cp's OEIS Frontend

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

%I A062318 #93 Oct 11 2023 11:33:55
%S A062318 0,1,2,5,8,17,26,53,80,161,242,485,728,1457,2186,4373,6560,13121,
%T A062318 19682,39365,59048,118097,177146,354293,531440,1062881,1594322,
%U A062318 3188645,4782968,9565937,14348906,28697813,43046720,86093441,129140162
%N A062318 Numbers of the form 3^m - 1 or 2*3^m - 1; i.e., the union of sequences A048473 and A024023.
%C A062318 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 A062318 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 A062318 The base 3 representation of these numbers is 222...222 or 122...222.
%C A062318 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 A062318 A138002(a(n)) = 0. - _Reinhard Zumkeller_, Feb 26 2008
%C A062318 Also, number of terms in S(n), where S(n) is defined in A114482. - _N. J. A. Sloane_, Nov 13 2014
%C A062318 a(n+1) is also the Moore lower bound on the order of a (4,g)-cage. - _Jason Kimberley_, Oct 30 2011
%H A062318 Vincenzo Librandi, <a href="/A062318/b062318.txt">Table of n, a(n) for n = 1..1000</a>
%H A062318 Daniel Birmajer, Juan B. Gil, Jordan O. Tirrell, and Michael D. Weiner, <a href="https://arxiv.org/abs/2306.03155">Pattern-avoiding stabilized-interval-free permutations</a>, arXiv:2306.03155 [math.CO], 2023.
%H A062318 Sayan Dutta, Lorenz Halbeisen, and Norbert Hungerbühler, <a href="https://arxiv.org/abs/2309.05048">Properties of Hesse derivatives of cubic curves</a>, arXiv:2309.05048 [math.AG], 2023. See p. 9.
%H A062318 Gyula Tasi and Fujio 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 A062318 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3).
%F A062318 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 A062318 From _Paul Curtz_, Feb 21 2008: (Start)
%F A062318 a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3).
%F A062318 Partial sums of A108411. (End)
%F A062318 G.f.: x^2*(1+x)/((1-x)*(1-3*x^2)). - _Colin Barker_, Apr 02 2012
%F A062318 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 A062318 a(n) = 2^((1+(-1)^n)/2) * 3^((2*n-3-(-1)^n)/4) - 1. - _Luce ETIENNE_, Aug 29 2014
%F A062318 a(n) = A052993(n-1) + A052993(n-2). - _R. J. Mathar_, Sep 10 2021
%F A062318 E.g.f.: (1 - 3*cosh(x) + 2*cosh(sqrt(3)*x) - 3*sinh(x) + sqrt(3)*sinh(sqrt(3)*x))/3. - _Stefano Spezia_, Apr 06 2022
%F A062318 a(n) = (1/3)*([n=0] - 3 + (1+(-1)^n)*3^(n/2) + ((1-(-1)^n)/2)*3^((n+1)/2)). - _G. C. Greubel_, Apr 17 2023
%e A062318 The first rows in Pascal's triangle with no multiples of 3 are:
%e A062318 row 0: 1;
%e A062318 row 1: 1, 1;
%e A062318 row 2: 1, 2,  1;
%e A062318 row 5: 1, 5, 10, 10,  5,  1;
%e A062318 row 8: 1, 8, 28, 56, 70, 56, 28, 8, 1;
%p A062318 A062318 :=proc(n)
%p A062318     if n mod 2 = 1 then
%p A062318         3^((n-1)/2)-1
%p A062318     else
%p A062318         2*3^(n/2-1)-1
%p A062318     fi
%p A062318 end proc:
%p A062318 seq(A062318(n), n=1..37); # _Emeric Deutsch_, Feb 03 2005, offset updated
%t A062318 CoefficientList[Series[x^2*(1+x)/((1-x)*(1-3*x^2)),{x,0,40}],x] (* _Vincenzo Librandi_, Apr 20 2012 *)
%t A062318 A062318[n_]:= (1/3)*(Boole[n==0] -3 +3^(n/2)*(2*Mod[n+1,2] +Sqrt[3] *Mod[n, 2]));
%t A062318 Table[A062318[n], {n, 50}] (* _G. C. Greubel_, Apr 17 2023 *)
%o A062318 (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 A062318 (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
%o A062318 (SageMath)
%o A062318 def A062318(n): return (1/3)*(int(n==0) - 3 + 2*((n+1)%2)*3^(n/2) + (n%2)*3^((n+1)/2))
%o A062318 [A062318(n) for n in range(1,41)] # _G. C. Greubel_, Apr 17 2023
%Y A062318 Cf. A062296, A024023, A048473, A114482. Pairwise sums of A052993.
%Y A062318 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 A062318 Cf. A037233 (actual order of a (4,g)-cage).
%Y A062318 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 A062318 nonn,easy
%O A062318 1,3
%A A062318 Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 05 2001
%E A062318 More terms from _Emeric Deutsch_, Feb 03 2005
%E A062318 Entry revised by _N. J. A. Sloane_, Jul 29 2011