A036561 -id:A036561 - cp's OEIS frontend

A175840 Mirror image of Nicomachus' table: T(n,k) = 3^(n-k)*2^k for n>=0 and 0<=k<=n.

1, 3, 2, 9, 6, 4, 27, 18, 12, 8, 81, 54, 36, 24, 16, 243, 162, 108, 72, 48, 32, 729, 486, 324, 216, 144, 96, 64, 2187, 1458, 972, 648, 432, 288, 192, 128, 6561, 4374, 2916, 1944, 1296, 864, 576, 384, 256, 19683, 13122, 8748, 5832, 3888, 2592, 1728, 1152, 768, 512

Offset: 0

Johannes W. Meijer, Sep 21 2010, Jul 13 2011, Jun 03 2012

Lenstra calls these numbers the harmonic numbers of Philippe de Vitry (1291-1361). De Vitry wanted to find pairs of harmonic numbers that differ by one. Levi ben Gerson, also known as Gersonides, proved in 1342 that there are only four pairs with this property of the form 2^n*3^m. See also Peterson’s story ‘Medieval Harmony’.

This triangle is the mirror image of Nicomachus' table A036561. The triangle sums, see the crossrefs, mirror those of A036561. See A180662 for the definitions of these sums.

			1;
3, 2;
9, 6, 4;
27, 18, 12, 8;
81, 54, 36, 24, 16;
243, 162, 108, 72, 48, 32;

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
J. O'Connor and E.F. Robertson, Nicomachus of Gerasa, The MacTutor History of Mathematics archive, 2010.
Jay Kappraff, The Arithmetic of Nicomachus of Gerasa and its Applications to Systems of Proportion, Nexus Network Journal, vol. 2, no. 4 (October 2000).
Hendrik Lenstra, Aeternitatem Cogita, Nieuw Archief voor Wiskunde, 5/2, maart 2001, pp. 23-28.
Ivars Peterson, Medieval Harmony, Math Trek, Mathematical Association of America, 1998.

Triangle sums: A001047 (Row1), A015441 (Row2), A016133 (Kn1 & Kn4), A005061 (Kn2 & Kn3), A016153 (Fi1& Fi2), A180844 (Ca1 & Ca4), A016140 (Ca2, Ca3), A180846 (Gi1 & Gi4), A180845 (Gi2 & Gi3), A016185 (Ze1 & Ze4), A180847 (Ze2 & Ze3).

Cf. A000079, A000244, A000400, A003586.

a175840 n k = a175840_tabf !! n !! k
a175840_row n = a175840_tabf !! n
a175840_tabf = iterate (\xs@(x:_) -> x * 3 : map (* 2) xs) [1]
-- Reinhard Zumkeller, Jun 08 2013

A175840 := proc(n,k): 3^(n-k)*2^k end: seq(seq(A175840(n,k),k=0..n),n=0..9);

Flatten[Table[3^(n-k) 2^k,{n,0,10},{k,0,n}]] (* Harvey P. Dale, May 08 2013 *)

T(n,k) = 3^(n-k)*2^k for n>=0 and 0<=k<=n.

T(n,n-k) = T(n,n-k+1) + T(n-1,n-k) for n>=1 and 1<=k<=n with T(n,n) = 2^n for n>=0.

A036567 Basic numbers used in Sedgewick-Incerpi upper bound for shell sort.

Original entry on oeis.org

1, 3, 7, 16, 41, 101, 247, 613, 1529, 3821, 9539, 23843, 59611, 149015, 372539, 931327, 2328307, 5820767, 14551919, 36379789, 90949471, 227373677, 568434193, 1421085473, 3552713687, 8881784201, 22204460497, 55511151233, 138777878081, 346944695197, 867361737989

Offset: 0

N. J. A. Sloane

			2.5^4 = 39.0625, and 41 is the next integer that is relatively prime to 1, 3, 7 and 16.

