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A168083
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Fibonacci 12-step numbers.
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6
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0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4095, 8189, 16376, 32748, 65488, 130960, 261888, 523712, 1047296, 2094336, 4188160, 8375296, 16748544, 33492993, 66977797, 133939218, 267845688, 535625888, 1071120816
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OFFSET
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1,14
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COMMENTS
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a(n+12) is the number of compositions of n with no part greater than 12.
a(n+12) is the number of ways of throwing n with an unstated number of dodecahedra (here dice with numbers from 1 to 12).
(End)
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1,1,1,1,1,1).
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FORMULA
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Another form of the g.f. f: f(z)= (z^(k-1)-z^(k))/(1-2*z+z^(k+1)) with k=12. a(n)=sum((-1)^i*binomial(n-k+1-k*i,i)*2^(n-k+1-(k+1)*i),i=0..floor((n-k+1)/(k+1)))-sum((-1)^i*binomial(n-k-k*i,i)*2^(n-k-(k+1)*i),i=0..floor((n-k)/(k+1))) with k=12 and convention sum(alpha(i),i=m..n)=0 for m>n. - Richard Choulet, Feb 22 2010
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MAPLE
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k:=12:for n from 0 to 50 do l(n):=sum((-1)^i*binomial(n-k+1-k*i, i)*2^(n-k+1-(k+1)*i), i=0..floor((n-k+1)/(k+1)))-sum((-1)^i*binomial(n-k-k*i, i)*2^(n-k-(k+1)*i), i=0..floor((n-k)/(k+1))):od:seq(l(n), n=0..50); a:=taylor((z^(k-1)-z^(k))/(1-2*z+z^(k+1)), z=0, 51); for p from 0 to 50 do j(p):=coeff(a, z, p):od :seq(j(p), p=0..50); # Richard Choulet, Feb 22 2010
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MATHEMATICA
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a={1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}; Flatten[Prepend[Table[s=Plus@@a; a=RotateLeft[a]; a[[ -1]]=s, {n, 60}], Table[0, {m, Length[a]-1}]]]
LinearRecurrence[{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, 50]
With[{nn=12}, LinearRecurrence[Table[1, {nn}], Join[Table[0, {nn-1}], {1}], 50]] (* Harvey P. Dale, Aug 17 2013 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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