A151779 -id:A151779 - cp's OEIS frontend

A151781 Partial sums of A151779.

1, 7, 13, 43, 49, 79, 109, 259, 265, 295, 325, 475, 505, 655, 805, 1555, 1561, 1591, 1621, 1771, 1801, 1951, 2101, 2851, 2881, 3031, 3181, 3931, 4081, 4831, 5581, 9331, 9337, 9367, 9397, 9547, 9577, 9727, 9877, 10627, 10657, 10807, 10957, 11707, 11857, 12607

Offset: 1

N. J. A. Sloane, Jun 25 2009

Total number of ON cells after n-th generation of cellular automaton based on Z^3 lattice in the same way that A147562 is based on the Z^2 lattice. Here each cell has six neighbors.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 32-33.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A147562, A151779, A151790, A151792, A151793.

a[n_] := 6*5^(Total@ IntegerDigits[n - 1, 2] - 1); a[1] = 1; Accumulate@ Array[a, 46] (* Michael De Vlieger, Oct 31 2022 *)

a(n)=sum(k=1,n,6*5^(hammingweight(k-1)-1)\1) \\ Charles R Greathouse IV, Sep 14 2015

A147582 First differences of A147562.

Original entry on oeis.org

1, 4, 4, 12, 4, 12, 12, 36, 4, 12, 12, 36, 12, 36, 36, 108, 4, 12, 12, 36, 12, 36, 36, 108, 12, 36, 36, 108, 36, 108, 108, 324, 4, 12, 12, 36, 12, 36, 36, 108, 12, 36, 36, 108, 36, 108, 108, 324, 12, 36, 36, 108, 36, 108, 108, 324, 36, 108, 108, 324, 108, 324, 324, 972, 4

Offset: 1

N. J. A. Sloane, Apr 29 2009

nonn

Bisection of A323651. - Omar E. Pol, Mar 04 2019

			From _Omar E. Pol_, Jun 14 2009: (Start)
When written as a triangle:
.1;
.4;
.4,12;
.4,12,12,36;
.4,12,12,36,12,36,36,108;
.4,12,12,36,12,36,36,108,12,36,36,108,36,108,108,324;
.4,12,12,36,12,36,36,108,12,36,36,108,36,108,108,324,12,36,36,108,36,108,...
The rows converge to A161411. (End)

D. Singmaster, On the cellular automaton of Ulam and Warburton, M500 Magazine of the Open University, #195 (December 2003), pp. 2-7.
S. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4*3^(wt(n-1)-1) for n >= 2.]
David Applegate, The movie version
Omar E. Pol, Illustration of initial terms (Fig. 1: one-step rook), (Fig. 2: one-step bishop), (Fig. 3: overlapping squares), (Fig. 4: overlapping X-toothpicks), 2009
Omar E. Pol, Illustration of initial terms of A139251, A160121, A147582 (Overlapping figures), 2009
D. Singmaster, On the cellular automaton of Ulam and Warburton, 2003 [Cached copy, included with permission]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021

Cf. A147562, A147610 (the sequence divided by 4), A048881, A000120.
Cf. A000079, A161411, A151779, A139250.
Cf. A048883, A139251, A160121, A162349. [Omar E. Pol, Nov 02 2009]
Cf. A323651.

A000120 := proc(n) local w,m,i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end: wt := A000120; A147582 := n-> if n <= 1 then n else 4*3^(wt(n-1)-1); fi; [seq(A147582(n),n=0..1000)]; # N. J. A. Sloane, Apr 07 2010

s = Plus @@ Flatten@ # & /@ CellularAutomaton[{686, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, 200]; f[n_] = If[n == 0, 1, s[[n + 1]] - s[[n]]]; Array[f, 120, 0] (* Michael De Vlieger, Apr 09 2015, after Nadia Heninger and N. J. A. Sloane at A147562 *)

a(1) = 1; for n > 1, a(n) = 4*3^(wt(n-1)-1) where wt() = A000120(). - R. J. Mathar, Apr 30 2009
This formula is (essentially) given by Singmaster. - N. J. A. Sloane, Aug 06 2009
G.f.: x + 4*x*(Product_{k >= 0} (1 + 3*x^(2^k)) - 1)/3. - N. J. A. Sloane, Jun 10 2009