D. E. Knuth, The Art of Computer Programming, Vol. 3, Sorting and Searching, 2nd ed, section 5.2.1, p. 91.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Janet Incerpi and Robert Sedgewick, Improved upper bounds on shellsort, Journal of Computer and System Sciences, Volume 31, Issue 2, October 1985, Pages 210-224
Robert Sedgewick, Analysis of shellsort and related algorithms, Fourth European Symposium on Algorithms, Barcelona, September, 1996.
Index entries for sequences related to sorting

Cf. A003462, A033622, A036561, A036562, A036564, A036566, A036567, A036569, A055875, A055876, A102549, A108870, A112262, A112263, A154393, A204772, A205669, A205670, A361506, A361507.

a:= proc(n) option remember; local l, m;
      l:= [seq(a(i), i=1..n-1)];
      for m from ceil((5/2)^n) while ormap(x-> igcd(m, x)>1, l) do od; m
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 06 2022

A036567[1] = 3;
A036567[q_] :=
With[{prev = A036567 /@ Range[q - 1]},
  Block[{n = Ceiling[(5/2)^q]},
   While[Nand @@ ((# == 1 &) /@ GCD[prev, n]), n++];
   n]]; (* Morgan Owens, Oct 08 2020 *)
Array[A036567, 10]

a036567(m)={my(v=vector(m)); for(n=1,m,my(b=ceil((5/2)^n));for(j=b,oo,my(g=1); for(k=1,n-1,if(gcd(j,v[k])>1,g=0;break));if(g,v[n]=j;break)));v};
a036567(28) \\ Hugo Pfoertner, Oct 15 2020

a(n) is the smallest number >= 2.5^n that is relatively prime to all previous terms in the sequence.

Better description and more terms from Jud McCranie, Jan 05 2001

a(0)=1 prepended by Alois P. Heinz, Dec 04 2023

A061930 Square array read by antidiagonals of T(n,k)=T(n-1,[k/2])+T(n-1,[k/3]) with T(0,0)=1.

Original entry on oeis.org

1, 0, 2, 0, 2, 4, 0, 1, 4, 8, 0, 0, 4, 8, 16, 0, 0, 4, 8, 16, 32, 0, 0, 3, 8, 16, 32, 64, 0, 0, 3, 8, 16, 32, 64, 128, 0, 0, 1, 8, 16, 32, 64, 128, 256, 0, 0, 1, 8, 16, 32, 64, 128, 256, 512, 0, 0, 1, 8, 16, 32, 64, 128, 256, 512, 1024, 0, 0, 0, 7, 16, 32, 64, 128, 256, 512, 1024

Offset: 0

Henry Bottomley, May 22 2001

			T(9, 7) = T(8, [7/2])+T(8, [7/3]) = T(8, 3)+T(8, 2) = 256+256 = 512. Rows start (1, 0, 0, 0, 0, ...), (2, 2, 1, 0, 0, ...), (4, 4, 4, 4, 3, ...) etc.

Row sums are 5^n, i.e. A000351. Each row starts with 2^n copies of 2^n, i.e. A000079 and then continues with A036561 copies of other terms in the rows of A055248. Cf. A061929.

A164092 Triangle by 2^n term rows, codes used for generating Petoukhov matrices in a Gray code format.

Original entry on oeis.org

0, 1, -1, 2, 0, -2, 0, 3, 1, -1, 1, -1, -3, -1, 1, 4, 2, 0, 2, 0, -2, 0, 2, 0, -2, -4, -2, 0, -2, 0, 2, 5, 3, 1, 3, 1, -1, 1, 3, 1, -1, -3, -1, 1, -1, 1, 3

