OFFSET
0,1
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.
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.)
FORMULA
a(n) = 4 + T(a(n-1)) where T(m) is the m-th triangular number.
EXAMPLE
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).
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Turniansky, Mar 16 2017
STATUS
approved