A115770 -id:A115770 - cp's OEIS frontend

A115872 Square array where row n gives all solutions k > 0 to the cross-domain congruence n*k = A048720(A065621(n),k), zero sequence (A000004) if no such solutions exist.

1, 2, 1, 3, 2, 3, 4, 3, 6, 1, 5, 4, 7, 2, 7, 6, 5, 12, 3, 14, 3, 7, 6, 14, 4, 15, 6, 7, 8, 7, 15, 5, 28, 7, 14, 1, 9, 8, 24, 6, 30, 12, 15, 2, 15, 10, 9, 28, 7, 31, 14, 28, 3, 30, 7, 11, 10, 30, 8, 56, 15, 30, 4, 31, 14, 3, 12, 11, 31, 9, 60, 24, 31, 5, 60, 15, 6, 3, 13, 12, 48, 10, 62, 28, 56, 6, 62, 28, 12, 6, 5, 14, 13, 51, 11, 63, 30, 60, 7, 63, 30, 15, 7, 10, 7

Offset: 1

Antti Karttunen, Feb 07 2006

Here * stands for ordinary multiplication and X means carryless (GF(2)[X]) multiplication (A048720).
Square array is read by descending antidiagonals, as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
Rows at positions 2^k are 1, 2, 3, ..., (A000027). Row 2n is equal to row n.
Numbers on each row give a subset of positions of zeros at the corresponding row of A284270. - Antti Karttunen, May 08 2019

			Fifteen initial terms of rows 1 - 19 are listed below:
   1:  1,  2,  3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,  14,  15, ...
   2:  1,  2,  3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,  14,  15, ...
   3:  3,  6,  7,  12,  14,  15,  24,  28,  30,  31,  48,  51,  56,  60,  62, ...
   4:  1,  2,  3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,  14,  15, ...
   5:  7, 14, 15,  28,  30,  31,  56,  60,  62,  63, 112, 120, 124, 126, 127, ...
   6:  3,  6,  7,  12,  14,  15,  24,  28,  30,  31,  48,  51,  56,  60,  62, ...
   7:  7, 14, 15,  28,  30,  31,  56,  60,  62,  63, 112, 120, 124, 126, 127, ...
   8:  1,  2,  3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,  14,  15, ...
   9: 15, 30, 31,  60,  62,  63, 120, 124, 126, 127, 240, 248, 252, 254, 255, ...
  10:  7, 14, 15,  28,  30,  31,  56,  60,  62,  63, 112, 120, 124, 126, 127, ...
  11:  3,  6, 12,  15,  24,  27,  30,  31,  48,  51,  54,  60,  62,  63,  96, ...
  12:  3,  6,  7,  12,  14,  15,  24,  28,  30,  31,  48,  51,  56,  60,  62, ...
  13:  5, 10, 15,  20,  21,  30,  31,  40,  42,  45,  47,  60,  61,  62,  63, ...
  14:  7, 14, 15,  28,  30,  31,  56,  60,  62,  63, 112, 120, 124, 126, 127, ...
  15: 15, 30, 31,  60,  62,  63, 120, 124, 126, 127, 240, 248, 252, 254, 255, ...
  16:  1,  2,  3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,  14,  15, ...
  17: 31, 62, 63, 124, 126, 127, 248, 252, 254, 255, 496, 504, 508, 510, 511, ...
  18: 15, 30, 31,  60,  62,  63, 120, 124, 126, 127, 240, 248, 252, 254, 255, ...
  19:  7, 14, 28,  31,  56,  62,  63, 112, 119, 124, 126, 127, 224, 238, 248, ...

Transpose: A114388. First column: A115873.
Cf. A048720, A065621, A115857, A115871, A325565, A325566, A325567, A325568, A325570, A325571.
Cf. also arrays A277320, A277810, A277820, A284270.
A few odd-positioned rows: row 1: A000027, Row 3: A048717, Row 5: A115770 (? Checked for all values less than 2^20), Row 7: A115770, Row 9: A115801, Row 11: A115803, Row 13: A115772, Row 15: A115801 (? Checked for all values less than 2^20), Row 17: A115809, Row 19: A115874, Row 49: A114384, Row 57: A114386.

X[a_, b_] := Module[{A, B, C, x},
     A = Reverse@IntegerDigits[a, 2];
     B = Reverse@IntegerDigits[b, 2];
     C = Expand[
        Sum[A[[i]]*x^(i-1), {i, 1, Length[A]}]*
        Sum[B[[i]]*x^(i-1), {i, 1, Length[B]}]];
     PolynomialMod[C, 2] /. x -> 2];
T[n_, k_] := Module[{x = BitXor[n-1, 2n-1], k0 = k},
     For[i = 1, True, i++, If[n*i == X[x, i],
     If[k0 == 1, Return[i], k0--]]]];
Table[T[n-k+1, k], {n, 1, 14}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jan 04 2022 *)

up_to = 120;
A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1,n+n-1);
A115872sq(n, k) = { my(x = A065621(n)); for(i=1,oo,if((n*i)==A048720(x,i),if(1==k,return(i),k--))); };
A115872list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A115872sq(col,(a-(col-1))))); (v); };
v115872 = A115872list(up_to);
A115872(n) = v115872[n]; \\ (Slow) - Antti Karttunen, May 08 2019

Example section added and the data section extended up to n=105 by Antti Karttunen, May 08 2019

A115772 Integers i such that 13*i = A048720bi(21,i).