Offset: 0

Gary W. Adamson, Aug 09 2009

			First few rows of the triangle =
0;
1, -1;
2, .0, -2, 0;
3, .1, -1, 1, -1, -3, -1, 1;
4, .2, .0, 2, .0, -2, .0, 2, 0, -2, -4, -2, 0, -2, 0, 2;
...
We present examples of Petoukhov matrices (Cf. A164091) using rows 2 and 3.
.
Row 3 = [2, 0, -2, 0] = A. We crease an "alternating column circulant. If by convention such matrices have an upper left term (1,1), then odd rows cycle from term (n,n) downward using A. Even rows circulate from (n,n) upwards (Cf. A164057). Using these rules, we obtain the exponents for constants k in 4 X 4 Petoukhov matrices:
.
[2, 0, -2, 0;
.0, 2, 0, -2;
-2, 0, 2, .0;
.0,-2, 0, .2]
.
Let the Petoukhov constant k = phi, 1.6180339,... then insert k into the matrix using the exponents shown, getting [phi^2, 1, 1/phi^2, 1; 1, phi^2, 1, 1/phi^2; 1/phi^2, 1, phi^2, 1; 1, 1/phi^2, 1, phi^2] = M.
.
Then square matrix: M^2 =
9, 6, 4, 6;
6, 9, 6, 4;
4, 6, 9, 6;
6, 4, 6, 9;
...
The terms (4, 6, 9) may be obtained from a 2 X 3 multiplication table, (Cf. A036561, A164057):
.
1,..3,..9,..27,...
2,..6,.18,..54,...
4,.12,.36.........
8..24.............
16................
.
As antidiagonals of this array, we see the terms (4, 6, 9). Similarly, for the 8 X 8 matrix, we apply exponents to phi in the next row using the same circulant rule. As indicated by the next antidiagonal of the 2 X 3 table, the 8 X 8 matrix uses the terms (8, 12, 18, 27), but with a binomial frequency of (1, 3, 3, 1). The 8 X 8 matrix is likewise a square of the corresponding matrix using the exponents [3, 1, -1, 1, -1, -3, -1, 1], then applying the circulant rule. Let this 8 X 8 phi matrix = Q. Then Q^2 = the 8 X 8 Petoukhov matrix (Cf. A164057):
.
27...18...12...18...12...08...12...18;
18...27...18...12...08...12...18...12;
12...18...27...18...12...18...12...08;
18...12...18...27...18...12...08...12;
12...08...12...18...27...18...12...18;
08...12...18...12...18...27...18...12;
12...18...12...08...12...18...27...18;
18...12...08...12...18...12...18...27;
.
Note the binomial distribution of (by rows and columns) one 27, three 18's three 12's and one 8. A harmonic relationship is preserved by Knight's moves in any direction including wrap arounds; any neighbor = (2/3) or (3/2) * another neighbor.

Cf. A164057, A036561, A164056, A164057, A147995.

Let a(0) = 0. Add "1" to each term in n-th row, then bring down to create the first half of the next row. Reverse terms of n-th row and subtract "1", then append, as the right half of row (n+1).

A164279 Triangle of 2^n terms per row, a Petoukhov sequence generated from (3,2).

Original entry on oeis.org

1, 3, 2, 9, 6, 4, 6, 27, 18, 12, 18, 12, 8, 12, 18, 81, 54, 36, 54, 36, 24, 36, 54, 36, 24, 16, 24, 36, 24, 36, 54

Offset: 0

Gary W. Adamson, Aug 11 2009

Row sums = powers of 5: (1, 5, 25, 125,...).
Petoukhov has pioneered the investigation of a class of matrices that are squares of other matrices composed of entirely irrational terms. A164279 terms = top rows, left columns of the Petoukhov matrices shown in A164092.
The Petoukhov matrices associated with A164279 are shown in A164092 along with their derivation from phi, 1.618033989...
The original Petoukhov matrices were in a binary Karnaugh map format.
I have standardized the matrices and sequences, mapping them on the Gray code format shown in A147995. This allows for a ("1 operation" change from one term to the next. For example, in A164279, the next term is either (3/2)*(current term) or (2/3)*(current term) depending on the corresponding positional code of A164057: (a 1 or 0).
Note the binomial frequence of terms per row: (e.g. one 27, three 18's, three 12's, and one 8) in row 3.

