A283793 - OEIS

%I #29 Oct 19 2024 18:10:25

%S 4,14,109,5999,17997004,161946085486514,13113267302202731189080679359,

%T 85978889669509647874887802052390686151982448025024665124

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

%C 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 This list is akin to A006894, which is the sequence if we started with 1 element instead of 4.

%H Michael Turniansky, <a href="/A283793/b283793.txt">Table of n, a(n) for n = 0..9</a>

%H Anton Rybakov as Joybits Ltd., <a href="http://doodlegod.com/">Doodle God game homepage</a>

%H 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 a(n) = 4 + T(a(n-1)) where T(m) is the m-th triangular number.

%e 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 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 (PARI) a(n) = if(n<1, 4, 4 + a(n - 1) * (a(n - 1) + 1) / 2);

%o for(n=0, 7, print1(a(n),", ")) \\ _Indranil Ghosh_, Mar 16 2017

%Y Cf. A006894.

%K nonn

%O 0,1

%A _Michael Turniansky_, Mar 16 2017