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A001169
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Number of board-pile polyominoes with n cells.
(Formerly M1636 N0639)
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10
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1, 2, 6, 19, 61, 196, 629, 2017, 6466, 20727, 66441, 212980, 682721, 2188509, 7015418, 22488411, 72088165, 231083620, 740754589, 2374540265, 7611753682, 24400004911, 78215909841, 250726529556, 803721298537, 2576384425157, 8258779154250, 26474089989299
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OFFSET
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1,2
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COMMENTS
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The inverse binomial transform is 1,1,3,6,..., i.e., the unsigned version of A077926. - R. J. Mathar, May 15 2008
a(n+1)/a(n) tends to a limit which is equal to the largest real root of the denominator of the g.f., 3.20556943040... = A246773 . - Robert G. Wilson v, Feb 01 2015
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REFERENCES
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W. F. Lunnon, Counting polyominoes, pp. 347-372 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics I, p. 259.
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LINKS
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FORMULA
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G.f.: x*(1-x)^3/(1 - 5*x + 7*x^2 - 4*x^3). - Simon Plouffe in his 1992 dissertation
a(n) = 5*a(n-1) - 7*a(n-2) + 4*a(n-3) for n >= 5.
a(n) = sum(k=0..n-1, sum(i=0..k, binomial(k,i)*binomial(n+2*i-1,4*k-i))). - Emanuele Munarini, May 19 2011
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MATHEMATICA
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a[n_] := a[n] = If[n<5, {1, 2, 6, 19}[[n]], 5a[n-1] - 7a[n-2] + 4a[n-3]]; Table[a[n], {n, 30}]
Join[{1}, LinearRecurrence[{5, -7, 4}, {2, 6, 19}, 40]] (* Harvey P. Dale, Sep 11 2014 *)
Rest@ CoefficientList[ Series[x (1 - x)^3/(1 - 5x + 7x^2 - 4x^3), {x, 0, 28}], x] (* Robert G. Wilson v, Feb 01 2015 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, polcoeff( x * (1 - x)^3 / (1 - 5*x + 7*x^2 - 4*x^3) + x * O(x^n), n))}; /* Michael Somos, Jun 02 2016 */
(Maxima) makelist(sum(sum(binomial(k, i)*binomial(n+2*i-1, 4*k-i), i, 0, k), k, 0, n-1), n, 0, 24); /* Emanuele Munarini, May 19 2011 */
(Magma) I:=[1, 2, 6, 19, 61]; [n le 5 select I[n] else 5*Self(n-1)-7*Self(n-2)+4*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Feb 15 2015
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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