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 1-5 of 5 results.

A307048 Permutation of the positive integers derived from the terms of A322469 having the form 6*k - 2.

Original entry on oeis.org

2, 1, 6, 5, 10, 4, 14, 7, 18, 13, 22, 8, 26, 3, 30, 21, 34, 12, 38, 9, 42, 29, 46, 16, 50, 23, 54, 37, 58, 20, 62, 19, 66, 45, 70, 24, 74, 17, 78, 53, 82, 28, 86, 39, 90, 61, 94, 32, 98, 15, 102, 69, 106, 36, 110, 25, 114, 77, 118, 40, 122, 55
Offset: 1

Views

Author

Georg Fischer, Mar 21 2019

Keywords

Comments

The sequence is the flattened form of an irregular table U(i, j) similar to table T(i, j) in A322469. U(i, j) = k is defined only for the elements T(i, j) which have the form 6*k - 2, so the table is sparsely filled.
Like in A322469, the columns in table U contain arithmetic progressions.
a(n) is a permutation of the positive integers, since A322469 is one, and since there is a one-to-one mapping between any a(n) = k and some A322469(m) = 6*k - 2.
There is a hierarchy of such permutations of the positive integers derived by mapping the terms of the form 6*k - 2 to k:
Level 1: A322469
Level 2: A307048 (this sequence)
Level 3: A160016 = 2, 1, 4, 6, 8, 3, ... period of (3 even, 1 odd number)
Level 4: A000027 = 1, 2, 3, 4 ... (the positive integers)
Level 5: A000027

Examples

			Table U(i, j) begins:
   i\j   1  2  3  4  5  6  7
   -------------------------
   1:
   4:          2
   7:                   1
  10:
  13:          6
  16:                5
  19:
  22:         10
  25:             4
  28:
  31:         14
-----
T(4, 3) = 10 = 6*2 - 2, therefore U(4, 3) = 2.
T(7, 6) =  4 = 6*1 - 2, therefore U(7, 6) = 1.

Crossrefs

Cf. A000027, A160016, A322469.

Programs

  • Perl
    # Derived from A322469
use integer; my $n = 1; my $i = 1; my $an;
while ($i <= 1000) { # next row
  $an = 4 * $i - 1; &term();
  while ($an % 3 == 0) {
    $an /= 3; &term();
    $an *= 2; &term();
  } # while divisible by 3
  $i ++;
} # while next row
sub term {
  if (($an + 2) % 6 == 0) {
    my $bn = ($an + 2) / 6;
    print "$n $bn\n"; $n ++;
  }
}

A338208 Inverse permutation to A322469.

Original entry on oeis.org

2, 3, 1, 16, 7, 14, 4, 17, 12, 8, 5, 15, 21, 33, 6, 126, 26, 13, 9, 69, 31, 50, 10, 124, 38, 22, 11, 34, 43, 67, 18, 127, 48, 27, 19, 122, 55, 86, 20, 70, 60, 32, 23, 51, 65, 103, 24, 125, 74, 39, 25, 179, 79, 120, 28, 287, 84, 44, 29, 68, 91, 143, 30, 1100, 96, 49, 35, 232, 101, 160, 36, 123, 108, 56, 37, 87, 113, 177, 40, 611
Offset: 1

Views

Author

Georg Fischer, Oct 16 2020

Keywords

Comments

Permutation of the positive integers.
There is a hierarchy of such permutations derived by selecting and mapping the terms of the form 6*k - 2 to k:
Level 0: A307407
Level 1: A322469, inverse is A338208 (this sequence)
Level 2: A307048 A338207
Level 3: A160016 A338206
Level 4: A000027 (the positive integers)

Crossrefs

Cf. A160016, A307048, A307407, A322469, A338206, A338207.

A307407 Irregular table read by rows: rows list terms that map to the nodes in the graph of the "3x+1" (or Collatz) problem.

Original entry on oeis.org

16, 4, 5, 1, 10, 2, 3, 40, 12, 13, 64, 20, 21, 88, 28, 29, 9, 58, 112, 36, 37, 136, 44, 45, 160, 52, 53, 17, 106, 34, 35, 11, 70, 22, 23, 7, 46, 14, 15, 184, 60, 61, 208, 68, 69, 232, 76, 77, 25, 154, 50, 51, 256, 84, 85, 280, 92, 93
Offset: 1

