A060867 a(n) = (2^n - 1)^2.

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

Offset: 1

Author

Ahmed Fares (ahmedfares(AT)my-deja.com), May 04 2001

Keywords

• nonn
• easy

Comments

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

Partial sums of A165665. - J. M. Bergot, Dec 06 2014

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

Examples

			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.
		

References

• Richard P. Stanley, Enumerative Combinatorics: Volume 1: Wadsworth & Brooks: 1986: p. 11.

Links

• Harry J. Smith, Table of n, a(n) for n = 1..200
• Michael Baake, Franz Gähler and Uwe Grimm, Examples of substitution systems and their factors, Journal of Integer Sequences, Vol. 16 (2013), Article 13.2.14; arXiv preprint, arXiv:1211.5466 [math.DS], 2012-2013. - From _N. J. A. Sloane_, Jan 03 2013
• Franck Ramaharo, A one-variable bracket polynomial for some Turk's head knots, arXiv:1807.05256 [math.CO], 2018.
• Eric Weisstein's World of Mathematics, Near-Square Prime.
• Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Crossrefs

Cf. A000225, A060704, A065443, A165665 (first differences)

Programs

Maple

seq((Stirling2(n+1, 2))^2, n=1..25); # Zerinvary Lajos, Dec 20 2006

Mathematica

(2^Range[30] - 1)^2 (* Harvey P. Dale, Sep 15 2013 *)
LinearRecurrence[{7, -14, 8}, {1, 9, 49}, 30] (* Harvey P. Dale, Sep 15 2013 *)
Table[(2^n - 1)^2, {n, 30}] (* Eric W. Weisstein, Jun 29 2017 *)

PARI

a(n) = (2^n - 1)^2; \\ Michel Marcus, Mar 11 2016

Sage

[stirling_number2(n+1,2)^2 for n in range(1,25)] # Zerinvary Lajos, Mar 14 2009

Formula

a(n) = (2^n - 1)^2 = A000225(n)^2.

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

E.g.f.: (1 - 2*exp(x) + exp(3*x))*exp(x). - Ilya Gutkovskiy, May 23 2016

Sum_{n>=1} 1/a(n) = A065443. - Amiram Eldar, Nov 12 2020

Extensions

Description changed to formula by Eric W. Weisstein, Jun 29 2017