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A020988 a(n) = (2/3)*(4^n-1). 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 (list; graph; refs; listen; history; text; internal format)
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 base-4 representation consists entirely of 2's; twice base-4 repunits. - Franklin T. Adams-Watters, 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 5|1234, 45|123, 345|12, 2345|1, 5|4|123, 5|34|12, 45|23|1, 345|2|1, 5|4|23|1, 5|34|2|1 (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(n-1), 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 odd-indexed 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 n-th 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 Golay-Rudin-Shapiro sequence, Amer. Math. Monthly, 103 (1996) 854-869.

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(n-1) + 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/((1-x)(1-4x)). - R. J. Mathar, Sep 17 2008

a(n) = a(n-1) + 2^(2n-1), a(0) = 0. - Washington Bomfim, Jan 22 2011

a(n) = A193652(2*n). - Reinhard Zumkeller, Aug 08 2011

a(n) = 5*a(n-1) - 4*a(n-2) (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^n-1)/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*n-1); print1(an, ", ")) \\ Washington Bomfim, Jan 22 2011

(MAGMA) [(2/3)*(4^n-1): n in [0..40] ]; // Vincenzo Librandi, Apr 28 2011

(PARI) vector(100, n, n--; (2/3)*(4^n-1)) \\ Altug Alkan, Oct 06 2015

(PARI) Vec(2*z/((1-z)*(1-4*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(n-1)+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

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Last modified December 7 23:12 EST 2016. Contains 278900 sequences.