Views

Author

Georg Fischer, Apr 14 2019

Keywords

Comments

The construction is similar to that in A322469. The sequence is the flattened form of an irregular table S(i, j) (see the example below) which has rows i >= 1 consisting of subsequences of varying length.
Like Truemper (cf. link), we denote the mapping x -> 2*x by "m" ("multiply"), the mapping x -> (x - 1)/3 by "d" ("divide"), and the combined mapping "dm" x -> (x - 1)/3 * 2 by "s" ("squeeze"). The d mapping is defined only if it is possible, that is, if x - 1 is divisible by 3. We write m, d and s as infix operation words, for example "4 mms 10", and we use exponents for repeated operations, for example "mms^2 = mmss".
Row i in table S is constructed by the following algorithm: Start with 6 * i - 2 in column j = 1. The following columns j are defined in groups of four by the operations:
k j=4*k+2 j=4*k+3 j=4*k+4 j=4*k+5
--------------------------------------------------
0 mm dmm mmd dmmd
1 mms dmms mmsd dmmsd
2 mms^2 dmms^2 mms^2d dmms^2d
...
k mms^k dmms^k mm(s^k)d dmm(s^k)d
The construction for the row terminates at the first column where a d operation is no longer possible. This point is always reached. This can be proved by the observation that, for any row i in S, there is a unique mapping x -> (x + 2)/6 of the terms in column 1, 2, 5, 9, 13, ... 4*m+1 to the terms in row i of table T in A322469. The row construction process in A322469 stops, therefore it stops also in the sequence defined here.
Conjecture: The sequence is a permutation of the positive numbers.

Examples

			Table S(i, j) begins:
  i\j    1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
  ----------------------------------------------------------------
  1:    16   4   5   1  10   2   3
  2:    40  12  13
  3:    64  20  21
  4:    88  28  29   9  58
  5:   112  36  37
  6:   136  44  45
  7:   160  52  53  17 106  34  35  11  70  22  23   7  46  14  15
  8:   184  60  61

Links

Crossrefs

Cf. A160016 (level 3), A307048 (level 2), A322469 (level 1).

Programs

  • Perl
    cf. link.

A308709 Start with 3, divide by 3, multiply by 2, multiply by 3, multiply by 2, repeat.

Original entry on oeis.org

3, 1, 2, 6, 12, 4, 8, 24, 48, 16, 32, 96, 192, 64, 128, 384, 768, 256, 512, 1536, 3072, 1024, 2048, 6144, 12288, 4096, 8192, 24576, 49152, 16384, 32768, 98304, 196608, 65536, 131072, 393216, 786432, 262144, 524288, 1572864, 3145728, 1048576
Offset: 1

Views

Author

Georg Fischer, Aug 05 2019

Keywords

Comments

The division by 3 is always possible since it is always preceded by a multiplication by 3.
This sequence arises in the "3x+1" (Collatz) problem. In the rows of A322469, the terms of this sequence appear at the end of any first row which is longer than all previous rows.

Examples

			3; /3 => 1; *2 => 2; *3 => 6;  *2 => 12;
   /3 => 4; *2 => 8; *3 => 24; *2 => 48 ...

Links

Crossrefs

Cf. A307407, A322469.

Programs

  • Mathematica
    LinearRecurrence[{0, 0, 0, 4},{3, 1, 2, 6}, 50]

Formula

G.f.: x*(3+x+2*x^2+6*x^3)/(1-4*x^4).

A338186 Expansion of (2-6*x-12*x^2)/((1-x)^2*(1-9*x)).

Original entry on oeis.org

2, 16, 126, 1100, 9850, 88584, 797174, 7174468, 64570098, 581130752, 5230176622, 47071589436, 423644304746, 3812798742520, 34315188682470, 308836698142004, 2779530283277794, 25015772549499888, 225141952945498718, 2026277576509488172, 18236498188585393242, 164128483697268538856
Offset: 0

Views

Author

Georg Fischer, Oct 15 2020

Keywords

Comments

The locally small terms 4^k in A322469 occur at the positions a(k) (for k = 0..9, and probably in general; cf. conjectures in A322469).

Examples

			A322469(a(4)) = A322469(9850) = 256 = 4^4.

Links

Crossrefs

Cf. A322469.

Programs

  • Maple
    f:= gfun:-rectoproc({a(n)=11*a(n-1)-19*a(n-2)+9*a(n-3), a(0)=2, a(1)=16, a(2)=126}, a(n), remember): map(f, [$0..21]);
  • Mathematica
    CoefficientList[Series[(2-6*x-12*x^2)/((1-x)^2*(1-9*x)), {x,0,21}], x]
  • PARI
    my(x='x+O('x^22)); Vec((2-6*x-12*x^2)/((1-x)^2*(1-9*x)))

Formula

a(n) = 11*a(n-1) - 19*a(n-2) + 9*a(n-3) for n >= 3.
Showing 1-5 of 5 results.

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