Search: id:A006516 Results 1-1 of 1 results found. %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, Table of n, a(n) for n=0..200 %H A006516 Index entries for sequences related to linear recurrences with constant coefficients %H A006516 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. %H A006516 S. Plouffe, 1031 Generating Functions and Conjectures, 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, Applied Mathematics Electronic Notes, vol. 8 (2008). %H A006516 MathSciNet, Descartes numbers. (English summary), 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] [Should be replaced by a real reference! - njas, Nov 11 2009] %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 A037131 A026851 A002693 %Y A006516 Adjacent sequences: A006513 A006514 A006515 this_sequence A006517 A006518 A006519 %K A006516 nonn,nice,easy,new %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 Search completed in 0.002 seconds