			The distinct terms per row are (Cf. A036561): (1; 2,3; 4,6,9; 8,12,18,27; 16,24,36,54,81;) while the codes of A164057 begin:
.
1;
1, 0;
1, 0, 0, 1;
1, 0, 0, 1, 0, 0, 1, 1;
1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1;
...
Given (1, 3, 9, 27,...) as leftmost row terms and following the operational rules: (multiply current term by (3/2) if the corresponding code = 1; (or by (2/3) if 0). This generates A164279: .
1;
3, 2;
9, 6, 4, 6;
27, 18, 12, 18, 12, 8, 12, 18;
81, 54, 36, 54, 36, 24, 36, 54, 36, 24, 16, 24, 36, 24, 36, 54;
...

Sergei Petoukhov & Matthew He, "Symmetrical Analysis Techniques for Genetic Systems and Bioinformatics - Advanced Patterns and Applications"; IGI Global, 978-1-60566-127-9, October, 2009; Chapters 2, 4, and 6.

Cf. A036561, A164057, A147995.

Using the row terms of A036562 (a 2x3 multiplication table): (1, 3,2; 4,6,9;, 8,12,18,27;...), rows of A164279 have leftmost terms extracting the power of 9 from A036562: (1, 3, 9, 27,...). Then accessing the corresponding row codes from A164057, and starting from the left, first term = a power of 9, then given the codes of A164057 (0 or 1), the next row term of A164279 = (3/2)*current term) if the corresponding term of A164057 = 1, and (2/3)*current term if 0.

A061929 Triangle with n >= k >= 0 where a(n,k) = 2^k3^(n-k)(C(n+1,0)+C(n+1,1)+...C(n+1,k)).

Original entry on oeis.org

1, 3, 6, 9, 24, 28, 27, 90, 132, 120, 81, 324, 576, 624, 496, 243, 1134, 2376, 3024, 2736, 2016, 729, 3888, 9396, 13824, 14256, 11520, 8128, 2187, 13122, 35964, 60264, 70416, 63072, 47424, 32640, 6561, 43740, 134136, 252720, 331776, 330048, 268416

Offset: 0

Henry Bottomley, May 22 2001

			Rows start (1), (3,6), (9,24,68), (27,90,132,120) etc.

Row sums are 5^(n+1)-4^(n+1), i.e. A005060. Cf. A061930.

a(n, k) =A054143(n, k)*A036561(n, n-k)

A120027 Triangle, generated from (3^(n-k) * 5^k) table.

Original entry on oeis.org

1, 3, 5, 9, 15, 25, 27, 45, 75, 125, 81, 135, 225, 375, 625, 243, 405, 675, 1125, 1875, 3125, 729, 1215, 2025, 3375, 5625, 9375, 15625, 2187, 3645, 6075, 10125, 16875, 28125, 46875, 78125, 6561, 10935, 18225, 30375, 50625, 84375, 140625, 234375

Offset: 0

Gary W. Adamson, Jun 04 2006

Row 1 of the array (3, 15, 75, 375, ...) = A005053, (3 * 5^n), deleting the "1".

			First few rows of the array:
  1,  5,  25,  125, ...
  3, 15,  75,  375, ...
  9, 45, 225, 1125, ...
First few rows of the triangle are:
   1;
   3,  5;
   9, 15, 25;
  27, 45, 75, 125;
  ...
Example: a(17) = 675 = (3,2) in the array, = 3^3 * 5^2.

Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv preprint arXiv:1212.2732 [math.CO], 2012.

Cf. A005053, A036561, A036565, A036566.

