OFFSET
0,2
COMMENTS
Also, a(n-1) is the number of topologically inequivalent opening moves in the Sprouts game on n nodes [Browne]. - Andrey Zabolotskiy, Feb 12 2020
Also the number of symmetrically distinct faces in the 1 x 1 x (n+1) polycube. - Eric W. Weisstein, Sep 02 2022
REFERENCES
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
LINKS
Robert Price, Table of n, a(n) for n = 0..1000
Cameron B. Browne, Boundary Matching for Interactive Sprouts, in: ACG 2015, pp. 147-159, LNCS 9525, Springer, doi:10.1007/978-3-319-27992-3_14.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
FORMULA
From Colin Barker, Jan 08 2016 and Apr 16 2019: (Start)
a(n) = (2*n+(-1)^n+7)/4 for n>0.
a(n) = a(n-1)+a(n-2)-a(n-3) for n>3.
G.f.: (1+x-x^3) / ((1-x)^2*(1+x)). (End)
From Stefano Spezia, Aug 08 2021: (Start)
E.g.f.: ((4 + x)*cosh(x) + (3 + x)*sinh(x))/2 - 1.
a(n) = 2 + A004526(n) for n > 0. (End)
MATHEMATICA
rule=78; rows=20; ca=CellularAutomaton[rule, {{1}, 0}, rows-1, {All, All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]], {rows-k+1, rows+k-1}], {k, 1, rows}]; (* Truncated list of each row *) Table[Total[catri[[k]]], {k, 1, rows}] (* Number of Black cells in stage n *)
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
Robert Price, Jan 07 2016
STATUS
approved