Darrah Chavey - cp's OEIS frontend

User: Darrah Chavey

Darrah Chavey has authored 3 sequences.

A323711 Numbers k such that k, 2k, and 3k are anagrams of each other.

142857, 285714, 1402857, 1428570, 1428597, 1429857, 2857014, 2857140, 2859714, 2985714, 14002857, 14028570, 14028597, 14029857, 14285700, 14285970, 14285997, 14298570, 14298597, 14299857, 15623784, 15843762, 17438256, 17562438, 18243756, 21584376, 23784156, 24375618, 24381756

Offset: 1

Darrah Chavey, Jan 24 2019

We assume entries have no leading zeros, so that n = 53617824 is not in the sequence, even though 2*n = 107235648 and 3*n = 160853472 are anagrams of 053617824.
From Chai Wah Wu, Feb 01 2019: (Start)
The first digit of terms is either 1, 2 or 3. Numbers of the form 140..028570..0 and 29..98570..0140..0 are terms where the number of 9's and 0's can be zero.
More generally, let a number n be written in decimal as xxxzzz where x and z are arbitrary digits and xxx, zzz are not empty strings. Let m be the number that is written as zzz in decimal and k be the least power of 10 that is strictly greater than m.
If 3*m < k, then n is a term if and only if xxx0..0zzz0..0 is a term. Note that this condition is satisfied if the first digit of m is 0, 1 or 2.
If 2*k <= 3*m, then n is a term if and only if xxx9..9zzz0..0 is a term. Note that this condition is satisfied if the first digit of m is 7, 8, or 9.
Not all terms with digits 0 and 9 are formed this way, see for instance the terms 137965842 and 157836042.
The first term where the first digit is 3 is a(1507) = 3051267489.
(End)
From David A. Corneth, Feb 02 2019: (Start)
Terms are multiples of 9.
Proof: as 3*k and k have the same digits, k is divisible by 3. If k isn't divisible by 9 then it has a different digital sum from 3*k. Therefore, k is divisible by 9. (End)

			The first entry, 142857, is well known for having n, 2*n, 3*n, 4*n, 5*n and 6*n all being anagrams. The next two numbers for which that happens are 1428570 and 1429857.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Subsequence of A023086, numbers where n and 2*n are anagrams.

char[] digits1, digits2, digits3;
int val1, val2, val3;
for (int value=10; value<25000000; value++) {
     digits1 = Integer.toString(value).toCharArray();
     digits2 = Integer.toString(2*value).toCharArray();
     digits3 = Integer.toString(3*value).toCharArray();
     if (digits1.length == digits3.length) {
          Arrays.sort(digits1);
          Arrays.sort(digits2);
          Arrays.sort(digits3);
          val1 = Integer.parseInt(new String(digits1));
          val2 = Integer.parseInt(new String(digits2));
          val3 = Integer.parseInt(new String(digits3));
          if ((val1 == val2) && (val1 == val3)) {
               System.out.print(value + ",");
          }
     }
}

A323711_list = [n for n in range(9,10**7,9) if sorted(str(n)) == sorted(str(2*n)) == sorted(str(3*n))] # Chai Wah Wu, Feb 02 2019

A250122 Coordination sequence for planar net 3.12.12.

Original entry on oeis.org

1, 3, 4, 6, 8, 12, 14, 15, 18, 21, 22, 24, 28, 30, 30, 33, 38, 39, 38, 42, 48, 48, 46, 51, 58, 57, 54, 60, 68, 66, 62, 69, 78, 75, 70, 78, 88, 84, 78, 87, 98, 93, 86, 96, 108, 102, 94, 105, 118, 111, 102, 114, 128, 120, 110, 123, 138, 129

Offset: 0

Darrah Chavey, Nov 23 2014

Also, growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^6 = 1 >. See Magma program in A298805. - N. J. A. Sloane, Feb 06 2018

Maurizio Paolini, Table of n, a(n) for n = 0..1021
Agnes Azzolino, Regular and Semi-Regular Tessellation Paper, 2011.
Agnes Azzolino, Illustration of 3.12.12 tiling [From previous link]
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also on arXiv, arXiv:1803.08530 [math.CO], 2018-2019. See section 10 The 3.12^2 tiling.
Rostislav Grigorchuk and Cosmas Kravaris, On the growth of the wallpaper groups, arXiv:2012.13661 [math.GR], 2020. See section 4.6 p. 23.
Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227-247.
Tom Karzes, Tiling Coordination Sequences
Maurizio Paolini, C program for A250122
Reticular Chemistry Structure Resource, hca
N. J. A. Sloane, The uniform planar nets and their A-numbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]
Index entries for linear recurrences with constant coefficients, signature (2,-3,4,-3,2,-1).

