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A182929 The rows of the binomial triangle reduced to balanced ternary lists encoded as decimal numbers. 1
0, 1, 4, 7, 16, 61, 160, 547, 1456, 5110, 13120, 44287, 118096, 398581, 1075840, 3720094, 9565936, 32285041, 86093440, 290565367, 774840976, 2711943430, 7059662080, 23535794707, 61987278400, 212693848522, 564945153280, 1979718703900, 5083731656656 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Define an operation ~: ZxZ -> {-1,0,1} (Z integers) by b ~ n = (b sigmod n) [|b sigmod n|=1]. Here [] is the Iverson bracket and sigmod is the signed mod operation defined as b sigmod n = b - n*ceil(b/n - 1/2) if n <> 0 and b otherwise. Further let T(n) = list_{0<=k<n} binomial(n-1,k) ~ n for n > 1 and n if n is 0 or 1. We call T(n) the binomial notation of n.

A non-obvious arithmetical property of the binomial triangle becomes apparent from these balanced ternary lists: the rows which have an odd prime number as an index and only these rows are represented by a ternary list where 1 and -1 are alternating. One might also say that an odd integer is prime iff n > 1 and its binomial notation is zerofree.

Finally a(n) = sum_{0<=k<n} T(n)[k]*3^k.

The sequence starts at n=0 although the definitions can be extended to the negative integers by flipping the signs of the ternary digits. To illustrate the definitions:

  n          T(n)           a(n)

---------------------------------

[-6] -1, 1, 0, 0, 1, -1   [-160]

[-5]  -1, 1, -1, 1, -1    [ -61]

[-4]    -1, 1, 1, -1      [ -16]

[-3]     -1, 1, -1        [  -7]

[-2]       -1, -1         [  -4]

[-1]         -1           [  -1]

[ 0]         0            [   0]

[ 1]         1            [   1]

[ 2]       1, 1           [   4]

[ 3]     1, -1, 1         [   7]

[ 4]   1, -1, -1, 1       [  16]

[ 5]  1, -1, 1, -1, 1     [  61]

[ 6] 1, -1, 0, 0, -1, 1   [ 160]

LINKS

Table of n, a(n) for n=0..28.

Wikipedia, Balanced ternary

MAPLE

A182929 := proc(n) local lop, k, Tlist;

lop := proc(a, n) if n = 0 then a else mods(a, n);

`if`(abs(%)=1, %, 0) fi end;

Tlist := proc(n) `if`(abs(n)<2, n, seq(signum(n)*

lop(binomial(abs(n)-1, k), n), k=0..abs(n)-1)) end:

[Tlist(n)]; signum(n)*add(3^k*%[k+1], k=0..abs(n)-1) end:

seq(A182929(n), n=0..30);

PROG

(PARI)

a(n) = {

  if (n == 0, 0,

  subst(apply(r->if(r==1, 1, r==n-1, -1, 0), lift(Mod(1+'x, n)^(n-1))), 'x, 3));

};

vector(29, n, a(n-1))  \\ Gheorghe Coserea, Nov 21 2016

CROSSREFS

Cf. A007318, A047999, A001317.

Sequence in context: A182561 A246915 A013625 * A124402 A216552 A034736

Adjacent sequences:  A182926 A182927 A182928 * A182930 A182931 A182932

KEYWORD

nonn

AUTHOR

Peter Luschny, Mar 16 2011

STATUS

approved

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Last modified December 9 14:26 EST 2016. Contains 278971 sequences.