%I A006516 M4183
%S A006516 0,1,6,28,120,496,2016,8128,32640,130816,523776,2096128,8386560,
%T A006516 33550336,134209536,536854528,2147450880,8589869056,34359607296,
%U A006516 137438691328,549755289600,2199022206976,8796090925056,35184367894528
%N A006516 2^(n-1)*(2^n - 1).
%C A006516 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
%C A006516 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
%C A006516 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
%C A006516 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
%C A006516 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
%C A006516 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
%C A006516 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
%C A006516 Number of monic irreducible polynomials of degree 2 in GF(2^n)[x]. -
Max Alekseyev (maxale(AT)gmail.com), Jan 23 2006
%C A006516 (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
%C A006516 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
%C A006516 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
%C A006516 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. - see A109241(n-1).
- Artur Jasinski (grafix(AT)csl.pl), Nov 26 2007
%C A006516 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
%C A006516 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
%C A006516 If n is a member of A000043 then a(n) is also a perfect number A000396.
[From Omar E. Pol (info(AT)polprimos.com), Aug 30 2008]
%C A006516 a(n) is also the number whose binary representation is A109241(n-1),
for n>0. [From Omar E. Pol (info(AT)polprimos.com), Aug 31 2008]
%C A006516 Contribution from Daniel Forgues (squid(AT)zensearch.com), Nov 10 2009:
(Start)
%C A006516 If we define a spoof-perfect number as:
%C A006516 A spoof-perfect number is a number that would be perfect if some (one
or more) of its odd composite factors were wrongly assumed to be
prime, i.e. taken as a spoof prime.
%C A006516 And if we define a "strong" spoof-perfect number as:
%C A006516 A "strong" spoof-perfect number is a spoof-perfect number where sigma(n)
does not reveal the compositeness of the odd composite factors of
n which are wrongly assumed to be prime, i.e. taken as a spoof prime.
%C A006516 The odd composite factors of n which are wrongly assumed to be prime
then have to be obtained additively in sigma(n) and not multiplicatively.
%C A006516 Then:
%C A006516 If 2^n-1 is odd composite but taken as a spoof prime then 2^(n-1)*(2^n-1)
is an even spoof perfect number (and moreover "strong" spoof-perfect.)
%C A006516 For example:
%C A006516 a(8) = 2^(8-1)*(2^8-1) = 128*255 = 19840 (where 255 (with factors 3*5*17)
is taken as a spoof prime)
%C A006516 sigma(a(8)) = (2^8-1)*(255+1) = 255*256 = 2*(128*255) = 2*19840 = 2n
is spoof-perfect (and also "strong" spoof-perfect since 255 is obtained
additively)
%C A006516 a(11) = 2^(11-1)*(2^11-1) = 1024*2047 = 2096128 (where 2047 (with factors
23*89) is taken as a spoof prime)
%C A006516 sigma(a(11)) = (2^11-1)*(2047+1) = 2047*2048 = 2*(1024*2047) = 2*2096128
= 2n is spoof-perfect (and also "strong" spoof-perfect since 2047
is obtained additively)
%C A006516 I did a Google search and didn't find anything about the distinction
between "strong" versus "weak" spoof-perfect numbers. Maybe some
other terminology is used.
%C A006516 An example of an even "weak" spoof-perfect number would be:
%C A006516 n = 90 = 2*5*9 (where 9 (with factors 3^2) is taken as a spoof prime)
%C A006516 sigma(n) = (1+2)*(1+5)*(1+9) = 3*(2*3)*(2*5) = 2*(2*5*(3^2)) = 2*90 =
2n is spoof-perfect (but is not "strong" spoof-perfect since 9 is
obtained multiplicatively as 3^2 and is thus revealed composite)
%C A006516 Euler proved:
%C A006516 If 2^k-1 is a prime number, then 2^(k-1)*(2^k-1) is a perfect number
and every even perfect number has this form.
%C A006516 The following seems to be true (is there a proof?):
%C A006516 If 2^k-1 is an odd composite number taken as a spoof prime, then 2^(k-1)*(2^k-1)
is a "strong" spoof-perfect number and every even "strong" spoof-perfect
number has this form?
%C A006516 There is only one known odd spoof-perfect number (found by Rene Descartes)
but it is a "weak" spoof-perfect number (Cf. 'Descartes numbers'
and 'Unsolved problems in number theory' links below.) (End)
%D A006516 M. Gardner, Mathematical Carnival, "Pascal's Triangle", pp. 201 Alfred
A. Knopf NY 1975.
%D A006516 Richard K. Guy, Unsolved problems in number theory, (p 72.) [From Daniel
Forgues (squid(AT)zensearch.com), Nov 10 2009]
%D A006516 R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 113.
