A301293 Expansion of (x^2+x+1)^2 / ((x^2+1)*(x-1)^2).

1, 4, 9, 14, 18, 22, 27, 32, 36, 40, 45, 50, 54, 58, 63, 68, 72, 76, 81, 86, 90, 94, 99, 104, 108, 112, 117, 122, 126, 130, 135, 140, 144, 148, 153, 158, 162, 166, 171, 176, 180, 184, 189, 194, 198, 202, 207, 212, 216, 220, 225, 230, 234, 238, 243, 248, 252

Offset: 0

Author

N. J. A. Sloane, Mar 23 2018

Keywords

• nonn
• easy

Comments

Appears to be coordination sequence for node of type 4^4 in "krm" 2-D tiling (or net).

Also appears to be coordination sequence for tetravalent node in "krk" 2-D tiling (or net).

Linear recurrence and g.f. confirmed by Shutov/Maleev link. - Ray Chandler, Aug 30 2023

References

• Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 67, row 3, first tiling; also p. 66, row 3, first tiling.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Brian Galebach, Collection of n-Uniform Tilings. See Numbers 3 and 8 from the list of 20 2-uniform tilings.
• 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 arXiv:1803.08530.
• Reticular Chemistry Structure Resource (RCSR), The krm tiling (or net)
• Reticular Chemistry Structure Resource (RCSR), The krk tiling (or net)
• Anton Shutov and Andrey Maleev, Coordination sequences of 2-uniform graphs, Z. Kristallogr., 235 (2020), 157-166. See supplementary material, krb, vertex u_1.
• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Crossrefs

Cf. A301291.

Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.

Programs

Maple

f:=proc(n) if n=0 then 1
elif (n mod 2) = 0 then 9*n/2
elif (n mod 4) = 1 then 18*(n-1)/4+4
else 18*(n-3)/4+14; fi; end;
s1:=[seq(f(n),n=0..60)];

Mathematica

Join[{1}, LinearRecurrence[{2, -2, 2, -1}, {4, 9, 14, 18}, 60]] (* Jean-François Alcover, Jan 08 2019 *)

PARI

Vec((x^2+x+1)^2 / ((x^2+1)*(x-1)^2) + O(x^60)) \\ Colin Barker, Mar 23 2018

Formula

For explicit formula for a(n) see Maple code.

a(n) = 9*n/2 + (1 - (-1)^n)*i^(n*(n + 1))/4 for n>0, a(0)=1 and i=sqrt(-1). Therefore, for even n>0 a(n) = 9*n/2, otherwise a(n) = 9*n/2 - (-1)^((n-1)/2)/2. - Bruno Berselli, Mar 23 2018

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4. - Colin Barker, Mar 23 2018