A151781 -id:A151781 - cp's OEIS frontend

A151779 a(1)=1; for n > 1, a(n)=6*5^{wt(n-1)-1}.

1, 6, 6, 30, 6, 30, 30, 150, 6, 30, 30, 150, 30, 150, 150, 750, 6, 30, 30, 150, 30, 150, 150, 750, 30, 150, 150, 750, 150, 750, 750, 3750, 6, 30, 30, 150, 30, 150, 150, 750, 30, 150, 150, 750, 150, 750, 750, 3750, 30, 150, 150, 750, 150, 750, 750, 3750, 150, 750, 750, 3750

Offset: 1

N. J. A. Sloane, Jun 25 2009

Number of cells turned ON in 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.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A147582, A151780, A151781, A151782.

wt := 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:
f:=d->[seq((2*d)*(2*d-1)^(wt(n-1)-1),n=2..120)];
f2:=d->[1,op(f(d))];
f2(3);

a(n)=6*5^(hammingweight(n-1)-1)\1 \\ Charles R Greathouse IV, Mar 07 2015

A255741 Square array read by antidiagonals upwards: T(n,k), n>=1, k>=1, in which row n lists the partial sums of the n-th row of the square array of A255740.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 3, 1, 1, 5, 7, 7, 4, 1, 1, 6, 9, 13, 9, 4, 1, 1, 7, 11, 21, 16, 11, 4, 1, 1, 8, 13, 31, 25, 22, 13, 4, 1, 1, 9, 15, 43, 36, 37, 28, 15, 5, 1, 1, 10, 17, 57, 49, 56, 49, 40, 17, 5, 1, 1, 11, 19, 73, 64, 79, 76, 85, 43, 19, 5, 1, 1, 12, 21, 91, 81, 106, 109, 156, 89, 49, 21, 5, 1

Offset: 1

Omar E. Pol, Mar 05 2015

			The corner of the square array with the first 15 terms of the first 12 rows looks like this:
-------------------------------------------------------------------------
A000012: 1, 1, 1,  1,  1,  1,  1,   1,   1,   1,   1,   1,   1,   1,   1
A070941: 1, 2, 3,  3,  4,  4,  4,   4,   5,   5,   5,   5,   5,   5,   5
A005408: 1, 3, 5,  7,  9, 11, 13,  15,  17,  19,  21,  23,  25,  27,  29
A151788: 1, 4, 7, 13, 16, 22, 28,  40,  43,  49,  55,  67,  73,  85,  97
A147562: 1, 5, 9, 21, 25, 37, 49,  85,  89, 101, 113, 149, 161, 197, 233
A151790: 1, 6,11, 31, 36, 56, 76, 156, 161, 181, 201, 281, 301, 381, 461
A151781: 1, 7,13, 43, 49, 79,109, 259, 265, 295, 325, 475, 505, 655, 805
A151792: 1, 8,15, 57, 64,106,148, 400, 407, 449, 491, 743, 785,1037,1289
A151793: 1, 9,17, 73, 81,137,193, 585, 593, 649, 705,1097,1153,1545,1937
A255764: 1,10,19, 91,100,172,244, 820, 829, 901, 973,1549,1621,2197,2773
A255765: 1,11,21,111,121,211,301,1111,1121,1211,1301,2111,2201,3011,3821
A255766: 1,12,23,133,144,254,364,1464,1475,1585,1695,2795,2905,4005,5105
...

Index entries for sequences related to cellular automata

Rows 1-12: A000012, A070941, A005408, A151788, A147562, A151790, A151781, A151792, A151793, A255764, A255765, A255766.
Columns 1-10: A000012, A000027, A005408, A002061, A000290, A084849, A056107, A053698, A100705, A100104.
Cf. A000120, A255740.

A255764 Partial sums of A255743.

Original entry on oeis.org

1, 10, 19, 91, 100, 172, 244, 820, 829, 901, 973, 1549, 1621, 2197, 2773, 7381, 7390, 7462, 7534, 8110, 8182, 8758, 9334, 13942, 14014, 14590, 15166, 19774, 20350, 24958, 29566, 66430, 66439, 66511, 66583, 67159, 67231, 67807, 68383, 72991, 73063, 73639, 74215

Offset: 1

Omar E. Pol, Mar 05 2015

nonn

Also, this is a row of the square array A255741.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
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. A005408, A151788, A147562, A151790, A151781, A151792, A151793, A255740, A255741, A255743, A255764, A255765, A255766.

