A165988 -id:A165988 - cp's OEIS frontend

A298028 Coordination sequence of Dual(3.6.3.6) tiling with respect to a trivalent node.

1, 3, 12, 9, 24, 15, 36, 21, 48, 27, 60, 33, 72, 39, 84, 45, 96, 51, 108, 57, 120, 63, 132, 69, 144, 75, 156, 81, 168, 87, 180, 93, 192, 99, 204, 105, 216, 111, 228, 117, 240, 123, 252, 129, 264, 135, 276, 141, 288, 147, 300, 153, 312, 159, 324, 165, 336, 171, 348, 177, 360, 183, 372, 189, 384, 195

Offset: 0

N. J. A. Sloane, Jan 21 2018

Also known as the kgd net.

This is one of the Laves tilings.

Robert Israel, Table of n, a(n) for n = 0..10000
Tom Karzes, Tiling Coordination Sequences
Reticular Chemistry Structure Resource (RCSR), The kgd tiling (or net)
N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A008579, A135556 (partial sums), A298026 (trivalent point).
If the initial 1 is changed to 0 we get A165988 (but we need both sequences, just as we have both A008574 and A008586).
List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.

f3:=proc(n) if n=0 then 1 elif (n mod 2) = 0 then 6*n else 3*n; fi; end;
[seq(f3(n),n=0..80)];

Join[{1}, LinearRecurrence[{0, 2, 0, -1}, {3, 12, 9, 24}, 80]] (* Jean-François Alcover, Mar 23 2020 *)

a(0)=1; a(2*k) = 12*k, a(2*k+1) = 6*k+3.
G.f.: 1 + 3*x*(x^2+4*x+1)/(1-x^2)^2. - Robert Israel, Jan 21 2018
a(n) = 3*A022998(n), n>0. - R. J. Mathar, Jan 29 2018

A166304 Third trisection of A022998.

Original entry on oeis.org

4, 5, 16, 11, 28, 17, 40, 23, 52, 29, 64, 35, 76, 41, 88, 47, 100, 53, 112, 59, 124, 65, 136, 71, 148, 77, 160, 83, 172, 89, 184, 95, 196, 101, 208, 107, 220, 113, 232, 119, 244, 125, 256, 131, 268, 137, 280, 143, 292, 149, 304, 155, 316, 161, 328, 167, 340, 173, 352, 179

Offset: 0

Paul Curtz, Oct 11 2009

The sequence read modulo 9 is the periodic sequence 4, 5, 7, 2, 1, 8 (repeat..)
The same set of numbers in a period of length 6 is in A153130,
A165355 read modulo 9, A165367 read modulo 9, and A166138 read modulo 9.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0, 2, 0, -1).

Cf. A165988 (first trisection), A166138 (2nd trisection).

LinearRecurrence[{0, 2, 0, -1}, {4, 5, 16, 11}, 100] (* G. C. Greubel, May 09 2016 *)

a(n) = A022998(3*n+2).
a(n) = 2*a(n-2)-a(n-4).
G.f.: (4+5*x+8*x^2+x^3)/((x-1)^2 *(1+x)^2 ).
a(2*n) = A017569(n). a(2n+1) = A016969(n) .

Edited and extended by R. J. Mathar, Oct 14 2009

A166138 Trisection A022998(3n+1).

Original entry on oeis.org

1, 8, 7, 20, 13, 32, 19, 44, 25, 56, 31, 68, 37, 80, 43, 92, 49, 104, 55, 116, 61, 128, 67, 140, 73, 152, 79, 164, 85, 176, 91, 188, 97, 200, 103, 212, 109, 224, 115, 236, 121, 248, 127, 260, 133, 272, 139, 284, 145, 296, 151, 308, 157, 320, 163, 332, 169, 344, 175, 356, 181, 368, 187, 380, 193, 392

Offset: 0

Paul Curtz, Oct 08 2009

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A165988, A166304.

LinearRecurrence[{0,2,0,-1},{1,8,7,20},70] (* Harvey P. Dale, Aug 15 2012 *)
Table[If[OddQ@ #, #, 2 #] &[3 n + 1], {n, 0, 65}] (* or *)
CoefficientList[Series[(1 + 8 x + 5 x^2 + 4 x^3)/((1 - x)^2 (1 + x)^2), {x, 0, 65}], x] (* Michael De Vlieger, Apr 27 2016 *)

a(2n) = 6n+1 = A016921(n).
a(2n+1) = 12n+8 = A017617(n).
a(n) = 2*a(n-2)-a(n-4) = (3n+1)*(3-(-1)^n)/2.
From G. C. Greubel, Apr 26 2016: (Start)
O.g.f.: (1 + 8*x + 5*x^2 + 4*x^3)/((1 - x)^2*(1 + x)^2).
E.g.f.: (1/2)*(-1 + 3*x + (3+9*x)*exp(2*x))*exp(-x). (End)

A281098 a(n) is the GCD of the sequence d(n) = A261327(k+n) - A261327(k) for all k.

