A331047 T(n,k) = -(-1)^k*ceiling(k/2)^2 mod p, where p is the n-th prime congruent to 2 or 3 mod 4; triangle T(n,k), n>=1, 0<=k<=p-1, read by rows.

0, 1, 0, 1, 2, 0, 1, 6, 4, 3, 2, 5, 0, 1, 10, 4, 7, 9, 2, 5, 6, 3, 8, 0, 1, 18, 4, 15, 9, 10, 16, 3, 6, 13, 17, 2, 11, 8, 7, 12, 5, 14, 0, 1, 22, 4, 19, 9, 14, 16, 7, 2, 21, 13, 10, 3, 20, 18, 5, 12, 11, 8, 15, 6, 17, 0, 1, 30, 4, 27, 9, 22, 16, 15, 25, 6, 5

Offset

1,5

Comments

Row n is a permutation of {0, 1, ..., A045326(n)-1}.

Links

Alois P. Heinz, Rows n = 1..75, flattened

Example

Triangle T(n,k) begins:
  0, 1;
  0, 1,  2;
  0, 1,  6, 4,  3, 2,  5;
  0, 1, 10, 4,  7, 9,  2,  5, 6, 3,  8;
  0, 1, 18, 4, 15, 9, 10, 16, 3, 6, 13, 17, 2, 11, 8, 7, 12, 5, 14;
  ...

Maple

b:= proc(n) option remember; local p;
      p:= 1+`if`(n=1, 1, b(n-1));
      while irem(p, 4)<2 do p:= nextprime(p) od; p
    end:
T:= n-> (p-> seq(modp(-(-1)^k*ceil(k/2)^2, p), k=0..p-1))(b(n)):
seq(T(n), n=1..8);

Mathematica

b[n_] := b[n] = Module[{p}, p = 1+If[n == 1, 1, b[n-1]]; While[Mod[p, 4]<2, p = NextPrime[p]]; p];
T[n_] := With[{p = b[n]}, Table[Mod[-(-1)^k*Ceiling[k/2]^2, p], {k, 0, p-1}]];
Table[T[n], {n, 1, 8}] // Flatten (* Jean-François Alcover, Oct 29 2021, after Alois P. Heinz *)

Crossrefs

Columns k=0-2 give: A000004, A000012, A281664 (for n>1).

Last elements of rows give A190105(n-1) for n>1.

Row lengths give A045326.

Row sums give A000217(A281664(n)).

Cf. A000040, A230077, A331103.

Sequence in context: A323837 A114709 A293147 * A264550 A089949 A085845

Adjacent sequences: A331044 A331045 A331046 * A331048 A331049 A331050

Keyword

nonn,look,tabf

Author

Alois P. Heinz, Jan 08 2020

Status

approved