
COMMENTS

a(n+1)/2 = A000225(n). Binomial transform is A002783. Binomial transform of 2  2*0^n + (1)^n or 1,1,3,1,3,1,3,1,...
From Peter C. Heinig (algorithms(AT)gmx.de), Apr 17 2007: (Start)
Number of ntuples where each entry is chosen from the subsets of {1,2} such that the intersection of all n entries contains exactly one element.
There is the following general formula: The number T(n,k,r) of ntuples where each entry is chosen from the subsets of {1,2,..,k} such that the intersection of all n entries contains exactly r elements is: T(n,k,r) = binomial(k,r) * (2^n  1)^(kr). This may be shown by exhibiting a bijection to a set whose cardinality is obviously binomial(k,r) * (2^n  1)^(kr), namely the set of all ktuples where each entry is chosen from subsets of {1,..,n} in the following way: Exactly r entries must be {1,..,n} itself (there are binomial(k,r) ways to choose them) and the remaining (kr) entries must be chosen from the 2^n1 proper subsets of {1,..,n}, i.e., for each of the (kr) entries, {1,..,n} is forbidden (there are, independent of the choice of the full entries, (2^n  1)^(kr) possibilities to do that, hence the formula). The bijection into this set is given by (X_1,..,X_n) > (Y_1,..,Y_k) where for each j in {1,..,k} and each i in {1,..,n}, i is in Y_j if and only if j is in X_i.
Examples: a(1)=2 because the two tuples of length one are: ({1}) and ({2}).
a(3)=14 because the fourteen tuples of length three are: ({1},{1},{1}), ({2},{2},{2}), ({1,2},{1},{1}), ({1},{1,2},{1}), ({1},{1},{1,2}), ({1,2},{2},{2}), ({2},{1,2},{2}), ({2},{2},{1,2}), ({1,2},{1,2},{1}), ({1,2},{1},{1,2}), ({1},{1,2},{1,2}), ({1,2}{1,2}{2}), ({1,2}{2}{1,2}), ({2}{1,2}{1,2}).
The image of this set of tuples under the bijection described in the comment is: ({1,2,3},{}), ({},{1,2,3}), ({1,2,3},{1}), ({1,2,3},{2}), ({1,2,3},{3}), ({1},{1,2,3}), ({2},{1,2,3}), ({3},{1,2,3}), ({1,2,3},{1,2}), ({1,2,3},{1,3}), ({1,2,3},{2,3}), ({1,2},{1,2,3}), ({1,3},{1,2,3}), ({2,3},{1,2,3}). Here exactly one entry is {1,..,n}={1,2,3} and the other is a proper subset. (End)
An elephant sequence, see A175654. For the corner squares just one A[5] vector, with decimal value 170, leads to this sequence. For the central square this vector leads to the companion sequence A151821.  Johannes W. Meijer, Aug 15 2010
Conjecture: a(n) is the least m>0 such that A007814(A000108(m)) = n, where A000108 gives the Catalan numbers and A007814(x) is the 2adic valuation of x (cf. A048881).  L. Edson Jeffery, Nov 26 2015
Also, the decimal representation of the diagonal from the corner to the origin of the nth stage of growth of the twodimensional cellular automaton defined by "Rule 645", based on the 5celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero.  Robert Price, Jul 19 2017
