login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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_{k=0..n-1} 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_{k=0..n-1} 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);

MATHEMATICA

lop[a_, n_] := Module[{m}, m = If[n == 0, a, Mod[a, n, -Quotient[Abs[n]-1, 2]]]; If[Abs[m] == 1, m, 0]];

Tlist[n_] := If[Abs[n]<2, {n}, Table[Sign[n]*lop[Binomial[Abs[n]-1, k], n], {k, 0, Abs[n]-1}]];

a[n_] := Module[{t=Tlist[n]}, Sign[n]*Sum[3^k*t[[k+1]], {k, 0, Abs[n]-1}]];

Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 22 2019, from Maple *)

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: A246915 A340600 A013625 * A124402 A216552 A034736

Adjacent sequences:  A182926 A182927 A182928 * A182930 A182931 A182932

KEYWORD

nonn

AUTHOR

Peter Luschny, Mar 16 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 20 06:56 EDT 2021. Contains 345157 sequences. (Running on oeis4.)