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A242026
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Number of non-palindromic n-tuples of 4 distinct elements.
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3
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0, 12, 48, 240, 960, 4032, 16128, 65280, 261120, 1047552, 4190208, 16773120, 67092480, 268419072, 1073676288, 4294901760, 17179607040, 68719214592, 274876858368, 1099510579200, 4398042316800, 17592181850112, 70368727400448, 281474959933440, 1125899839733760
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OFFSET
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1,2
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COMMENTS
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Non-palindromic vs palindromic (DNA) sequences (e.g., {a,c,a,c} is a non-palindromic sequence but {a,c,c,a} is palindromic). Useful in bioinformatics.
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LINKS
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Table of n, a(n) for n=1..25.
Index entries for linear recurrences with constant coefficients, signature (4,4,-16).
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FORMULA
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a(n) = 2^(n-1) * (2^(n+1) + (-1)^n - 3).
a(n) = 4^n - 4^ceiling(n/2) = A000302(n) - A056450(n).
a(n) = 4*a(n-1) + 4*a(n-2) - 16*a(n-3). - Colin Barker, Aug 12 2014
G.f.: 12*x^2 / ((2*x-1)*(2*x+1)*(4*x-1)). - Colin Barker, Aug 12 2014
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EXAMPLE
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For n=2 the a(2)=12 solutions (non-palindromic 2-tuples over 4 distinct elements {a,c,g,t}) are: {a,c}, {a,g}, {a,t}, {c,a}, {c,g}, {c,f}, {g,a},{g,c}, {g,t}, {t,a}, {t,c}, {t,g}.
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MATHEMATICA
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Table[2^(n-1) * (2^(n+1) + (-1)^n - 3), {n, 66}]
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PROG
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(PARI) a(n) = ((-1)^n - 3)*2^(n-1) + 4^n; \\ Michel Marcus, Aug 12 2014
(PARI) concat(0, Vec(12*x^2 / ((2*x-1)*(2*x+1)*(4*x-1)) + O(x^100))) \\ Colin Barker, Aug 12 2014
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CROSSREFS
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Cf. A000302, A056450, A233411, A242278, A240437.
Sequence in context: A007200 A061148 A221908 * A052601 A003498 A269030
Adjacent sequences: A242023 A242024 A242025 * A242027 A242028 A242029
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KEYWORD
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nonn,easy
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AUTHOR
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Mikk Heidemaa, Aug 12 2014
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EXTENSIONS
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Typos in formula fixed by Colin Barker, Aug 12 2014
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STATUS
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approved
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