Original entry on oeis.org

0, 5, 10, 15, 20, 21, 30, 31, 40, 42, 45, 47, 60, 61, 62, 63, 80, 84, 85, 90, 94, 95, 120, 122, 124, 125, 126, 127, 160, 165, 168, 170, 173, 175, 180, 181, 188, 189, 190, 191, 240, 244, 245, 248, 250, 252, 253, 254, 255, 320, 330, 336, 340, 341, 346, 350, 351

Offset: 0

Antti Karttunen, Jan 30 2006

nonn

Here * stands for ordinary multiplication and A048720 is the carryless (GF(2)[X]) multiplication.

a(n) appears to be the set of all n that can be expressed as x OR 4x for the bitwise OR operation. [From Gary Detlefs Dec 20 2010]

Index entries for sequences defined by congruent products between domains N and GF(2)[X]

Row 13 of A115872. Cf. A048717, A115767, A115770. Superset of A115774 ? A115776 gives the terms which are not in A115774. A115773 shows this sequence in binary.

Cf. A178891

A115774 Integers i such that 15*i = A048720bi(23,i).

Original entry on oeis.org

0, 5, 10, 20, 21, 40, 42, 80, 84, 85, 160, 168, 170, 320, 336, 340, 341, 640, 645, 672, 680, 682, 1280, 1285, 1290, 1344, 1360, 1364, 1365, 2560, 2565, 2570, 2580, 2581, 2688, 2693, 2720, 2728, 2730, 5120, 5125, 5130, 5140, 5141, 5160, 5162, 5376

Offset: 0

Antti Karttunen, Jan 30 2006

nonn

Here * stands for ordinary multiplication and A048720 is the carryless (GF(2)[X]) multiplication.

Index entries for sequences defined by congruent products between domains N and GF(2)[X]

Cf. A048717, A115767, A115770. Subset of A115772 ? A115776 gives the terms of A115772 which do not occur here. Differs from A062052 for the first time at n=18, where A115774(18)=645 while A062052(18)=672. A115775 shows this sequence in binary.

A085588 Eventual period of a single cell in rule 150 cellular automaton in a cyclic universe of width n.

Original entry on oeis.org

1, 2, 3, 1, 7, 4, 7, 6, 31, 2, 21, 14, 15, 8, 15, 14, 511, 12, 63, 62, 2047, 4, 1023, 42, 511, 28, 16383, 30, 31, 16, 31, 30, 4095, 28, 29127, 1022, 4095, 24, 1023, 126, 127, 124, 4095, 4094, 8388607, 8, 2097151, 2046, 255, 84, 67108863, 1022, 1048575, 56, 511, 32766, 536870911, 60, 17043521, 62, 63, 32, 63, 62

Offset: 3

N. J. A. Sloane, Jul 03 2003

nonn

From Roman Khrabrov, Aug 17 2024: (Start)
It appears that 2^A007814(n) * (2^A309786(n) - 1) divides a(n). For rule 90, it follows from Lemma 3.5 and Theorem 3.5 from Martin & Odlyzko & Wolfram's paper, and the definition of A309786. Rule 150 appears to have the same behavior (verified for n <= 1000).
The numbers for which a(n) differs from 2^A007814(n) * (2^A309786(n) - 1), are the powers of 2 and the numbers in the form 6*2^k, 13*2^k, 37*2^k, 61*2^k, 67*2^k, 95*2^k and so on (there is no corresponding OEIS sequence).
It seems that in 2D case (totalistic rule 34 on a toroidal grid) the formula 2^A007814(n) * (2^A309786(n) - 1) gives the correct maximum cycle lengths in all cases except powers of 2. Replacing A007814(n) with A091090(n) appears to always provide the correct maximum cycle lengths, even at powers of 2.
Conjecture: a(n) = n only if n belongs to A115770. The inverse does not hold true in general; the first exception is 445. (End)

O. Martin, A. M. Odlyzko and S. Wolfram, Algebraic properties of cellular automata, Comm Math. Physics, 93 (1984), pp. 219-258, Reprinted in Theory and Applications of Cellular Automata, S Wolfram, Ed., World Scientific, 1986, pp. 51-90 and in Cellular Automata and Complexity: Collected Papers of Stephen Wolfram, Addison-Wesley, 1994, pp. 71-113 See Table 1.

Roman Khrabrov, Table of n, a(n) for n = 3..1000
Shin-ichi Tadaki, Orbits in one-dimensional finite linear cellular automata, arXiv:cond-mat/9305012, 1993.

Cf. A085587-A085595.

More terms from Sean A. Irvine, Jun 10 2018

Name clarified by Roman Khrabrov, Aug 17 2024

cp's OEIS Frontend

A115771 Sequence A115770 in binary.

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A115872 Square array where row n gives all solutions k > 0 to the cross-domain congruence n*k = A048720(A065621(n),k), zero sequence (A000004) if no such solutions exist.

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A115772 Integers i such that 13*i = A048720bi(21,i).

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A115774 Integers i such that 15*i = A048720bi(23,i).

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A085588 Eventual period of a single cell in rule 150 cellular automaton in a cyclic universe of width n.

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