%I
%S 1,9,49,225,961,3969,16129,65025,261121,1046529,4190209,16769025,
%T 67092481,268402689,1073676289,4294836225,17179607041,68718952449,
%U 274876858369,1099509530625,4398042316801,17592177655809
%N a(n) = (2^n  1)^2.
%C Number of n X n matrices over GF(2) with rank 1.
%C Let M_2(n) be the 2 X 2 matrix M_2(n)(i,j)=i^n+j^n; then a(n)=det(M_2(n)).  _Benoit Cloitre_, Apr 21 2002
%C Number of distinct lines through the origin in the ndimensional lattice of side length 3. A001047 gives lines in the ndimensional lattice of side length 2, A049691 gives lines in the 2dimensional lattice of side length n.  _Joshua Zucker_, Nov 19 2003
%C a(n) is also the number of ntuples with each entry chosen from the subsets of {1,2} such that the intersection of all n entries is empty. See example. This may be shown by exhibiting a bijection to a set whose cardinality is obviously (2^n1)^2, namely the set of all pairs with each entry chosen from the 2^n1 proper subsets of {1,..,n}, i.e., for both entries {1,..,n} is forbidden. The bijection is given by (X_1,..,X_n) > (Y_1,Y_2) where for each j in {1,2} and each i in {1,..,n}, i is in Y_j if and only if j is in X_i. For example, a(2)=9, because the nine pairs of subsets of {1,2} with empty intersection are: ({},{}), ({},{1}), ({},{2}), ({},{1,2}), ({1},{}), ({2},{}), ({1,2},{}), ({1},{2}), ({2},{1}).  Peter C. Heinig (algorithms(AT)gmx.de), Apr 13 2007
%C Partial sums of A165665.  _J. M. Bergot_, Dec 06 2014
%C Except for a(1)=4, the number of active (ON,black) cells at stage 2^n1 of the twodimensional cellular automaton defined by "Rule 737", based on the 5celled von Neumann neighborhood.  _Robert Price_, May 23 2016
%C Apparently (with offset 0) also the number of active cells at state 2^n1 of the automaton defined by "Rule 7".  _Robert Price_, Apr 12 2016
%C a(n) is the difference xy where positive integer x has binary form of n leading ones followed by n zeros and nonnegative integer y has binary form of n leading zeros followed by n ones. For example, a(4) = (111100000001111)(base 2) = 24015 = 225 = 15^2. The result follows readily by noting y=2^n1 and x=2^(2*n)1y. Therefore xy=2^(2*n)2^(n+1)+1=(2^n1)^2.  _Dennis P. Walsh_, Sep 19 2016
%C Also the number of dominating sets in the nbarbell graph.  _Eric W. Weisstein_, Jun 29 2017
%C For n > 1, also the number of connected dominating sets in the complete bipartite graph K_n,n.  _Eric W. Weisstein_, Jun 29 2017
%D Stanley, R. P., Enumerative Combinatorics: Volume 1: Wadsworth & Brooks: 1986: p. 11.
%H Harry J. Smith, <a href="/A060867/b060867.txt">Table of n, a(n) for n = 1..200</a>
%H M. Baake, F. Gahler and U. Grimm, <a href="http://arxiv.org/abs/1211.5466">Examples of substitution systems and their factors</a>, arXiv preprint arXiv:1211.5466 [math.DS], 2012.  From _N. J. A. Sloane_, Jan 03 2013
%H Michael Baake, Franz GĂ¤hler, and Uwe Grimm, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Baake/baake3.html">Examples of Substitution Systems and Their Factors</a>, Journal of Integer Sequences, Vol. 16 (2013), #13.2.14.
%H Franck Ramaharo, <a href="https://arxiv.org/abs/1807.05256">A onevariable bracket polynomial for some Turk's head knots</a>, arXiv:1807.05256 [math.CO], 2018.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NearSquarePrime.html">NearSquare Prime</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7, 14, 8).
%F a(n) = (2^n  1)^2 = A000225(n)^2.
%F a(n) = sum_{j=1..n} sum_{k=1..n} binomial(n+j,nk).  _Yalcin Aktar_, Dec 28 2011
%F G.f.: x*(1+2*x)/((1x)(12*x)(14*x)). a(n) = 7*a(n1)14*a(n2)+8*a(n3).  _Colin Barker_, Feb 03 2012
%F E.g.f.: (1  2*exp(x) + exp(3*x))*exp(x).  _Ilya Gutkovskiy_, May 23 2016
%e a(2) = 9 because there are 10 (the second element in sequence A060704) singular 2 X 2 matrices over GF(2), that have rank <= 1 of which only the zero matrix has rank zero so a(2) = 10  1 = 9.
%p [seq ((stirling2(n,2))^2,n=2..23)]; # _Zerinvary Lajos_, Dec 20 2006
%t (2^Range[30]  1)^2 (* _Harvey P. Dale_, Sep 15 2013 *)
%t LinearRecurrence[{7, 14, 8}, {1, 9, 49}, 30] (* _Harvey P. Dale_, Sep 15 2013 *)
%t Table[(2^n  1)^2, {n, 30}] (* _Eric W. Weisstein_, Jun 29 2017 *)
%o (Sage) [stirling_number2(n,2)^2 for n in xrange(2,24)] # _Zerinvary Lajos_, Mar 14 2009
%o (PARI) for (n=1, 200, write("b060867.txt", n, " ", (2^n  1)^2)) \\ _Harry J. Smith_, Jul 13 2009
%o (PARI) a(n) = (2^n  1)^2; \\ _Michel Marcus_, Mar 11 2016
%Y Cf. A000225, A060704, A165665 (first differences)
%K nonn,easy
%O 1,2
%A Ahmed Fares (ahmedfares(AT)mydeja.com), May 04 2001
%E Description changed to formula by _Eric W. Weisstein_, Jun 29 2017
