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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_{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 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

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