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A238823 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. 13
2, 3, 6, 14, 34, 84, 208, 515, 1272, 3138, 7734, 19055, 46940, 115631, 284846, 701709, 1728662, 4258604, 10491218, 25845514, 63671404, 156856887, 386422704, 951966378, 2345203554, 5777493461, 14233063160, 35063663603, 86380598122, 212801715171, 524244692006, 1291495687122 (list; graph; refs; listen; history; text; internal format)
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

Cf. A238824-A238833.

Sequence in context: A331875 A010357 A190166 * A002995 A093467 A246640

Adjacent sequences:  A238820 A238821 A238822 * A238824 A238825 A238826

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Mar 08 2014

STATUS

approved

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Last modified June 25 10:28 EDT 2021. Contains 345453 sequences. (Running on oeis4.)