Extended by R. J. Mathar, Apr 30 2009

A255740 Square array read by antidiagonals upwards: T(n,1) = 1; for k > 1, T(n,k) = (n-1)*(n-2)^(A000120(k-1)-1) with n >= 1.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 2, 0, 0, 1, 4, 3, 2, 1, 0, 1, 5, 4, 6, 2, 0, 0, 1, 6, 5, 12, 3, 2, 0, 0, 1, 7, 6, 20, 4, 6, 2, 0, 0, 1, 8, 7, 30, 5, 12, 6, 2, 1, 0, 1, 9, 8, 42, 6, 20, 12, 12, 2, 0, 0, 1, 10, 9, 56, 7, 30, 20, 36, 3, 2, 0, 0, 1, 11, 10, 72, 8, 42, 30, 80, 4, 6, 2, 0, 0, 1, 12, 11, 90, 9, 56, 42, 150, 5, 12, 6, 2, 0, 0

Offset: 1

Omar E. Pol, Mar 05 2015

The partial sums of row n give the n-th row of the square array A255741.

			The corner of the square array with the first 16 terms of the first 12 rows looks like this:
-------------------------------------------------------------------------
A000007: 1, 0, 0,  0, 0,  0,  0,   0, 0,  0,  0,   0,  0,   0,   0,    0
A255738: 1, 1, 1,  0, 1,  0,  0,   0  1,  0,  0,   0,  0,   0,   0,    0
A040000: 1, 2, 2,  2, 2,  2,  2,   2, 2,  2,  2,   2,  2,   2,   2,    2
A151787: 1, 3, 3,  6, 3,  6,  6,  12, 3,  6,  6,  12,  6,  12,  12,   24
A147582: 1, 4, 4, 12, 4, 12, 12,  36, 4, 12, 12,  36, 12,  36,  36,  108
A151789: 1, 5, 5, 20, 5, 20, 20,  80, 5, 20, 20,  80, 20,  80,  80,  320
A151779: 1, 6, 6, 30, 6, 30, 30, 150, 6, 30, 30, 150, 30, 150, 150,  750
A151791: 1, 7, 7, 42, 7, 42, 42, 252, 7, 42, 42, 252, 42, 252, 252, 1512
A151782: 1, 8, 8, 56, 8, 56, 56, 392, 8, 56, 56, 392, 56, 392, 392, 2744
A255743: 1, 9, 9, 72, 9, 72, 72, 576, 9, 72, 72, 576, 72, 576, 576, 4608
A255744: 1,10,10, 90,10, 90, 90, 810,10, 90, 90, 810, 90, 810, 810, 7290
A255745: 1,11,11,110,11,110,110,1100,11,110,110,1100,110,1100,1100,11000
...

Index entries for sequences related to cellular automata

Cf. A000120, A255741.
Rows 1-12: A000007, A255738, A040000, A151787, A147582, A151789, A151779, A151791, A151782, A255743, A255744, A255745.
Column 1 is A000012.
Columns 2^k+1, for k >=0: A011477.
Columns 4, 6, 7, 10, 11, 13...: 0 together with A002378.

tabl(nn) = {for (n=1, nn, for (k=1, nn, if (k==1, x = 1, x= (n-1)*(n-2)^(hammingweight(k-1)-1)); print1(x, ", ");); print(););} \\ Michel Marcus, Mar 15 2015

T(n,1) = 1; for k > 1, T(n,k) = (n-1)*(n-2)^(A000120(k-1)-1) with n >= 1.

A151787 a(1)=1; for n > 1, a(n)=3*2^{wt(n-1)-1}.

Original entry on oeis.org

1, 3, 3, 6, 3, 6, 6, 12, 3, 6, 6, 12, 6, 12, 12, 24, 3, 6, 6, 12, 6, 12, 12, 24, 6, 12, 12, 24, 12, 24, 24, 48, 3, 6, 6, 12, 6, 12, 12, 24, 6, 12, 12, 24, 12, 24, 24, 48, 6, 12, 12, 24, 12, 24, 24, 48, 12, 24, 24, 48, 24, 48, 48, 96, 3, 6, 6, 12, 6, 12, 12, 24, 6, 12, 12, 24, 12, 24, 24, 48

Offset: 1

N. J. A. Sloane, Jun 25 2009

nonn

wt(n) is the Hamming weight = binary weight of n (A000120).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000120, A001316, A147582, A151779.

wt:= n -> convert(convert(n,base,2),`+`):
[1, seq(3*2^(wt(n-1)-1),n=2..100)]; # Robert Israel, Feb 27 2018

a[n_] := If[n == 1, 1, 3*2^(Total[IntegerDigits[n-1, 2]]-1)];
Array[a, 100] (* Jean-François Alcover, Mar 27 2019 *)

a(n) = if (n==1, 1, 3*2^(hammingweight(n-1)-1)); \\ Michel Marcus, Feb 27 2018

From Robert Israel, Feb 27 2018: (Start)
For n > 1, a(2*n)=2*a(n).
a(2*n+1)=a(n+1).
G.f. g(x) satisfies g(x) = (2+1/x)*g(x^2) + x^2. (End)
a(n) = 3*A001316(n-1)/2 for n >= 2. - Georg Fischer, Jun 23 2021

Definition corrected by Omar E. Pol, Mar 03 2015

A151780 a(n) = 5^(wt(n) - 1) where wt(n) = A000120(n).

