A186101 -id:A186101 - cp's OEIS frontend

A066393 Coordination sequence for (9^3, 3.9^2) net with respect to a vertex of type 9^3.

1, 3, 6, 6, 12, 15, 12, 21, 24, 18, 30, 33, 24, 39, 42, 30, 48, 51, 36, 57, 60, 42, 66, 69, 48, 75, 78, 54, 84, 87, 60, 93, 96, 66, 102, 105, 72, 111, 114, 78, 120, 123, 84, 129, 132, 90, 138, 141, 96, 147, 150, 102, 156, 159, 108, 165, 168, 114

Offset: 0

N. J. A. Sloane, Dec 24 2001

nonn

This net may be regarded as a tiling of the plane by 9-gons and triangles. There are two kinds of vertices: (a) 9^3 vertices, where three 9-gons meet, and (b) 3.9^2 vertices, where a triangle and two 9-gons meet. The present sequence is the coordination sequence with respect to a vertex of type 9^3. See also A319980.

Muniru A Asiru, Table of n, a(n) for n = 0..300
Jean-Guillaume Eon, Geometrical relationships between nets mapped on isomorphic quotient graphs: examples, Journal of Solid State Chemistry 138.1 (1998): 55-65. See Fig. 1.
Jean-Guillaume Eon, Algebraic determination of generating functions for coordination sequences in crystal structures, Acta Cryst. A58 (2002), 47-53. See Section 8.
N. J. A. Sloane, A portion of the (9^3, 3.9^2) net

Cf. A319980, A109044, A186101.

seq(coeftayl((1+3*x+6*x^2+4*x^3+6*x^4+3*x^5+x^6)/(1-x^3)^2, x = 0, k), k=0..60); # Muniru A Asiru, Feb 13 2018

G.f.: (1+3*x+6*x^2+4*x^3+6*x^4+3*x^5+x^6)/(1-x^3)^2.
a(n) = (3*n + lcm(n,3))/2, for n>=1. - Ridouane Oudra, Jan 22 2021
a(n) = 3*A186101(n), for n>=1. - Ridouane Oudra, Jun 11 2025

A255368 a(n) = -(-1)^n * 2 * n / 3 if n divisible by 3, a(n) = -(-1)^n * n otherwise.

Original entry on oeis.org

0, 1, -2, 2, -4, 5, -4, 7, -8, 6, -10, 11, -8, 13, -14, 10, -16, 17, -12, 19, -20, 14, -22, 23, -16, 25, -26, 18, -28, 29, -20, 31, -32, 22, -34, 35, -24, 37, -38, 26, -40, 41, -28, 43, -44, 30, -46, 47, -32, 49, -50, 34, -52, 53, -36, 55, -56, 38, -58, 59

Offset: 0

Michael Somos, May 04 2015

			G.f. = x - 2*x^2 + 2*x^3 - 4*x^4 + 5*x^5 - 4*x^6 + 7*x^7 - 8*x^8 + 6*x^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Rational function multiplicative coefficients.
Index entries for linear recurrences with constant coefficients, signature (0,0,-2,0,0,-1).

Cf. A186101.

m:=60; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1-x)^2*(1+x^2)/(1+x^3)^2)); // G. C. Greubel, Aug 02 2018

a[ n_] := -(-1)^n If[ Divisible[ n, 3], 2 n/3, n];
a[ n_] := n {1, -1, 2/3, -1, 1, -2/3}[[Mod[n, 6, 1]]];
CoefficientList[Series[x*(1-x)^2*(1+x^2)/(1+x^3)^2, {x,0,60}], x] (* G. C. Greubel, Aug 02 2018 *)

PARI
```
{a(n) = -(-1)^n * if( n%3, n, 2*n/3)};
```

my(x='x+O('x^60)); concat([0], Vec(x*(1-x)^2*(1+x^2)/(1+x^3)^2)) \\ G. C. Greubel, Aug 02 2018

Euler transform of length 6 sequence [-2, 1, -2, -1, 0, 2].
a(n) is multiplicative with a(2^e) = -(2^e) if e>0, a(3^e) = 2 * 3^(e-1) if e>0, otherwise a(p^e) = p^e.
G.f.: f(x) - f(x^3) where f(x) := x / (1 + x)^2.
G.f.: x * (1 - x)^2 * (1 + x^2) / (1 + x^3)^2.
G.f.: x * (1 - x)^2 * (1 - x^3)^2 * (1 - x^4) / ((1 - x^2) * (1 - x^6)^2).
a(n) = -a(-n) = -(-1)^n * A186101(n) for all n in Z.
Dirichlet g.f.: zeta(s-1)*(2^s-4)*(3^s-1)/6^s. - Amiram Eldar, Dec 29 2022

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A066393 Coordination sequence for (9^3, 3.9^2) net with respect to a vertex of type 9^3.

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A255368 a(n) = -(-1)^n * 2 * n / 3 if n divisible by 3, a(n) = -(-1)^n * n otherwise.

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