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A062318 Numbers of the form 3^m - 1 or 2*3^m - 1; i.e., the union of sequences A048473 and A024023. 39


%S 0,1,2,5,8,17,26,53,80,161,242,485,728,1457,2186,4373,6560,13121,

%T 19682,39365,59048,118097,177146,354293,531440,1062881,1594322,

%U 3188645,4782968,9565937,14348906,28697813,43046720,86093441,129140162

%N Numbers of the form 3^m - 1 or 2*3^m - 1; i.e., the union of sequences A048473 and A024023.

%C WARNING: The offset of this sequence has been changed from 0 to 1 without correcting the formulas and programs, many of them correspond to the original indexing a(0)=0, a(1)=1, ... - _M. F. Hasler_, Oct 06 2014

%C Numbers n such that no entry in n-th row of Pascal's triangle is divisible by 3, i.e., such that A062296(n) = 0.

%C The base 3 representation of these numbers is 222...222 or 122...222.

%C a(n+1) is the smallest number with ternary digit sum = n: A053735(a(n+1)) = n and A053735(m) <> n for m < a(n+1). - _Reinhard Zumkeller_, Sep 15 2006

%C A138002(a(n)) = 0. - _Reinhard Zumkeller_, Feb 26 2008

%C Also, number of terms in S(n), where S(n) is defined in A114482. - _N. J. A. Sloane_, Nov 13 2014

%C a(n+1) is also the Moore lower bound on the order of a (4,g)-cage. - _Jason Kimberley_, Oct 30 2011

%H Vincenzo Librandi, <a href="/A062318/b062318.txt">Table of n, a(n) for n = 1..1000</a>

%H Gy. Tasi and F. Mizukami, <a href="http://dx.doi.org/10.1023/A:1019163812482">Quantum algebraic-combinatoric study of the conformational properties of n-alkanes</a>, J. Math. Chemistry, 25, 1999, 55-64 (see p. 60).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3).

%F a(n) = 2*3^(n/2-1)-1 if n is even; a(n) = 3^(n/2-1/2)-1 if n is odd. - _Emeric Deutsch_, Feb 03 2005, offset updated

%F a(n) = a(n-1)+3*a(n-2)-3*a(n-3). Differences: A108411. - _Paul Curtz_, Feb 21 2008

%F G.f.: x^2*(1+x)/((1-x)*(1-3*x^2)). - _Colin Barker_, Apr 02 2012

%F a(2n+1) = 3*a(2n-1) + 2; a(2n) = ( a(2n-1) + a(2n+1) )/2. See A060647 for case where a(1)= 1. - _Richard R. Forberg_, Nov 30 2013

%F a(n) = 2^((1+(-1)^n)/2) * 3^((2*n-3-(-1)^n)/4) - 1. - _Luce ETIENNE_, Aug 29 2014

%e The first rows in Pascal's triangle with no multiples of 3 are:

%e row 0: 1;

%e row 1: 1,1;

%e row 2: 1,2,1;

%e row 5: 1,5,10,10,5,1;

%e row 8: 1,8,28,56,70,56,28,8,1;

%p A062318 :=proc(n)

%p if n mod 2 = 1 then

%p 3^((n-1)/2)-1

%p else

%p 2*3^(n/2-1)-1

%p fi

%p end proc:

%p seq(A062318(n), n=1..37); # _Emeric Deutsch_, Feb 03 2005, offset updated

%t CoefficientList[Series[x^2*(1+x)/((1-x)*(1-3*x^2)),{x,0,40}],x] (* _Vincenzo Librandi_, Apr 20 2012 *)

%o (MAGMA) I:=[0, 1, 2]; [n le 3 select I[n] else Self(n-1)+3*Self(n-2)-3*Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Apr 20 2012

%o (PARI) a(n)=3^(n\2)<<bittest(n,0)-1 \\ [Program corresponds to offset=0, a(0)=0, a(1)=1.] - _M. F. Hasler_, Oct 06 2014

%Y Cf. A062296, A024023, A048473, A114482.

%Y Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), this sequence (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7). - _Jason Kimberley_, Oct 30 2011

%Y Cf. A037233 (actual order of a (4,g)-cage).

%Y Smallest number whose base b sum of digits is n: A000225 (b=2), this sequence (b=3), A180516 (b=4), A181287 (b=5), A181288 (b=6), A181303 (b=7), A165804 (b=8), A140576 (b=9), A051885 (b=10).

%K nonn,easy

%O 1,3

%A Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 05 2001

%E More terms from _Emeric Deutsch_, Feb 03 2005

%E Entry revised by _N. J. A. Sloane_, Jul 29 2011

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Last modified December 8 01:46 EST 2019. Contains 329850 sequences. (Running on oeis4.)