Original entry on oeis.org

1, 1, 5, 1, 5, 5, 25, 1, 5, 5, 25, 5, 25, 25, 125, 1, 5, 5, 25, 5, 25, 25, 125, 5, 25, 25, 125, 25, 125, 125, 625, 1, 5, 5, 25, 5, 25, 25, 125, 5, 25, 25, 125, 25, 125, 125, 625, 5, 25, 25, 125, 25, 125, 125, 625, 25, 125, 125, 625, 125, 625, 625, 3125, 1, 5, 5, 25, 5, 25, 25, 125, 5

Offset: 1

N. J. A. Sloane, Jun 25 2009

nonn

			From _Omar E. Pol_, Jul 21 2009: (Start)
If written as a triangle:
  1;
  1,5;
  1,5,5,25;
  1,5,5,25,5,25,25,125;
  1,5,5,25,5,25,25,125,5,25,25,125,25,125,125,625;
  1,5,5,25,5,25,25,125,5,25,25,125,25,125,125,625,5,25,25,125,25,125,125,625,...
(End)

Essentially A151779/6.
Cf. A000120, A048881, A048896, A147610, A151783.
Cf. A000351. - Omar E. Pol, Jul 21 2009

a(n) = 5^(hammingweight(n)-1); \\ Michel Marcus, Nov 15 2022

A151782 a(1)=1; for n > 1, a(n)=8*7^{wt(n-1)-1}.

Original entry on oeis.org

1, 8, 8, 56, 8, 56, 56, 392, 8, 56, 56, 392, 56, 392, 392, 2744, 8, 56, 56, 392, 56, 392, 392, 2744, 56, 392, 392, 2744, 392, 2744, 2744, 19208, 8, 56, 56, 392, 56, 392, 392, 2744, 56, 392, 392, 2744, 392, 2744, 2744, 19208, 56, 392, 392, 2744, 392, 2744, 2744, 19208, 392

Offset: 1

N. J. A. Sloane, Jun 25 2009

nonn

Cf. A151779, A151780, etc.

A255743 a(1) = 1; for n > 1, a(n) = 9*8^{A000120(n-1)-1}.

Original entry on oeis.org

1, 9, 9, 72, 9, 72, 72, 576, 9, 72, 72, 576, 72, 576, 576, 4608, 9, 72, 72, 576, 72, 576, 576, 4608, 72, 576, 576, 4608, 576, 4608, 4608, 36864, 9, 72, 72, 576, 72, 576, 576, 4608, 72, 576, 576, 4608, 576, 4608, 4608, 36864, 72, 576, 576, 4608, 576, 4608, 4608

Offset: 1

Omar E. Pol, Mar 05 2015

nonn

Also, this is a row of the square array A255740.

Partial sums give A255764.

			Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
   1;
   9;
   9, 72;
   9, 72, 72, 576;
   9, 72, 72, 576, 72, 576, 576, 4608;
   ...

Felix Fröhlich, Table of n, a(n) for n = 1..10000
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 33.

Cf. A000120, A151787, A147582, A151789, A151779, A151791, A151782, A255740, A255741, A255744, A255745, A255764.

MapAt[Floor, Array[9*8^(DigitCount[# - 1, 2, 1] - 1) &, 55], 1] (* Michael De Vlieger, Nov 03 2022 *)

a(n) = if (n==1, 1, 9*8^(hammingweight(n-1)-1)); \\ Michel Marcus, Mar 15 2015

# Python 3.10+
def A255743(n): return 1 if n == 1 else 9*(1<<((n-1).bit_count()-1)*3) # Chai Wah Wu, Nov 15 2022

More terms from Michel Marcus, Mar 15 2015

A255744 a(1) = 1; for n > 1, a(n) = 10*9^(A000120(n-1)-1).

