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.

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A246647 Starting numbers k = a(n) for the (modified) Collatz map iteration which survive in the sieved Collatz table of A246646.

Original entry on oeis.org

2, 3, 6, 7, 9, 12, 15, 16, 18, 19, 21, 24, 25, 27, 28, 30, 33, 34, 36, 37, 39, 42, 43, 45, 48, 51, 52, 54, 55, 57, 60, 63, 64, 66, 69, 70, 72, 73, 75, 78, 79, 81, 82, 84, 87, 88, 90, 93, 96, 97, 99, 100, 102, 105, 106, 108, 109, 111, 114, 115
Offset: 1

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Author

Wolfdieter Lang, Oct 02 2014

Keywords

Comments

The (modified or Terras) Collatz map is T(k) = (3*k+1)/2 if k is odd and T(k) = k/2 if k is even, where k is a positive integer.
1 is not listed because it appears in row n=1 for k=2 of A246646 . 4 does not appear because 4 = A246646(2,4). Similarly 5 = A246646(2,2), 8 = A246646(2,3), 10 = A246646(4,7), 11 = A246646(4,2), 13 = A246646(4,5), 14 = A246646(5,2), etc.

Crossrefs

Cf. A246646.

A248154 Row length of the irregular triangle A246646 (sieved modified Collatz table).

Original entry on oeis.org

2, 5, 2, 8, 3, 2, 7, 2, 2, 5, 3, 2, 3, 60, 2, 2, 3, 2, 2, 3, 9, 2, 6, 3, 2, 5, 2, 2, 6, 3, 2, 9, 2, 2, 3, 2, 2, 3, 6, 2, 7, 2, 2, 2, 6, 2, 2, 3, 2, 3, 5, 2, 2, 3, 2, 2, 3, 2, 2, 5, 3, 2, 2, 9, 2, 2, 16, 3, 2, 3, 8, 2, 2, 3, 2, 2, 3, 5, 2, 6, 3, 2, 8, 2, 2, 5, 3, 2, 3, 6, 2, 3, 2, 2, 3, 6, 2, 8, 3, 2
Offset: 1

Views

Author

Wolfdieter Lang, Oct 02 2014

Keywords

Examples

			Row n = 1 of A246646(n,m) is 2, 1 with length 2.
Row n = 14 for the starting number k = A246647(14) = 27 of the (modified) Collatz iteration has 60 entries in the sieved table A246646.

Crossrefs

Cf. A246646, A246647.

A246645 Expansion of 1/(1 - 22*x + 81*x^2), used in A246643.

Original entry on oeis.org

1, 22, 403, 7084, 123205, 2136706, 37027927, 641541208, 11114644489, 192557340910, 3335975296411, 57794311907332, 1001260862952013, 17346399720450394, 300518663950795615, 5206352229561021616, 90197737270328030737, 1562635689352773925318, 27071968446864455867299
Offset: 0

Views

Author

Wolfdieter Lang, Sep 30 2014

Keywords

Comments

This sequence is used in the formula for the curvature in a touching circle problem considered in A247512 and A246643.

Links

Crossrefs

Cf. A247512, A246643, A246646.

Programs

  • Magma
    I:=[1, 22]; [n le 2 select I[n] else 22*Self(n-1) - 81*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 20 2017
  • Mathematica
    CoefficientList[Series[1/(1 - 22*x + 81*x^2), {x, 0, 50}], x] (* or *) LinearRecurrence[{22,-81}, {1,22}, 50] (* G. C. Greubel, Dec 20 2017 *)
  • PARI
    Vec(1/(1 - 22*x + 81*x^2) + O(x^40)) \\ Michel Marcus, Sep 30 2014

Formula

O.g.f.: 1/(1 - 22*x + 81*x^2).
a(n) = 9^n*S(n, 22/9) with Chebyshev's S-polynomials (see A049310).
a(n) = 22*a(n-1) - 81*a(n-2), n >= 1, a(-1) = 0 and a(0) = 1.
a(n) = 9^n*(ap^(n+1) - am^(n+1))/(ap - am), n >= 1, with ap := (11 + 2*sqrt(10))/9 and am = 1/ap = (11 - 2*sqrt(10))/9 (Binet - de Moivre formula). a(0) = 1 (via L'Hopital's rule).
a(n) = 9^(n+1)*sinh(2*(n + 1)*arccsch(3))/(2*sqrt(10)). - Federico Provvedi, Feb 02 2021
Showing 1-3 of 3 results.

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