0,3

a(n) is also the number of different lines determined by pair of vertices in an n-dimensional hypercube. The number of these lines modulo being parallel is in A003462. - Ola Veshta (olaveshta(AT)my-deja.com), Feb 15 2001

Let G_n be the elementary abelian group G_n = (C_2)^n for n >= 1: A006516 is the number of times the number -1 appears in the character table of G_n and A007582 is the number of times the number 1. Together the two sequences cover all the values in the table i.e. A006516(n) + A007582(n) = 2^(2n). - Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 01 2001

a(n) counts the n-lettered words formed using four distinct letters, one of which appears an odd number of times. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 22 2003

Number of 0's making up the central triangle in a Pascal's triangle mod 2 gasket. - Lekraj Beedassy (blekraj(AT)yahoo.com), May 14 2004

m-th triangular number, where m is the n-th Mersenne number, i.e. a(n)=A000217(A000225(n)) - Lekraj Beedassy (blekraj(AT)yahoo.com), May 25 2004

Number of walks of length 2n+1 between two nodes at distance 3 in the cycle graph C_8. - Herbert Kociemba (kociemba(AT)t-online.de), Jul 02 2004

The sequence of fractions a(n+1)/(n+1) is the 3rd binomial transform of (1,0,1/3,0,1/5,0,1/7,...). - Paul Barry (pbarry(AT)wit.ie), Aug 05 2005

Number of monic irreducible polynomials of degree 2 in GF(2^n)[x]. - Max Alekseyev (maxal(AT)cs.ucsd.edu), Jan 23 2006

(A007582(n))^2 + a(n)^2 = A007582(2n). E.g. A007582(3) = 36, a(3) = 28; A007582(6) = 2080. 36^2 + 28^2 = 2080. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 17 2006

The sequence 6*a(n), n>=1, gives the number of edges of the Hanoi graph H_4^{n} with 4 pegs and n>=1 discs. - Daniele Parisse (daniele.parisse(AT)eads.com), Jul 28 2006

8*a(n) is the total border length of the 4*n masks used when making an order n regular DNA chip, using the bidimensional Gray code suggested by Pevzner in the book "Computational Molecular Biology" - Bruno Petazzoni (bruno(AT)enix.org), Apr 05 2007

If we start with 1 in binary and at each step we prepend 1 and append 0, we construct this sequence: 1 110 11100 1111000 etc. - Artur Jasinski (grafix(AT)csl.pl), Nov 26 2007

Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which x does not equal y. - Ross La Haye (rlahaye(AT)new.rr.com), Jan 02 2008

Wieder calls these "conjoint usual k-combinations." - Ross La Haye (rlahaye(AT)new.rr.com), Jan 12 2008

Wieder calls these "conjoint usual 2-combinations." The set of "conjoint strict k-combinations" is the subset of conjoint usual k-combinations where the empty set and the set itself are excluded from possible selection. These number C(2^n - 2,k), which for k = 2 (i.e., {x,y} of the power set of a set) gives {1, 0, 1, 15, 91, 435, 1891, 7875, 32131, 129795, 521731 ...} - Ross La Haye (rlahaye(AT)new.rr.com), Jan 15 2008 Ross

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 113.

M. Gardner, Mathematical Carnival, "Pascal's Triangle", pp. 201 Alfred A. Knopf NY 1975.

C. A. Pickover, Wonders of Numbers, Chap. 55, Oxford Univ. Press NY 2000.

T. D. Noe, Table of n, a(n) for n=0..200

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008).

G.f.: x/((1-2*x)*(1-4*x)). E.g.f. for a(n+1), n>=0: 2*exp(4*x)-exp(2*x).

a(n)=2^(n-1)*stirling2(n+1, 2), n>=0, with stirling2(n, m)=A008277(n, m). Second column of triangle A075497.

a(n) = StirlingS2(2^n,2^n - 1) = C(2^n,2). - Ross La Haye (rlahaye(AT)new.rr.com), Jan 12 2008

a(n+1) = 4*a(n) + 2^n . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 20 2004

Convolution of 4^n and 2^n - Ross La Haye (rlahaye(AT)new.rr.com), Oct 29 2004

a(n+1)=sum{k=0..n, sum{j=0..n, 4^(n-j)*binomial(j, k)}}; - Paul Barry (pbarry(AT)wit.ie), Aug 05 2005

a(n+2) = 6*a(n-1)-8*a(n), a(1)=1, a(2)=6 - Daniele Parisse (daniele.parisse(AT)eads.com), Jul 28 2006

Row sums of triangle A134346. Also, binomial transform of A048473: (1, 5, 17, 53, 161,...); double bt of A046055: (1, 4, 8, 16, 32, 64,...) and triple bt of A010684: (1, 3, 1, 3, 1, 3,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 21 2007

a(n) = StirlingS2(2^n,2^n - 1) = C(2^n,2). - Ross La Haye (rlahaye(AT)new.rr.com), Jan 15 2008 Ross

GBC := proc(n, k, q) local i; mul( (q^(n-i)-1)/(q^(k-i)-1), i=0..k-1); end; # define q-ary Gaussian binomial coefficient [ n, k ]_q

[ seq(GBC(n+1, 2, 2)-GBC(n, 2, 2), n=0..30) ]; # produces A006516

A006516:=1/(4*z-1)/(2*z-1); [Conjectured by S. Plouffe in his 1992 dissertation.]

seq(binomial(2^n, 2), n=0..19); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 22 2008

b = {}; a = {1}; Do[AppendTo[b, FromDigits[a, 2]]; a = Prepend[a, 1]; a = AppendTo[a, 0]; , {n, 1, 50}]; b - Artur Jasinski (grafix(AT)csl.pl), Nov 26 2007

Cf. A016290, A003462, A079598.

A006095(n+1)-A006095(n), i.e. the differences between Gaussian binomial coefficients [ n+1,2 ]-[ n,2 ] (n >= 0).

Cf. A010036.

Cf. A007582.

Cf. A134346, A048473, A046055, A010684.

Adjacent sequences: A006513 A006514 A006515 this_sequence A006517 A006518 A006519

Sequence in context: A002694 A007691 A065997 this_sequence A037131 A026851 A002693

nonn,nice,easy

njas