1,2

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 (benoit7848c(AT)orange.fr), 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 (joshua.zucker(AT)stanfordalumni.org), 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

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

Harry J. Smith, Table of n, a(n) for n=1,...,200

Eric Weisstein's World of Mathematics, Carol Number

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

a(2) = 9 because there are 10 (the second element in sequence A060704) singular 2X2 matrices over GF(2), that have rank <= 1 of which only the zero matrix has rank zero so a(2) = 10 - 1 = 9.

[seq ((stirling2(n, 2))^2, n=2..23)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 20 2006

(Other) SAGE: [stirling_number2(n, 2)^2for n in xrange(2, 24)] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 14 2009]

(PARI) { for (n=1, 200, write("b060867.txt", n, " ", (2^n - 1)^2); ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Jul 13 2009]

A000225, A060704.

Cf. A000225.

Sequence in context: A003297 A012248 A080026 this_sequence A081655 A146798 A055428

Adjacent sequences: A060864 A060865 A060866 this_sequence A060868 A060869 A060870

nonn,easy

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

More terms from Larry Reeves (larryr(AT)acm.org), May 07 2001