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0, 2, 12, 56, 240, 992, 4032, 16256, 65280, 261632, 1047552, 4192256, 16773120, 67100672, 268419072, 1073709056, 4294901760, 17179738112, 68719214592, 274877382656, 1099510579200, 4398044413952, 17592181850112, 70368735789056, 281474959933440
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OFFSET
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0,2
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COMMENTS
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Number of walks of length 2n+2 between any two diametrically opposite vertices of the cycle graph C_8. - Herbert Kociemba, Jul 02 2004
If we consider a(4k+2), then 2^4 == 3^4 == 3 (mod 13); 2^(4k+2) + 3^(4k+2) == 3^k(4+9) == 3*0 == 0 (mod 13). So a(4k+2) can never be prime. - Jose Brox, Dec 27 2005
If k is odd, then a(nk) is divisible by a(n), since: a(nk) = (2^n)^k + (3^n)^k = (2^n + 3^n) [(2^n)^(k-1) - (2^n)^(k-2) (3^n) + - ... + (3^n)^(k-1)]. So the only possible primes in the sequence are a(0) and a(2^n) for n>=1. I've checked that a(2^n) is composite for 3 <= n <= 15. As with Fermat primes, a probabilistic argument suggests that there are only finitely many primes in the sequence. - Dean Hickerson, Dec 27 2005
Let x,y,z be elements from some power set P(n), i.e. the power set of a set of n elements. Define a function f(x,y,z) in the following manner: f(x,y,z) = 1 if x is a subset of y and y is a subset of z and x does not equal z; f(x,y,z) = 0 if x is not a subset of y or y is not a subset of z or x equals z. Now sum f(x,y,z) for all x,y,z of P(n). This gives a(n). - Ross La Haye (rlahaye(AT)new.rr.com), Dec 26 2005
Number of monic (irreducible) polynomials of degree 1 over GF(2^n). - Max Alekseyev, Jan 13 2006
a(n) = A099393(n)-A000225(n+1) = A083420(n)-A099393(n); in binary representation, n>0: n ones followed by n zeros (A138147(n)); A000120(a(n))=n; A023416(a(n))=n; A070939(a(n))=2*n; 2*a(n)+1=A030101(A099393(n)). - Reinhard Zumkeller, Feb 07 2006, Jaroslav Krizek, Aug 02 2009
Let P(A) be the power set of an n-element set A and B be the Cartesian product of P(A) with itself. Then a(n) = the number of (x,y) of B for which x does not equal y. - Ross La Haye (rlahaye(AT)new.rr.com), Jan 02 2008
For n>1: central terms of the triangle in A173787. [From Reinhard Zumkeller, Feb 28 2010]
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REFERENCES
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Putnam Exam. Question A6, Amer. Math. Monthly 107 (Oct 2000), 721-732; see p. 725.
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), Mar 13 2009]
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..170
Index to sequences with linear recurrences with constant coefficients, signature (6,-8).
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FORMULA
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G.f.: 2x/((-1 + 2x)(-1 + 4x)), a(n)=6a(n-1)-8a(n-2). - Herbert Kociemba, Jul 02 2004
E.g.f.: e^(4*x)-e^(2*x). [From Mohammad K. Azarian, Jan 14 2009]
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EXAMPLE
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n=5: a(5)=4^5-2^5=1024-32=992 -> '1111100000'.
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MATHEMATICA
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Table[4^n - 2^n, {n, 40}] (* or *) LinearRecurrence[{6, -8}, {0, 2}, 40] (* From Vladimir Joseph Stephan Orlovsky, Feb 20 2012 *)
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PROG
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(Sage) [4^n - 2^n for n in xrange(0, 23)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 05 2009]
(MAGMA) [4^n - 2^n: n in [0..60]]; // Vincenzo Librandi, Apr 26 2011
(PARI) a(n)=4^n-2^n \\ Charles R Greathouse IV, Jan 30 2012
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CROSSREFS
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Ratio of successive terms of A028365.
Cf. A000225, A060867.
Cf. A161168, A006516, A059153. [From Reinhard Zumkeller, Feb 28 2010]
Sequence in context: A006659 A194771 A127221 * A037130 A181298 A078543
Adjacent sequences: A020519 A020520 A020521 * A020523 A020524 A020525
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KEYWORD
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nonn,easy,changed
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AUTHOR
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N. J. A. Sloane, Simon Plouffe
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STATUS
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approved
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