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A259278
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Number of compositions of n into parts 1, 6, and 7.
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1
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1, 1, 1, 1, 1, 1, 2, 4, 6, 8, 10, 12, 15, 21, 31, 45, 63, 85, 112, 148, 200, 276, 384, 532, 729, 989, 1337, 1813, 2473, 3389, 4650, 6368, 8694, 11844, 16130, 21992, 30031, 41049, 56111, 76649, 104623, 142745, 194768, 265848, 363008, 495768, 677040, 924408, 1261921
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OFFSET
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0,7
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COMMENTS
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Suppose A is a subset of {1,2,3,...,n} having the following property: if A includes an integer k, then A includes none of the integers k+2, k+3, k+4, or k+5. The number of subsets having this property is a(n+5).
The terms of this sequence also give us this coloring problem's answer: suppose that, given an n-section board, if we paint the k-th section, we can't paint the (k+2)-th, (k+3)-th, (k+4)-th, or (k+5)-th section. In how many different ways can we paint this n-section board (where painting none of the sections is considered one of the ways)? Similarly the answer is a(n+5).
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LINKS
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FORMULA
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a(n) = a(n-1) + a(n-6) + a(n-7).
G.f.: 1/(1-x-x^6-x^7).
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EXAMPLE
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G.f. = 1 + x + x^2 + x^3 + x^4 + x^5 + 2*x^6 + 4*x^7 + 6*x^8 + 8*x^9 + ...
For n=3 so {1,2,3}, the answer is a(3+5) = a(8), so the answer is 6.
It can be checked easily. Here are the subsets: {},{1},{2},{3},{1,2},{2,3}.
For n=4, the number of ways of painting a 4-section board is a(4+5)=a(9)=8; here are the 8 situations:
situation 1: none
situation 2: painted only 1st section
situation 3: painted only 2nd section
situation 4: painted only 3rd section
situation 5: painted only 4th section
situation 6: painted 1st and 2nd sections
situation 7: painted 2nd and 3rd sections
situation 8: painted 3rd and 4th sections
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MAPLE
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F:= gfun:-rectoproc({a(n)=a(n-1)+a(n-6)+a(n-7), seq(a(i)=1, i=0..5), a(6)=2}, a(n), remember):
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MATHEMATICA
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LinearRecurrence[{1, 0, 0, 0, 0, 1, 1}, {1, 1, 1, 1, 1, 1, 2}, 50] (* Vincenzo Librandi, Jun 27 2015 *)
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PROG
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(PARI) Vec(1/(1-x-x^6-x^7) + O(x^50)) \\ Michel Marcus, Jun 26 2015
(Magma) I:=[1, 1, 1, 1, 1, 1, 2]; [n le 7 select I[n] else Self(n-1)+Self(n-6)+Self(n-7): n in [1..60]]; // Vincenzo Librandi, Jun 27 2015
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CROSSREFS
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KEYWORD
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nonn,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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