

A020988


a(n) = (2/3)*(4^n1).


45



0, 2, 10, 42, 170, 682, 2730, 10922, 43690, 174762, 699050, 2796202, 11184810, 44739242, 178956970, 715827882, 2863311530, 11453246122, 45812984490, 183251937962, 733007751850, 2932031007402, 11728124029610, 46912496118442
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OFFSET

0,2


COMMENTS

Numbers whose binary representation is 10, n times (see A163662(n) for n >= 1).  Alexandre Wajnberg, May 31 2005
Numbers whose base4 representation consists entirely of 2's; twice base4 repunits.  Franklin T. AdamsWatters, Mar 29 2006
Expected time to finish a random Tower of Hanoi problem with 2n disks using optimal moves, so (since 2n is even and A010684(2n)=1) a(n)=A060590(2n).  Henry Bottomley, Apr 05 2001
a(n)=number of derangements of [2n+3] with runs consisting of consecutive integers. E.g., a(1)=10 because the derangements of {1,2,3,4,5} with runs consisting of consecutive integers are 51234, 45123, 34512, 23451, 54123, 53412, 45231, 34521, 54231, 53421 (the bars delimit the runs).  Emeric Deutsch, May 26 2003
For n>0, also smallest numbers having in binary representation exactly n+1 maximal groups of consecutive zeros: A087120(n)=a(n1), see A087116.  Reinhard Zumkeller, Aug 14 2003
Number of walks of length 2n+3 between any two diametrically opposite vertices of the cycle graph C_6. Example: a(0)=2 because in the cycle ABCDEF we have two walks of length 3 between A and D: ABCD and AFED.  Emeric Deutsch, Apr 01 2004
From Paul Barry, May 18 2003: (Start)
Row sums of triangle using cumulative sums of oddindexed rows of Pascal's triangle (start with zeros for completeness):
0 0
1 1
1 4 4 1
1 6 14 14 6 1
1 8 27 49 49 27 8 1 (End)
a(n) gives the position of the nth zero in A173732, i.e., A173732(a(n))=0 for all n and this gives all the zeros in A173732.  Howard A. Landman, Mar 14 2010
Smallest number having alternating bit sum n. Cf. A065359. For n=0,1,..., the last digit of a(n) is 0,2,0,2,... .  Washington Bomfim, Jan 22 2011
Number of toothpicks minus 1 in the toothpick structure of A139250 after 2^n stages.  Omar E. Pol, Mar 15 2012
For n>0 also partial sums of the odd powers of 2 (A004171).  K. G. Stier, Nov 04 2013
Values of m such that binomial(4*m + 2,m) is odd. Cf. A002450.  Peter Bala, Oct 06 2015
For a(n)>2, values of m, such that m is two steps away from a power of two under the Collatz problem.  Roderick MacPhee, Nov 10 2016


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..170
P. Bala, A characterization of A002450, A020988 and A080674.
J. Brillhart and P. Morton, A case study in mathematical research: the GolayRudinShapiro sequence, Amer. Math. Monthly, 103 (1996) 854869.
Andrei K. Svinin, Tuenter polynomials and a Catalan triangle, arXiv:1603.05748 [math.CO], 2016. See p.3.
Index entries for linear recurrences with constant coefficients, signature (5,4).


FORMULA

a(n) = 4a(n1) + 2, a(0)=0.
E.g.f. : (2/3)(exp(4x)exp(x)).  Paul Barry, May 18 2003
a(n) = A007583(n+1)  1 = A039301(n+2)  2 = A083584(n) + 1.  Ralf Stephan, Jun 14 2003
G.f.: 2x/((1x)(14x)).  R. J. Mathar, Sep 17 2008
a(n) = a(n1) + 2^(2n1), a(0) = 0.  Washington Bomfim, Jan 22 2011
a(n) = A193652(2*n).  Reinhard Zumkeller, Aug 08 2011
a(n) = 5*a(n1)  4*a(n2) (n>1), a(0)=0, a(1)=2.  L. Edson Jeffery, Mar 02 2012
a(n) = (2/3)*A024036(n).  Omar E. Pol, Mar 15 2012


MAPLE

A020988 := proc(n)
2*(4^n1)/3 ;
end proc: # R. J. Mathar, Feb 19 2015


MATHEMATICA

Table[ FromDigits[ Flatten[ Table[{1, 0}, {i, n}]], 2], {n, 0, 23}] (* Robert G. Wilson v, Jun 01 2005 *)
(2(4^Range[0, 30]1))/3 (* or *) LinearRecurrence[{5, 4}, {0, 2}, 30] (* Harvey P. Dale, Sep 25 2013 *)


PROG

(PARI) an=0; print1(an, ", "); for(n=1, 23, an+=2^(2*n1); print1(an, ", ")) \\ Washington Bomfim, Jan 22 2011
(MAGMA) [(2/3)*(4^n1): n in [0..40] ]; // Vincenzo Librandi, Apr 28 2011
(PARI) vector(100, n, n; (2/3)*(4^n1)) \\ Altug Alkan, Oct 06 2015
(PARI) Vec(2*z/((1z)*(14*z)) + O(z^30)) \\ Altug Alkan, Oct 11 2015


CROSSREFS

a(n) = A026644(2n).
a(n) = 2*A002450(n). These two sequences are both subsets of A000975.
a(n) = A007583(n)1 = A039301(n+1)2 = A083584(n1)+1.
Cf. A020989.
Sequence in context: A024483 A276666 A084180 * A177238 A084480 A099553
Adjacent sequences: A020985 A020986 A020987 * A020989 A020990 A020991


KEYWORD

nonn,easy,changed


AUTHOR

N. J. A. Sloane


EXTENSIONS

Edited by N. J. A. Sloane, Sep 06 2006


STATUS

approved



