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.
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
Offset: 1
Examples
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
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).
Programs
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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
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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)];
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Mathematica
LinearRecurrence[{3, 0, -4, 1, 1, 3, -1}, {2, 3, 6, 14, 34, 84, 208}, 40] (* Vincenzo Librandi, Mar 10 2014 *)
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]
Comments