Table[3^(n - k)*5^k, {n, 0, 8}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jun 06 2006 *)

Antidiagonals of the (3^i * 5^j) multiplication table, as an array.
From Boris Putievskiy, Jan 09 2013: (Start)
T(n,k) = 3^(k-1)*5^(n-1) n, k >0 read by antidiagonals.
a(n) = 3^(A004736(n)-1) * 5^(A002260(n)-1), n > 0, or
a(n) = 3^(j-1) * 5^(i-1), n > 0,
where i = n - t*(t+1)/2, j = (t*t + 3*t + 4)/2 - n, t = floor((-1+sqrt(8*n-7))/2). (End)
G.f.: 1/((1 - 3*x)(1 - 5*x*y)). - Ilya Gutkovskiy, Jun 03 2017

More terms from Robert G. Wilson v, Jun 06 2006

A155118 Array T(n,k) read by antidiagonals: the k-th term of the n-th iterated differences of A140429.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 3, 4, 6, 9, 5, 8, 12, 18, 27, 11, 16, 24, 36, 54, 81, 21, 32, 48, 72, 108, 162, 243, 43, 64, 96, 144, 216, 324, 486, 729, 85, 128, 192, 288, 432, 648, 972, 1458, 2187, 171, 256, 384, 576, 864, 1296, 1944, 2916, 4374, 6561, 341, 512, 768, 1152, 1728, 2592, 3888, 5832, 8748, 13122, 19683

Offset: 0

Paul Curtz, Jan 20 2009

Deleting column k=0 and reading by antidiagonals yields A036561.

Deleting column k=0 and reading the antidiagonals downwards yields A175840.

			The array starts in row n=0 with columns k>=0 as:
   0   1    3    9    27    81    243    729    2187  ... A140429;
   1   2    6   18    54   162    486   1458    4374  ... A025192;
   1   4   12   36   108   324    972   2916    8748  ... A003946;
   3   8   24   72   216   648   1944   5832   17496  ... A080923;
   5  16   48  144   432  1296   3888  11664   34992  ... A257970;
  11  32   96  288   864  2592   7776  23328   69984  ...
  21  64  192  576  1728  5184  15552  46656  139968  ...
Antidiagonal triangle begins as:
   0;
   1,   1;
   1,   2,   3;
   3,   4,   6,   9;
   5,   8,  12,  18,  27;
  11,  16,  24,  36,  54,  81;
  21,  32,  48,  72, 108, 162, 243;
  43,  64,  96, 144, 216, 324, 486, 729;
  85, 128, 192, 288, 432, 648, 972, 1458, 2187; - _G. C. Greubel_, Mar 25 2021

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

Cf. A001045, A004054, A046936.

t:= func< n,k | k eq 0 select (2^(n-k) -(-1)^(n-k))/3 else 2^(n-k)*3^(k-1) >;
[t(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 25 2021

T:=proc(n,k)if(k>0)then return 2^n*3^(k-1):else return (2^n - (-1)^n)/3:fi:end:
for d from 0 to 8 do for m from 0 to d do print(T(d-m,m)):od:od: # Nathaniel Johnston, Apr 13 2011

t[n_, k_]:= If[k==0, (2^(n-k) -(-1)^(n-k))/3, 2^(n-k)*3^(k-1)];
Table[t[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 25 2021 *)

def A155118(n,k): return (2^(n-k) -(-1)^(n-k))/3 if k==0 else 2^(n-k)*3^(k-1)
flatten([[A155118(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 25 2021

