OFFSET
1,1
COMMENTS
Number of horizontally convex polyiamonds with n triangles.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
K. A. Van'kov, V. M. Zhuravlyov, Regular tilings and generating functions, Mat. Pros. Ser. 3, issue 22, 2018 (127-157) [in Russian]. See page 128. - N. J. A. Sloane, Jan 09 2019
Kirill Vankov, Valerii Zhuravlev, Regular and semiregular (uniform) tilings and generating functions, hal-02535947, [math.CO], 2020.
Wikipedia, Polyiamond
V. M. Zhuravlev, Horizontally-convex polyiamonds and their generating functions, Mat. Pros. 17 (2013), 107-129 (in Russian). See the sequence g(n). [Note the recurrence for g(n) in Theorem 1 contains a typo]
Index entries for linear recurrences with constant coefficients, signature (3,0,-4,1,1,3,-1).
FORMULA
G.f.: x*(2 - 3*x - 3*x^2 + 4*x^3 + 2*x^4 + x^5 - 3*x^6) / (1 - 3*x + 4*x^3 - x^4 - x^5 - 3*x^6 + x^7). [Bruno Berselli, Mar 10 2014]
EXAMPLE
The initial values of Zhuravlev's sequences are as follows.
(The columns give n, A238829, A238828, A238824 (twice), A238830, A238833, A238832, A238825, A238831, A238827, A238826, A238823, respectively):
n a b c d i j e p q r h g
1 1 0 1 0 0 0 0 0 0 0 1 2
2 1 0 0 1 0 1 0 0 0 0 2 3
3 2 1 1 1 0 0 1 0 0 0 4 6
4 5 2 1 3 1 2 1 1 0 0 9 14
5 12 5 3 7 2 2 4 2 0 0 22 34
6 31 12 7 17 6 7 9 5 1 0 53 84
7 77 28 17 43 15 16 23 11 3 1 131 208
8 192 70 43 105 36 40 58 27 8 2 323 515
9 474 169 105 262 91 101 141 64 21 6 798 1272
MAPLE
g:=proc(n) option remember; local t1;
t1:=[2, 3, 6, 14, 34, 84, 208, 515];
if n <= 7 then t1[n] else
3*g(n-1)-4*g(n-3)+g(n-4)+g(n-5)+3*g(n-6)-g(n-7); fi; end proc;
[seq(g(n), n=1..32)];
MATHEMATICA
LinearRecurrence[{3, 0, -4, 1, 1, 3, -1}, {2, 3, 6, 14, 34, 84, 208}, 40] (* Vincenzo Librandi, Mar 10 2014 *)
PROG
(Magma) I:=[2, 3, 6, 14, 34, 84, 208, 515]; [n le 8 select I[n] else 3*Self(n-1)-4*Self(n-3)+Self(n-4)+Self(n-5)+3*Self(n-6)-Self(n-7): n in [1..40]]; // Vincenzo Librandi, Mar 10 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Mar 08 2014
STATUS
approved