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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. 37
0, 1, 2, 5, 8, 17, 26, 53, 80, 161, 242, 485, 728, 1457, 2186, 4373, 6560, 13121, 19682, 39365, 59048, 118097, 177146, 354293, 531440, 1062881, 1594322, 3188645, 4782968, 9565937, 14348906, 28697813, 43046720, 86093441, 129140162 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

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.

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

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

A138002(a(n)) = 0. - Reinhard Zumkeller, Feb 26 2008

Also, number of terms in S(n), where S(n) is defined in A114482. - N. J. A. Sloane, Nov 13 2014

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Gy. Tasi and F. Mizukami, Quantum algebraic-combinatoric study of the conformational properties of n-alkanes, J. Math. Chemistry, 25, 1999, 55-64 (see p. 60).

Index entries for linear recurrences with constant coefficients, signature (1,3,-3).

FORMULA

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

a(n) = a(n-1)+3*a(n-2)-3*a(n-3). Differences: A108411. - Paul Curtz, Feb 21 2008

G.f.: x^2*(1+x)/((1-x)*(1-3*x^2)). - Colin Barker, Apr 02 2012

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

a(n) = 2^((1+(-1)^n)/2) * 3^((2*n-3-(-1)^n)/4) - 1. - Luce ETIENNE, Aug 29 2014

EXAMPLE

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

row 0: 1;

row 1: 1,1;

row 2: 1,2,1;

row 5: 1,5,10,10,5,1;

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

MAPLE

A062318 :=proc(n)

    if n mod 2 = 1 then

        3^((n-1)/2)-1

    else

        2*3^(n/2-1)-1

    fi

end proc:

seq(A062318(n), n=1..37); # Emeric Deutsch, Feb 03 2005, offset updated

MATHEMATICA

CoefficientList[Series[x^2*(1+x)/((1-x)*(1-3*x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Apr 20 2012 *)

PROG

(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

(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

CROSSREFS

Cf. A062296, A024023, A048473, A114482.

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

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

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).

Sequence in context: A006827 A193992 A112346 * A034445 A285459 A259580

Adjacent sequences:  A062315 A062316 A062317 * A062319 A062320 A062321

KEYWORD

nonn,easy

AUTHOR

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

EXTENSIONS

More terms from Emeric Deutsch, Feb 03 2005

Entry revised by N. J. A. Sloane, Jul 29 2011

STATUS

approved

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Last modified September 20 21:08 EDT 2017. Contains 292293 sequences.