For the square array:
T(n,k) = 2^n*3^(k-1), k>0.
T(n,k) = T(n-1,k+1) - T(n-1,k), n>0.
Rows:
T(0,k) = A140429(k) = A000244(k-1).
T(1,k) = A025192(k).
T(2,k) = A003946(k).
T(3,k) = A080923(k+1).
T(4,k) = A257970(k+3).
Columns:
T(n,0) = A001045(n) (Jacobsthal numbers J_{n}).
T(n,1) = A000079(n).
T(n,2) = A007283(n).
T(n,3) = A005010(n).
T(n,4) = A175806(n).
T(0,k) - T(k+1,0) = 4*A094705(k-2).
From G. C. Greubel, Mar 25 2021: (Start)
For the antidiagonal triangle:
t(n, k) = T(n-k, k).
t(n, k) = (2^(n-k) - (-1)^(n-k))/3 (J_{n-k}) if k = 0 else 2^(n-k)*3^(k-1).
Sum_{k=0..n} t(n, k) = 3^n - J_{n+1}, where J_{n} = A001045(n).
Sum_{k=0..n} t(n, k) = A004054(n-1) for n >= 1. (End)

a(22) - a(57) from Nathaniel Johnston, Apr 13 2011

A230445 Triangle read by rows: T(n,m) = 3^m*2^(n-m)-1, the number of neighbors in an n-dimensional cubic array.

Original entry on oeis.org

0, 1, 2, 3, 5, 8, 7, 11, 17, 26, 15, 23, 35, 53, 80, 31, 47, 71, 107, 161, 242, 63, 95, 143, 215, 323, 485, 728, 127, 191, 287, 431, 647, 971, 1457, 2186, 255, 383, 575, 863, 1295, 1943, 2915, 4373, 6560, 511, 767, 1151, 1727, 2591, 3887, 5831, 8747, 13121

Offset: 0

Ron R. King, Oct 18 2013

Let n be the dimension of the cubic array.
Let m be the "placement depth" of the cell within the array. m = (number of horizontal or vertical neighbors)-n. 0 <= m <= n.
Let T(n,m) represent the number of neighbors (horizontally, vertically, or diagonally) a cell has in an n-dimensional cube that has at least 3^n cells.
The sequence forms a triangle structure similar to Pascal’s triangle: T(0,0) in row one, T(1,0), T(1,1) in row two, etc.
The triangle in A094615 is a subtriangle. - Philippe Deléham, Oct 31 2013
In a finite n-dimensional hypercube lattice, the sequence gives the number of nodes situated at a Chebyshev distance of 1 for a node, situated on an m-cube bound, which is not on an (m-1)-cube bound. The number of m-cube bounds for n-cube is given by A013609. In cellular automata theory, the cell surrounding with Chebyshev distance 1 is called Moore's neighborhood. For von Neumann neighborhood (with Manhattan distance 1), an analogous sequence is represented by A051162. - Dmitry Zaitsev, Oct 22 2015

			Triangle starts:
n \ m  0    1    2    3    4    5     6     7     8     9    10 ...
0:     0
1:     1    2
2:     3    5    8
3:     7   11   17   26
4:    15   23   35   53   80
5:    31   47   71  107  161  242
6:    63   95  143  215  323  485   728
7:   127  191  287  431  647  971  1457  2186
8:   255  383  575  863 1295 1943  2915  4373  6560
9:   511  767 1151 1727 2591 3887  5831  8747 13121 19682
10: 1023 1535 2303 3455 5183 7775 11663 17495 26243 39365 59048
... (reformatted (and extended) by _Wolfdieter Lang_, May 04 2022)
For a 3-d cube, at a corner, the number of horizontal and vertical neighbors is 3, hence m = 3-3 = 0.
Along the edge, the number of horizontal and vertical neighbors is 4, hence m = 4-3 = 1.
In a face, the number of horizontal and vertical neighbors is 5, hence m = 5-3 = 2.
In the interior, the number of horizontal and vertical neighbors is 6, hence m = 6-3 = 3.
T(3,2) = 17 because a cell on the face of a 3-d cube has 17 neighbors.

Dmitry A. Zaitsev, A generalized neighborhood for cellular automata, Theoretical Computer Science, 2016, Volume 666, 1 March 2017, Pages 21-35.

