)*M^n ). [Hinz et al.]
a(n) = (6/sqrt(17)) * Sum_{k=0..n-1} 3^k * (P^(n-1-k) - M^(n-1-k)) [Hinz et al.].
a(n) = 3*a(n-1) + 6*A107839(n-2), paths within and between subgraphs n-1.
a(n) = 8*a(n-1) - 17*a(n-2) + 6*a(n-3).
a(n) = (A052984(n) - 3^n)*3/2.
G.f.: 6*x^2/((1 - 5*x + 2*x^2)*(1 - 3*x)).
G.f.: (3/2 - 3*x)/(1 - 5*x + 2*x^2) - (3/2)/(1 - 3*x).
Showing 1-4 of 4 results.

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