%D A006516 Ross La Haye, Binary Relations on the Power Set of an n-Element Set,
Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6. [From
Ross La Haye (rlahaye(AT)new.rr.com), Feb 22 2009]
%D A006516 C. A. Pickover, Wonders of Numbers, Chap. 55, Oxford Univ. Press NY 2000.
%D A006516 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%H A006516 T. D. Noe, <a href="b006516.txt">Table of n, a(n) for n=0..200</a>
%H A006516 MathSciNet, <a href="http://www.ams.org/mathscinet/pdf/2437973.pdf?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=a\
ll&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=&s5=descartes%20number&s6=&s7=&s8=All\
&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq">Descartes
numbers. (English summary)</a>, Anatomy of integers, 167-173, CRM
Proc. Lecture Notes, 46, Amer. Math. Soc., Providence, RI, 2008.
[From Daniel Forgues (squid(AT)zensearch.com), Nov 10 2009]
%H A006516 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al,
1992.
%H A006516 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
1031 Generating Functions and Conjectures</a>, Universit\'{e} du
Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A006516 Thomas Wieder, The number of certain k-combinations of an n-set, <a href="http:/
/www.math.nthu.edu.tw/~amen/">Applied Mathematics Electronic Notes</
a>, vol. 8 (2008).
%H A006516 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%F A006516 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).
%F A006516 a(n)=2^(n-1)*stirling2(n+1, 2), n>=0, with stirling2(n, m)=A008277(n,
m). Second column of triangle A075497.
%F A006516 a(n) = StirlingS2(2^n,2^n - 1) = C(2^n,2). - Ross La Haye (rlahaye(AT)new.rr.com),
Jan 12 2008
%F A006516 a(n+1) = 4*a(n) + 2^n . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr),
Feb 20 2004
%F A006516 Convolution of 4^n and 2^n - Ross La Haye (rlahaye(AT)new.rr.com), Oct
29 2004
%F A006516 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
%F A006516 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 [Typo corrected by Yosu Yurramendi (yosu.yurramendi(AT)ehu.es),
Aug 06 2008]
%F A006516 Row sums of triangle A134346. Also, binomial transform of A048473: (1,
5, 17, 53, 161,...); double bt of A151821: (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
%F A006516 a(n) = StirlingS2(2^n,2^n - 1) = C(2^n,2). - Ross La Haye (rlahaye(AT)new.rr.com),
Jan 15 2008 Ross
%F A006516 a(n) = 3*StirlingS2(n+1,4) + StirlingS2(n+2,3). - Ross La Haye (rlahaye(AT)new.rr.com),
Jun 01 2008
%F A006516 a(n) = ((4^n - 2^n)/2-2^(n-1))/4 , n>=1 [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Jun 05 2009]
%p A006516 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
%p A006516 [ seq(GBC(n+1,2,2)-GBC(n,2,2), n=0..30) ]; # produces A006516
%p A006516 A006516:=1/(4*z-1)/(2*z-1); [S. Plouffe in his 1992 dissertation.]
%p A006516 seq(binomial(2^n, 2), n=0..19); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Feb 22 2008
%p A006516 with(finance):seq(add(futurevalue( 1, 1, n+k),k=0..n),n=-1..22); - Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Jun 16 2008
%t A006516 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
%o A006516 (Other) sage: [lucas_number1(n,6,8) for n in xrange(0, 24)]# [From Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
%o A006516 (Other) sage: [(4^n - 2^n)/2 for n in xrange(0,24)] # [From Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Jun 05 2009]
%o A006516 (Other) sage: [((4^n - 2^n)/2-2^(n-1))/4 for n in xrange(1,25)] # [From
Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 05 2009]
%Y A006516 A006095(n+1)-A006095(n), i.e. the differences between Gaussian binomial
coefficients [ n+1, 2 ]-[ n, 2 ] (n >= 0).
%Y A006516 Cf. A007582, A010036, A016290, A003462, A079598, A134346, A048473, A151821,
A010684.
%Y A006516 Cf. A000043, A000396. [From Omar E. Pol (info(AT)polprimos.com), Aug
30 2008]
%Y A006516 Cf. A109241. [From Omar E. Pol (info(AT)polprimos.com), Aug 31 2008]
%Y A006516 Sequence in context: A002694 A007691 A065997 this_sequence A171476 A171496
A037131
%Y A006516 Adjacent sequences: A006513 A006514 A006515 this_sequence A006517 A006518
A006519
%K A006516 nonn,nice,easy
%O A006516 0,3
%A A006516 N. J. A. Sloane (njas(AT)research.att.com).
%E A006516 Corrected by Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 17 2009
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