List of coordination sequences for uniform planar nets: A008458 (the planar net 3.3.3.3.3.3), A008486 (6^3), A008574 (4.4.4.4 and 3.4.6.4), A008576 (4.8.8), A008579 (3.6.3.6), A008706 (3.3.3.4.4), A072154 (4.6.12), A219529 (3.3.4.3.4), A250120 (3.3.3.3.6), A250122 (3.12.12).

Cf. A298805.

Join[{1, 3, 4}, LinearRecurrence[{2, -3, 4, -3, 2, -1}, {6, 8, 12, 14, 15, 18}, 100]] (* Jean-François Alcover, Aug 05 2018 *)

From Joseph Myers, Nov 28 2014: (Start)
Empirically,
a(4n) = 10n - 2 except for a(0) = 1
a(4n+1) = 9n + 3
a(4n+2) = 8n + 6 except for a(2) = 4
a(4n+3) = 9n + 6. (End)
If these are correct, the sequence has g.f.
-(-1 - x - x^2 - 3*x^3 + x^4 - 5*x^5 + 3*x^6 - 4*x^7 + 2*x^8)/((x - 1)^2*(x^2 + 1)^2). - N. J. A. Sloane, Nov 28 2014
All the above conjectures are true. - N. J. A. Sloane, Dec 31 2015
E.g.f.: (9*x*cosh(x) - 4*(2*cos(x) + x^2 - 3) + 9*x*sinh(x) - (x - 3)*sin(x))/4. - Stefano Spezia, Jan 05 2023

a(8) onwards from Maurizio Paolini and Joseph Myers (independently), Nov 28 2014

A007780 Losing initial configurations in 2-hole Tchuka Ruma.

Original entry on oeis.org

1, 2, 3, 6, 9, 11, 18, 20, 27, 30, 54, 81, 162, 168, 243, 486, 729, 1458, 2187, 4374, 6561, 13122, 19683, 39366, 59049, 118098, 177147, 354294, 531441, 1062882, 1594323, 3188646, 4782969, 9565938, 14348907, 28697814, 43046721, 86093442

Offset: 1

Darrah Chavey

nonn

The 2-hole Tchuka Ruma game cannot be won for the initial seeds 3^i (i>=1) or 2*3^i (i>=0). Though a sufficient condition, this is not necessary, as can be seen from the terms a(6)=11, a(8)=20, a(10)=30 and a(14)=168. - Pab Ter (pabrlos2(AT)yahoo.com), Nov 08 2005

If any further sporadic term exists, then it exceeds 6973568802. - Sean A. Irvine, Jan 25 2018

Sean A. Irvine, Table of n, a(n) for n = 1..46
Paul J. Campbell, and Darrah P. Chavey, Tchuka Ruma Solitaire, The UMAP Journal 16(4) (1995), 343-365.

Conjectures from Colin Barker, Jan 25 2018: (Start)
G.f.: x*(1 + 2*x - 7*x^5 - 9*x^6 - 13*x^7 - 27*x^8 - 30*x^9 - 27*x^10 - 9*x^11 - 75*x^13 - 243*x^14 - 18*x^15) / (1 - 3*x^2).
a(n) = 3*a(n-2) for n>2.
(End)

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 08 2005

More terms from Sean A. Irvine, Jan 25 2018

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User: Darrah Chavey

A323711 Numbers k such that k, 2k, and 3k are anagrams of each other.

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A250122 Coordination sequence for planar net 3.12.12.

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A007780 Losing initial configurations in 2-hole Tchuka Ruma.

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cp's OEIS Frontend

User: Darrah Chavey

A323711 Numbers k such that k, 2*k, and 3*k are anagrams of each other.

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

Java

Python

A250122 Coordination sequence for planar net 3.12.12.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A007780 Losing initial configurations in 2-hole Tchuka Ruma.

Views

Author

Keywords

Comments

Links

Formula

Extensions

A323711 Numbers k such that k, 2k, and 3k are anagrams of each other.