Original entry on oeis.org

0, 1, 1, 3, 4, 1, 3, 1, 8, 3, 5, 1, 12, 1, 7, 3, 16, 1, 9, 1, 20, 3, 11, 1, 24, 1, 13, 3, 28, 1, 15, 1, 32, 3, 17, 1, 36, 1, 19, 3, 40, 1, 21, 1, 44, 3, 23, 1, 48, 1, 25, 3, 52, 1, 27, 1, 56, 3, 29, 1, 60, 1, 31, 3, 64, 1, 33, 1, 68, 3, 35, 1, 72, 1, 37, 3, 76, 1, 39, 1

Offset: 0

Paul Curtz, Jan 14 2017

Successive sequences:
   0,   0,  0,    0, ...    = 0 * ( )
   4,  -3,  11,  -8, ...    = 1 * ( )
   1,   8,   3,  16, ...    = 1 * ( )                 A195161
  12,   0,  27,  -3, ...    = 3 * (4, 0, 9, -1, ...)
   4,  24,   8,  40, ...    = 4 * (1, 6, 2, 10, ...)  A064680
5;   28,   5,  51,   4, ...    = 1 * ( )
   9,  48,  15,  72, ...    = 3 * (3, 16, 5, 24, ...) A195161
  52,  12,  83,  13, ...    = 1 * ( )
  16,  80,  24, 112, ...    = 8 * (2, 10, 3, 14, ...) A064080
  84   21, 123,  24, ...    = 3 * (28, 7, 41, 8, ...)
 25, 120,  35, 160, ...    = 5 * (5, 24, 7, 32, ...) A195161

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for linear recurrences with constant coefficients, signature (0,-1,0,1,0,2,0,1,0,-1,0,-1).

Cf. A064680, A144433 or A195161.

Cf. A000012, A005408, A008586, A010701, A109007 (bisection), A016825, A165988 (via A022998), A166138, A166304, A280579.

CoefficientList[Series[(-x (-1 - x - 4 x^2 - 5 x^3 - 3 x^4 - 6 x^5 + 3 x^6 - 5 x^7 + 4 x^8 - x^9 + x^10))/((x^2 - x + 1) (1 + x + x^2) (x - 1)^2*(1 + x)^2*(1 + x^2)^2), {x, 0, 79}], x] (* Michael De Vlieger, Feb 02 2017 *)

f(n) = numerator((4 + n^2)/4);
a(n) = gcd(vector(1000, k, f(k+n) - f(k))); \\ Michel Marcus, Jan 15 2017

A281098(n) = if(n%2, gcd((n\2)-1,3), n>>(bitand(n,2)/2)); \\ Antti Karttunen, Feb 15 2023

G.f.: -x*( -1 - x - 4*x^2 - 5*x^3 - 3*x^4 - 6*x^5 + 3*x^6 - 5*x^7 + 4*x^8 - x^9 + x^10 )/( (x^2 - x + 1)*(1 + x + x^2)*(x - 1)^2*(1 + x)^2*(1 + x^2)^2 ). - R. J. Mathar, Jan 31 2017
a(2*k)   = A022998(k).
a(2*k+1) = A109007(k-1).
a(3*k)   = interleave 3*k*(3 +(-1)^k)/2, 3.
a(3*k+1) = interleave 1, A166304(k).
a(3*k+2) = interleave A166138(k), 1.
a(4*k)   = 4*k.
a(4*k+1) = period 3: repeat [1, 1, 3].
a(4*k+2) = 1 + 2*k.
a(4*k+3) = period 3: repeat [3, 1, 1].
a(n+12) - a(n) = 6*A131743(n+3).
a(n) = (18*n + 40 - 16*cos(n*Pi/3) + 9*n*cos(n*Pi/2) + 32*cos(2*n*Pi/3) + (18*n - 40)*cos(n*Pi) + 3*n*cos(3*n*Pi/2) - 16*cos(5*n*Pi/3))/48. - Wesley Ivan Hurt, Oct 04 2018

Corrected and extended by Michel Marcus, Jan 15 2017

A174012 a(n) = 3 * A064680(n).

Original entry on oeis.org

0, 6, 3, 18, 6, 30, 9, 42, 12, 54, 15, 66, 18, 78, 21, 90, 24, 102, 27, 114, 30, 126, 33, 138, 36, 150, 39, 162, 42, 174, 45, 186, 48, 198, 51, 210, 54, 222, 57, 234, 60, 246, 63, 258, 66, 270, 69, 282, 72, 294, 75, 306, 78, 318, 81, 330, 84, 342, 87, 354, 90, 366, 93, 378, 96

Offset: 0

Paul Curtz, Mar 05 2010

Cf. A008585, A017593, A064680, A080425, A165988.

a(n) = A064680(3*n), similar to A165988.
a(n) mod 9 = 3*A080425(n) (period length 3).
a(2n+1) = A017593(n).
a(2n) = A008585(n).

a(0) = 0 prepended by Georg Fischer, Jul 01 2020

cp's OEIS Frontend

A298028 Coordination sequence of Dual(3.6.3.6) tiling with respect to a trivalent node.

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A166304 Third trisection of A022998.

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A166138 Trisection A022998(3n+1).

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A281098 a(n) is the GCD of the sequence d(n) = A261327(k+n) - A261327(k) for all k.

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A174012 a(n) = 3 * A064680(n).

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