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A006516 2^(n-1)*(2^n - 1), n >= 0.
(Formerly M4183)
80
0, 1, 6, 28, 120, 496, 2016, 8128, 32640, 130816, 523776, 2096128, 8386560, 33550336, 134209536, 536854528, 2147450880, 8589869056, 34359607296, 137438691328, 549755289600, 2199022206976, 8796090925056, 35184367894528 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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, Jul 22 2003

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

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

Number of walks of length 2n+1 between two nodes at distance 3 in the cycle graph C_8. - Herbert Kociemba, 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, Aug 05 2005

Number of monic irreducible polynomials of degree 2 in GF(2^n)[x]. - Max Alekseyev, 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, 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.; see A109241(n-1). - Artur Jasinski, 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, Jan 02 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 numbers C(2^n - 2,k), which for k = 2 (i.e., {x,y} of the power set of a set) give {1, 0, 1, 15, 91, 435, 1891, 7875, 32131, 129795, 521731, ...}. - Ross La Haye, Jan 15 2008

If n is a member of A000043 then a(n) is also a perfect number (A000396). - Omar E. Pol, Aug 30 2008

a(n) is also the number whose binary representation is A109241(n-1), for n>0. - Omar E. Pol, Aug 31 2008

From Daniel Forgues, Nov 10 2009: (Start)

If we define a spoof-perfect number as:

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.

And if we define a "strong" spoof-perfect number as:

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.

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.

Then:

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).

For example:

a(8) = 2^(8-1)*(2^8-1) = 128*255 = 19840 (where 255 (with factors 3*5*17) is taken as a spoof prime);

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);

a(11) = 2^(11-1)*(2^11-1) = 1024*2047 = 2096128 (where 2047 (with factors 23*89) is taken as a spoof prime);

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).

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.

An example of an even "weak" spoof-perfect number would be:

n = 90 = 2*5*9 (where 9 (with factors 3^2) is taken as a spoof prime);

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).

Euler proved:

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.

The following seems to be true (is there a proof?):

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?

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)

a(n+1) = A173787(2*n+1,n); cf. A020522, A059153. - Reinhard Zumkeller, Feb 28 2010

Also, row sums of triangle A139251. - Omar E. Pol, May 25 2010

From Gary W. Adamson, Oct 26 2010: (Start)

Starting with "1" = (1, 1, 2, 4, 8,...) convolved with A002450: (1, 5, 21, 85, 341,...); and (1, 3, 7, 15, 31,...) convolved with A002001: (1, 3, 12, 48, 192,...). - Gary W. Adamson, Oct 26 2010

a(n) is also the number of toothpicks in the corner toothpick structure of A153006 after 2^n - 1 stages. - Omar E. Pol, Nov 20 2010

The number of n-dimensional odd theta functions of half-integral characteristic. (Gunning, p.22) - Michael Somos, Jan 03 2014

a(n) = A000217((2^n)-1) = 2^(2n-1) - 2^(n-1) is the nearest triangular number below 2^(2n-1); cf. A007582, A233327. - Antti Karttunen, Feb 26 2014

REFERENCES

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

Richard K. Guy, Unsolved problems in number theory, (p 72.) [From Daniel Forgues, Nov 10 2009]

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

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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

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

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata

William Banks, Ahmet Güloǧlu, Wesley Nevans and Filip Saidak, Descartes numbers, Anatomy of integers, 167-173, CRM Proc. Lecture Notes, 46, Amer. Math. Soc., Providence, RI, 2008. MathSciNet review (subscription required).

R. C. Gunning, Riemann Surfaces and Second-Order Theta Functions, Springer-Verlag, 1976. See page 22.

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.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

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

Index entries for sequences related to linear recurrences with constant coefficients, signature (6,-8).

FORMULA

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, Jan 12 2008

a(n+1) = 4*a(n) + 2^n. - Philippe Deléham, Feb 20 2004

Convolution of 4^n and 2^n. - Ross La Haye, Oct 29 2004

a(n+1) = sum{k=0..n, sum{j=0..n, 4^(n-j)*binomial(j, k)}}. - Paul Barry, 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 [Typo corrected by Yosu Yurramendi, Aug 06 2008]

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, Oct 21 2007

a(n) = StirlingS2(2^n,2^n - 1) = C(2^n,2). - Ross La Haye, Jan 15 2008

a(n) = 3*StirlingS2(n+1,4) + StirlingS2(n+2,3). - Ross La Haye, Jun 01 2008

a(n) = ((4^n - 2^n)/2-2^(n-1))/4 , n>=1. - Zerinvary Lajos, Jun 05 2009

a(n) = A153006(2^n - 1). - Omar E. Pol, Nov 20 2010

EXAMPLE

G.f. = x + 6*x^2 + 28*x^3 + 120*x^4 + 496*x^5 + 2016*x^6 + 8128*x^7 + 32640*x^8 + ...

MAPLE

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); # Simon Plouffe in his 1992 dissertation

seq(binomial(2^n, 2), n=0..19); # Zerinvary Lajos, Feb 22 2008

MATHEMATICA

b = {}; a = {1}; Do[AppendTo[b, FromDigits[a, 2]]; a = Prepend[a, 1]; a = AppendTo[a, 0]; , {n, 1, 50}]; b (* with offset 1 *) (* Artur Jasinski, Nov 26 2007 *)

Table[2^(n-1)(2^n-1), {n, 0, 30}] (* or *) LinearRecurrence[{6, -8}, {0, 1}, 30] (* Harvey P. Dale, Jul 15 2011 *)

PROG

(Sage) [lucas_number1(n, 6, 8) for n in xrange(0, 24)] # Zerinvary Lajos, Apr 22 2009

(Sage) [(4^n - 2^n)/2 for n in xrange(0, 24)] # Zerinvary Lajos, Jun 05 2009

(PARI) a(n)=(1<<n-1)<<(n-1) \\ Charles R Greathouse IV, Jun 10 2011

(Maxima) A006516(n):=2^(n-1)*(2^n - 1)$ makelist(A006516(n), n, 0, 30); /* Martin Ettl, Nov 15 2012 */

(Haskell)

a006516 n = a006516_list !! n

a006516_list = 0 : 1 :

    zipWith (-) (map (* 6) $ tail a006516_list) (map (* 8) a006516_list)

-- Reinhard Zumkeller, Oct 25 2013

CROSSREFS

Equals A006095(n+1)-A006095(n). In other words, A006095 gives the partial sums.

Cf. A007582, A010036, A016290, A003462, A079598, A134346, A048473, A151821, A010684.

Cf. A000043, A000396. - Omar E. Pol, Aug 30 2008

Cf. A109241, A139251, A153006. - Omar E. Pol, Aug 31 2008, May 25 2010, Nov 20 2010

Cf. A002450, A002001. - Gary W. Adamson, Oct 26 2010

Cf. A049072.

Sequence in context: A065997 * A171476 A171496 A037131 A225417 A026851

Adjacent sequences:  A006513 A006514 A006515 * A006517 A006518 A006519

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Corrected by Philippe Deléham, Sep 17 2009

STATUS

approved

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Last modified April 19 21:10 EDT 2014. Contains 240777 sequences.