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

4, 14, 109, 5999, 17997004, 161946085486514, 13113267302202731189080679359, 85978889669509647874887802052390686151982448025024665124

Offset: 0

Author

Michael Turniansky, Mar 16 2017

Keywords

• nonn

Comments

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.

Examples

			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).
		

Links

• Michael Turniansky, Table of n, a(n) for n = 0..9
• Anton Rybakov as Joybits Ltd., Doodle God game homepage
• Cary Kaiming Huang, Elements 3. An online version which is not bounded by predefined elements, so a(n) could theoretically be reached. (No longer works.)

Crossrefs

Cf. A006894.

Programs

Mathematica

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 *)

PARI

a(n) = if(n<1, 4, 4 + a(n - 1) * (a(n - 1) + 1) / 2);
for(n=0, 7, print1(a(n),", ")) \\ Indranil Ghosh, Mar 16 2017

Formula

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