Cf. A000225, A024023, A051162, A013609.
Sequence numbers are 1 less than A036561.
Cf. A048473, A094615.

void a10(){int p3[10], p2[10], n, m, a; p3[0]=1; p2[0]=1;
for(n=1;n<10;n++){ p2[n]=p2[n-1]*2; p3[n]=p3[n-1]*3;
  for(m=0;m<=d;m++){ a=p3[m]*p2[n-m]-1; printf("%d ",a); }
  printf("\n"); } } /* Dmitry Zaitsev, Oct 23 2015 */

Table[3^m 2^(n - m) - 1, {n, 0, 9}, {m, 0, n}] // Flatten (* Michael De Vlieger, Oct 23 2015 *)

T(n,m) = 3^m*2^(n-m)-1, 0 <= m <= n.
T(n,0) = 2^n-1. (A000225)
T(n,n) = 3^n-1. (A024023)
T(n,m) = (3*T(n,m-1)+1)/2, first part of the Collatz sequence for the number 2^n-1, for n >= 1.
T(n,m) = (T(n-1,m) + T(n,m+1))/2, 0 <= m <= n-1.
T(n,m) = 1 + T(n-1,m-1) + T(n,m-1), 1 <= m <= n.
m = T2(n,k)-n, where T2(n,k) is A051162.
From Wolfdieter Lang, May 04 2022: (Start)
G.f. for column m: G(m, x) = x^m*(3^m - 1 - (3^m - 2)*x)/((1 - 2*x)*(1 - x)).
G.f. for row polynomials R(n, x) = Sum_{m=1..n} T(n, m)*x^m, for n >= 0: G(z, x) = z*(1 + (2 - 5*z)*x)/((1 - 2*z)*(1 - z)*(1 - 3*x*z)*(1 - x*z)).
(End)

A232646 Sequence (or tree or triangle) generated by these rules: 1 is in S, and if x is in S, then 2x and 5x + 3 are in S, and duplicates are deleted as they occur.

Original entry on oeis.org

1, 2, 5, 4, 10, 25, 8, 20, 50, 125, 16, 40, 100, 250, 625, 32, 80, 200, 500, 1250, 3125, 64, 160, 400, 1000, 2500, 6250, 15625, 128, 320, 800, 2000, 5000, 12500, 31250, 78125, 256, 640, 1600, 4000, 10000, 25000, 62500, 156250, 390625, 512, 1280, 3200, 8000

Offset: 1

Clark Kimberling, Nov 28 2013

Let S be the set of numbers defined by these rules: 1 is in S, and if x is in S, then 2*x and 5*x 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) = (2,5), g(3) = (4,10,25), etc. Concatenating these gives A232646, a permutation of the positive integers. For n > 2, the number of numbers in g(n) is n. It is helpful to show the results as a tree with the terms of S as nodes, an edge from x to 2*x if 2*x has not already occurred, and an edge from x to 3*x if 3*x has not already occurred.

			Each x begets 2*x and 5*x, but if either has already occurred it is deleted.  Thus, 1 begets 2 and 5; then 2 begets 4 and 10, and 5 begets only 25, so that g(3) = (4,10,25).  Writing generations as rows results in a triangle whose first five rows are as follows:
1
2 .... 5
4 .... 10 ... 25
8 .... 20 ... 50 ... 125
16 ... 40 ... 100 .. 250 .. 625

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A232559, A232639, A036561.

x = {1}; Do[x = DeleteDuplicates[Flatten[Transpose[{x, 2*x, 5*x}]]], {12}]; x  (* Peter J. C. Moses, Nov 27 2013 *)

Counting the top row as row 0 and writing for (2^i)*(5*j) , the numbers in row n are , , ..., <0,n>.

cp's OEIS Frontend

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A232646 Sequence (or tree or triangle) generated by these rules: 1 is in S, and if x is in S, then 2x and 5x + 3 are in S, and duplicates are deleted as they occur.