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A283793 Number of elements formable in <= n steps, starting with 4 elements, combining 2 elements into a new element at each step.

%I A283793 #31 Jul 23 2025 15:41:44
%S A283793 4,14,109,5999,17997004,161946085486514,13113267302202731189080679359,
%T A283793 85978889669509647874887802052390686151982448025024665124
%N A283793 Number of elements formable in <= n steps, starting with 4 elements, combining 2 elements into a new element at each step.
%C A283793 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.
%C A283793 This list is akin to A006894, which is the sequence if we started with 1 element instead of 4.
%H A283793 Michael Turniansky, <a href="/A283793/b283793.txt">Table of n, a(n) for n = 0..9</a>
%H A283793 Anton Rybakov as Joybits Ltd., <a href="http://doodlegod.com/">Doodle God game homepage</a>
%H A283793 Cary Kaiming Huang, <a href="http://htwins.net/elem3/">Elements 3</a>. An online version which is not bounded by predefined elements, so a(n) could theoretically be reached. (No longer works.)
%F A283793 a(n) = 4 + T(a(n-1)) where T(m) is the m-th triangular number.
%e A283793 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).
%t A283793 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 *)
%o A283793 (PARI) a(n) = if(n<1, 4, 4 + a(n - 1) * (a(n - 1) + 1) / 2);
%o A283793 for(n=0, 7, print1(a(n),", ")) \\ _Indranil Ghosh_, Mar 16 2017
%Y A283793 Cf. A006894.
%K A283793 nonn
%O A283793 0,1
%A A283793 _Michael Turniansky_, Mar 16 2017