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A283793
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Number of elements formable by in <= n steps, starting with 4 elements, combining 2 elements into a new element at each step
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1
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OFFSET
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0,1
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COMMENTS
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In a game such as Doodle God (see links), you start with Earth, Air, Fire and Water, and combine them two at a time (including combining an element with itself) into new elements. A(n) is the hypothetical maximum number of different possible elements you could reach from clicking at most n times.
This list is akin to A006894, which is the sequence if we started with 1 element instead of 4.
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LINKS
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Cary Kaiming Huang, Elements 3 An online version which is not bounded by predefined elements, so a(n) could theoretically be reached.
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FORMULA
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a(n) = 4 + T(a(n-1)) where T(m) is the m-th triangular number.
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EXAMPLE
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Starting with {A, B, C, D}, we can make {AA, AB, AC, AD, BB, BC, BD, CC, CD, and DD}. The union of these two sets has cardinality 14 = a(1).
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MATHEMATICA
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a[0]=4; a[n_] := a[n] = 4 + a[n-1] (a[n-1] + 1)/2; a /@ Range[0, 7] (* Giovanni Resta, Mar 16 2017 *)
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PROG
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(PARI) a(n) = if(n<1, 4, 4 + a(n - 1) * (a(n - 1) + 1) / 2);
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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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