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A060867
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a(n) = (2^n - 1)^2.
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45
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1, 9, 49, 225, 961, 3969, 16129, 65025, 261121, 1046529, 4190209, 16769025, 67092481, 268402689, 1073676289, 4294836225, 17179607041, 68718952449, 274876858369, 1099509530625, 4398042316801, 17592177655809, 70368727400449, 281474943156225, 1125899839733761
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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Number of n X n matrices over GF(2) with rank 1.
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
Number of distinct lines through the origin in the n-dimensional lattice of side length 3. A001047 gives lines in the n-dimensional lattice of side length 2, A049691 gives lines in the 2-dimensional lattice of side length n. - Joshua Zucker, Nov 19 2003
a(n) is also the number of n-tuples 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^n-1)^2, namely the set of all pairs with each entry chosen from the 2^n-1 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
Except for a(1)=4, the number of active (ON,black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 737", based on the 5-celled von Neumann neighborhood. - Robert Price, May 23 2016
Apparently (with offset 0) also the number of active cells at state 2^n-1 of the automaton defined by "Rule 7". - Robert Price, Apr 12 2016
a(n) is the difference x-y 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) = (1111000-00001111)(base 2) = 240-15 = 225 = 15^2. The result follows readily by noting y=2^n-1 and x=2^(2*n)-1-y. Therefore x-y=2^(2*n)-2^(n+1)+1=(2^n-1)^2. - Dennis P. Walsh, Sep 19 2016
Also the number of dominating sets in the n-barbell graph. - Eric W. Weisstein, Jun 29 2017
For n > 1, also the number of connected dominating sets in the complete bipartite graph K_n,n. - Eric W. Weisstein, Jun 29 2017
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REFERENCES
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Richard P. Stanley, Enumerative Combinatorics: Volume 1: Wadsworth & Brooks: 1986: p. 11.
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LINKS
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FORMULA
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a(n) = sum_{j=1..n} sum_{k=1..n} binomial(n+j,n-k). - Yalcin Aktar, Dec 28 2011
G.f.: x*(1+2*x)/((1-x)(1-2*x)(1-4*x)). a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3). - Colin Barker, Feb 03 2012
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EXAMPLE
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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.
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MAPLE
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MATHEMATICA
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LinearRecurrence[{7, -14, 8}, {1, 9, 49}, 30] (* Harvey P. Dale, Sep 15 2013 *)
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PROG
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(Sage) [stirling_number2(n+1, 2)^2 for n in range(1, 25)] # Zerinvary Lajos, Mar 14 2009
(PARI) for (n=1, 200, write("b060867.txt", n, " ", (2^n - 1)^2)) \\ Harry J. Smith, Jul 13 2009
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Ahmed Fares (ahmedfares(AT)my-deja.com), May 04 2001
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EXTENSIONS
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STATUS
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approved
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