%I #29 Sep 08 2022 08:46:07
%S 2,3,6,14,34,84,208,515,1272,3138,7734,19055,46940,115631,284846,
%T 701709,1728662,4258604,10491218,25845514,63671404,156856887,
%U 386422704,951966378,2345203554,5777493461,14233063160,35063663603,86380598122,212801715171,524244692006,1291495687122
%N a(n) = 3*a(n-1)-4*a(n-3)+a(n-4)+a(n-5)+3*a(n-6)-a(n-7) for n >= 8, with initial values as shown.
%C Number of horizontally convex polyiamonds with n triangles.
%H Vincenzo Librandi, <a href="/A238823/b238823.txt">Table of n, a(n) for n = 1..1000</a>
%H K. A. Van'kov, V. M. Zhuravlyov, <a href="https://www.mccme.ru/free-books/matpros/pdf/mp-22.pdf#page=127">Regular tilings and generating functions</a>, Mat. Pros. Ser. 3, issue 22, 2018 (127-157) [in Russian]. See page 128. - _N. J. A. Sloane_, Jan 09 2019
%H Kirill Vankov, Valerii Zhuravlev, <a href="https://hal.archives-ouvertes.fr/hal-02535947">Regular and semiregular (uniform) tilings and generating functions</a>, hal-02535947, [math.CO], 2020.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Polyiamond">Polyiamond</a>
%H V. M. Zhuravlev, <a href="http://www.mccme.ru/free-books/matpros/mph.pdf">Horizontally-convex polyiamonds and their generating functions</a>, 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]
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-4,1,1,3,-1).
%F 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]
%e The initial values of Zhuravlev's sequences are as follows.
%e (The columns give n, A238829, A238828, A238824 (twice), A238830, A238833, A238832, A238825, A238831, A238827, A238826, A238823, respectively):
%e n a b c d i j e p q r h g
%e 1 1 0 1 0 0 0 0 0 0 0 1 2
%e 2 1 0 0 1 0 1 0 0 0 0 2 3
%e 3 2 1 1 1 0 0 1 0 0 0 4 6
%e 4 5 2 1 3 1 2 1 1 0 0 9 14
%e 5 12 5 3 7 2 2 4 2 0 0 22 34
%e 6 31 12 7 17 6 7 9 5 1 0 53 84
%e 7 77 28 17 43 15 16 23 11 3 1 131 208
%e 8 192 70 43 105 36 40 58 27 8 2 323 515
%e 9 474 169 105 262 91 101 141 64 21 6 798 1272
%p g:=proc(n) option remember; local t1;
%p t1:=[2,3,6,14,34,84,208,515];
%p if n <= 7 then t1[n] else
%p 3*g(n-1)-4*g(n-3)+g(n-4)+g(n-5)+3*g(n-6)-g(n-7); fi; end proc;
%p [seq(g(n),n=1..32)];
%t LinearRecurrence[{3, 0, -4, 1, 1, 3, -1}, {2, 3, 6, 14, 34, 84, 208}, 40] (* _Vincenzo Librandi_, Mar 10 2014 *)
%o (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
%Y Cf. A238824-A238833.
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_, Mar 08 2014