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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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A166060 a(n) = 4*3^n - 3*2^n.

Original entry on oeis.org

1, 6, 24, 84, 276, 876, 2724, 8364, 25476, 77196, 233124, 702444, 2113476, 6352716, 19082724, 57297324, 171990276, 516167436, 1548895524, 4647473004, 13943991876, 41835121356, 125511655524, 376547549484, 1129667814276, 3389053774476, 10167261986724, 30501987286764
Offset: 0

Views

Author

Philippe Deléham, Oct 05 2009

Keywords

Comments

Second binomial transform of A123932 = [1,4,4,4,4,4,4,4,...].

Links

Crossrefs

Cf. A082560, A123932, A217764, A257956.

Programs

  • Haskell
    a166060 n = a166060_list !! n
a166060_list = map fst $ iterate (\(u, v) -> (3 * (u + v), 2 * v)) (1, 1)
-- Reinhard Zumkeller, Jun 09 2013
  • Magma
    [4*3^n-3*2^n: n in [0..30]]; // Vincenzo Librandi, Dec 05 2012
  • Mathematica
    CoefficientList[Series[(1+x)/((1-2x)*(1-3x)), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 05 2012 *)
  • PARI
    a(n)=4*3^n-3<Charles R Greathouse IV, Jan 12 2012

Formula

a(n) = 5*a(n-1) - 6*a(n-2) for n > 1; a(0)= 1, a(1)= 6.
G.f.: (1+x)/(1-5x+6x^2).
a(n) = A217764(n,6). - Ross La Haye, Mar 27 2013
a(n) = Sum_{k = 1..2^n} A082560(n+1,k). - Reinhard Zumkeller, May 14 2015
E.g.f.: exp(2*x)*(4*exp(x) - 3). - Stefano Spezia, May 18 2024

Extensions

a(19) and a(22) corrected by Charles R Greathouse IV, Jan 12 2012

A232642 Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then x + 1 and 2*x + 2 are in S, and duplicates are deleted as they occur.

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 10, 8, 7, 14, 12, 11, 22, 9, 18, 16, 15, 30, 13, 26, 24, 23, 46, 20, 19, 38, 17, 34, 32, 31, 62, 28, 27, 54, 25, 50, 48, 47, 94, 21, 42, 40, 39, 78, 36, 35, 70, 33, 66, 64, 63, 126, 29, 58, 56, 55, 110, 52, 51, 102, 49, 98, 96, 95, 190, 44
Offset: 1

Views

Author

Clark Kimberling, Nov 28 2013

Keywords

Comments

Let S be the set of numbers defined by these rules: 1 is in S, and if x is in S, then x + 1 and 2*x + 2 are in S. Then S is the set of positive integers, which arise in generations. Deleting duplicates as they occur, the generations are given by g(1) = (1), g(2) = (2,4), g(3) = (3,6,5,10), etc. Concatenating these gives A232642, a permutation of the positive integers. For n > 1, the number of numbers in g(n) is 2*F(n+1), where F = A000045, the Fibonacci numbers. It is helpful to show the results as a tree with the terms of S as nodes, an edge from x to x + 1 if x + 1 has not already occurred, and an edge from x to 2*x + 2 if 2*x + 2 has not already occurred.
Seen as triangle read by rows: A082560 with duplicates removed. - Reinhard Zumkeller, May 14 2015

Examples

			Each x begets x + 1 and 2*x + 2, but if either has already occurred it is deleted. Thus, 1 begets 2 and 4; then 2 begets 3 and 6, and 4 begets 5 and 10, so that g(3) = (3,6,5,10).
First 5 generations, also showing the places where duplicates were removed:
.  1:                                1
.  2:                2                               4
.  3:        3              6               5                10
.  4:    _       8      7       14      _       12       11       22
.  5:  _  __   9  18  _  16  15   30  _  __  13   26  __   24  23   46
These are the corresponding complete rows of triangle A082560:
.  1:                                1
.  2:                2                               4
.  3:        3              6               5                10
.  4:    4       8      7       14      6       12       11       22
.  5:  5  10   9  18  8  16  15   30  7  14  13   26  12   24  23   46

Links

Crossrefs

Cf. A232559, A232639, A232643, A000045.
Cf. A128588 (row lengths), A033484 (right edges), A257956 (row sums), A082560.

Programs

  • Haskell
    import Data.List.Ordered (member); import Data.List (sort)
a232642 n k = a232642_tabf !! (n-1) !! (k-1)
a232642_row n = a232642_tabf !! (n-1)
a232642_tabf = f a082560_tabf [] where
   f (xs:xss) zs = ys : f xss (sort (ys ++ zs)) where
     ys = [v | v <- xs, not $ member v zs]
a232642_list = concat a232642_tabf
-- Reinhard Zumkeller, May 14 2015
  • Mathematica
    z = 14; g[1] = {1}; g[2] = {2}; g[n_] := Riffle[g[n - 1] + 1, 2 g[n - 1] + 2]; j[2] = Join[g[1], g[2]]; j[n_] := Join[j[n - 1], g[n]]; g1[n_] := DeleteDuplicates[DeleteCases[g[n], Alternatives @@ j[n - 1]]]; g1[1] = g[1]; g1[2] = g[2]; t = Flatten[Table[g1[n], {n, 1, z}]]  (* A232642 *)
Table[Length[g1[n]], {n, 1, z}]  (* A000045 *)
Flatten[Table[Position[t, n], {n, 1, 200}]]  (* A232643 *)

Extensions

Keyword tabf added, to bring out function g, by Reinhard Zumkeller, May 14 2015
Showing 1-2 of 2 results.

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