Original entry on oeis.org

1, 10, 10, 90, 10, 90, 90, 810, 10, 90, 90, 810, 90, 810, 810, 7290, 10, 90, 90, 810, 90, 810, 810, 7290, 90, 810, 810, 7290, 810, 7290, 7290, 65610, 10, 90, 90, 810, 90, 810, 810, 7290, 90, 810, 810, 7290, 810, 7290, 7290, 65610, 90, 810, 810, 7290, 810, 7290

Offset: 1

Omar E. Pol, Mar 05 2015

nonn

Also, this is a row of the square array A255740.

Partial sums give A255765.

			Also, written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
1;
10;
10, 90;
10, 90, 90, 810;
10, 90, 90, 810, 90, 810, 810, 7290;
...

Felix Fröhlich, Table of n, a(n) for n = 1..10000

Cf. A000120, A151787, A147582, A151789, A151779, A151791, A151782, A255740, A255741, A255743, A255745, A255765.

MapAt[Floor, Array[10*9^(DigitCount[# - 1, 2, 1] - 1) &, 54], 1] (* Michael De Vlieger, Nov 03 2022 *)

a(n) = if (n==1, 1, 10*9^(hammingweight(n-1)-1)); \\ Michel Marcus, Mar 15 2015

More terms from Michel Marcus, Mar 15 2015

A255745 a(1) = 1; for n > 1, a(n) = 11*10^{A000120(n-1)-1}.

Original entry on oeis.org

1, 11, 11, 110, 11, 110, 110, 1100, 11, 110, 110, 1100, 110, 1100, 1100, 11000, 11, 110, 110, 1100, 110, 1100, 1100, 11000, 110, 1100, 1100, 11000, 1100, 11000, 11000, 110000, 11, 110, 110, 1100, 110, 1100, 1100, 11000, 110, 1100, 1100, 11000, 1100, 11000, 11000

Offset: 1

Omar E. Pol, Mar 05 2015

nonn

Also, this is a row of the square array A255740.
Partial sums give A255766.
Is this also A151787 written in base 2?

			Also, written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
1;
11;
11, 110;
11, 110, 110, 1100;
11, 110, 110, 1100, 110, 1100, 1100, 11000;
...

Cf. A000120, A151787, A147582, A151789, A151779, A151791, A151782, A255740, A255741, A255743, A255744, A255766.

a(n) = if (n==1, 1, 11*10^(hammingweight(n-1)-1)); \\ Michel Marcus, Mar 13 2015

More terms from Michel Marcus, Mar 13 2015

A186411 First differences of A186410.

Original entry on oeis.org

0, 1, 5, 5, 21, 5, 21, 21, 101, 5, 21, 21, 101, 21, 101, 101, 501, 5, 21, 21, 101, 21, 101, 101, 501, 21, 101, 101, 501, 101, 501, 501, 2501, 5, 21, 21, 101, 21, 101, 101, 501, 21, 101, 101, 501, 101, 501, 501, 2501, 21, 101, 101, 501, 101, 501, 501, 2501, 101

Offset: 0

Omar E. Pol, Feb 21 2011

nonn

Number of cells turned "ON" at h-th stage of the three-dimensional cellular automaton of A186410. In other words: number of cubes added at n-th stage to the structure of A186410.

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS.
Index entries for sequences related to cellular automata.

Cf. A000120, A147582, A151779, A186410.

a[n_] := 1 + 4*5^(DigitCount[n-1, 2, 1]-1); a[0] = 0; a[1] = 1; Array[a, 100, 0] (* Amiram Eldar, Aug 01 2023 *)

a(n) = 1 + 4*5^(A000120(n-1)-1), n >= 2. - Nathaniel Johnston, Mar 22 2011

More terms from Nathaniel Johnston, Mar 22 2011

More terms from Amiram Eldar, Aug 01 2023

cp's OEIS Frontend

A151781 Partial sums of A151779.

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A147582 First differences of A147562.

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A255740 Square array read by antidiagonals upwards: T(n,1) = 1; for k > 1, T(n,k) = (n-1)*(n-2)^(A000120(k-1)-1) with n >= 1.

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A151787 a(1)=1; for n > 1, a(n)=3*2^{wt(n-1)-1}.

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A151780 a(n) = 5^(wt(n) - 1) where wt(n) = A000120(n).

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A151782 a(1)=1; for n > 1, a(n)=8*7^{wt(n-1)-1}.

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A255743 a(1) = 1; for n > 1, a(n) = 9*8^{A000120(n-1)-1}.

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A255744 a(1) = 1; for n > 1, a(n) = 10*9^(A000120(n-1)-1).

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