Accumulate@ MapAt[Floor, Array[9*8^(DigitCount[# - 1, 2, 1] - 1) &, 43], 1] (* Michael De Vlieger, Nov 03 2022 *)

lista(nn) = {s = 1; for (n=2, nn, print1(s, ", "); s += 9*8^(hammingweight(n-1)-1););} \\ Michel Marcus, Mar 15 2015

More terms from Michel Marcus, Mar 15 2015

A255765 Partial sums of A255744.

Original entry on oeis.org

1, 11, 21, 111, 121, 211, 301, 1111, 1121, 1211, 1301, 2111, 2201, 3011, 3821, 11111, 11121, 11211, 11301, 12111, 12201, 13011, 13821, 21111, 21201, 22011, 22821, 30111, 30921, 38211, 45501, 111111, 111121, 111211, 111301, 112111, 112201, 113011, 113821, 121111

Offset: 1

Omar E. Pol, Mar 05 2015

nonn

Also, this is a row of the square array A255741.

Is this sequence related to positive repunits? (see formula section).

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. A002275, A005408, A151788, A147562, A151790, A151781, A151792, A151793, A255740, A255741, A255744, A255764, A255766.

Accumulate@ MapAt[Floor, Array[10*9^(DigitCount[# - 1, 2, 1] - 1) &, 40], 1] (* Michael De Vlieger, Nov 03 2022 *)

lista(nn) = {s = 1; for (n=2, nn, print1(s, ", "); s += 10*9^(hammingweight(n-1)-1););} \\ Michel Marcus, Mar 15 2015

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

Question: a(2^k) = A002275(k+1), k >= 0. Is this true?

More terms from Michel Marcus, Mar 15 2015

A255766 Partial sums of A255745.

Original entry on oeis.org

1, 12, 23, 133, 144, 254, 364, 1464, 1475, 1585, 1695, 2795, 2905, 4005, 5105, 16105, 16116, 16226, 16336, 17436, 17546, 18646, 19746, 30746, 30856, 31956, 33056, 44056, 45156, 56156, 67156, 177156, 177167, 177277, 177387, 178487, 178597, 179697, 180797, 191797

Offset: 1

Omar E. Pol, Mar 05 2015

nonn

Also, this is a row of the square array A255741.

Cf. A005408, A151788, A147562, A151790, A151781, A151792, A151793, A255740, A255741, A255745, A255764, A255765.

a(n) = sum(k=1, n, if (k==1, 1, 11*10^(hammingweight(k-1)-1)));

More terms from Michel Marcus, Mar 19 2015

A186410 Number of "ON" cells at n-th stage of three-dimensional version of the cellular automaton A183060 using cubes.

Original entry on oeis.org

0, 1, 6, 11, 32, 37, 58, 79, 180, 185, 206, 227, 328, 349, 450, 551, 1052, 1057, 1078, 1099, 1200, 1221, 1322, 1423, 1924, 1945, 2046, 2147, 2648, 2749, 3250, 3751, 6252, 6257, 6278, 6299, 6400, 6421, 6522, 6623

Offset: 0

Omar E. Pol, Feb 21 2011

nonn

The sequence gives the total number of cells turned ON after n stages in a cellular automaton based on Z^3 lattice in the same way that A183060 is based on the Z^2 lattice. In general here each cell has six neighbors.

It appears that after 2^k stages the structure resembles a pyramid. For the first differences see A186411.

Michael De Vlieger, Table of n, a(n) for n = 0..1000
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
Index entries for sequences related to cellular automata

Cf. A139250, A147562, A151781, A183060.

a[n_] := n + (4/5) Sum[5^DigitCount[i, 2, 1], {i, n - 1}]; Array[a, 40, 0] (* Michael De Vlieger, Nov 02 2022 *)

From Nathaniel Johnston, Mar 14 2011: (Start)
a(n) = n + (4/5)*(Sum_{i=1..n-1} 5^A000120(i)).
a(2^n) = 2^n + (4/5)*(6^n - 1).
(End)

More terms from Nathaniel Johnston, Mar 14 2011

cp's OEIS Frontend

A151779 a(1)=1; for n > 1, a(n)=6*5^{wt(n-1)-1}.

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A255741 Square array read by antidiagonals upwards: T(n,k), n>=1, k>=1, in which row n lists the partial sums of the n-th row of the square array of A255740.

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A255764 Partial sums of A255743.

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A255765 Partial sums of A255744.

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A255766 Partial sums of A255745.

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A186410 Number of "ON" cells at n-th stage of three-dimensional version of the cellular automaton A183060 using cubes.

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Mathematica

Formula

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