A331047 - OEIS

%I #26 Oct 29 2021 05:09:17

%S 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,

%T 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,

%U 12,11,8,15,6,17,0,1,30,4,27,9,22,16,15,25,6,5

%N 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.

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

%H Alois P. Heinz, <a href="/A331047/b331047.txt">Rows n = 1..75, flattened</a>

%e Triangle T(n,k) begins:

%e   0, 1;

%e   0, 1,  2;

%e   0, 1,  6, 4,  3, 2,  5;

%e   0, 1, 10, 4,  7, 9,  2,  5, 6, 3,  8;

%e   0, 1, 18, 4, 15, 9, 10, 16, 3, 6, 13, 17, 2, 11, 8, 7, 12, 5, 14;

%e   ...

%p b:= proc(n) option remember; local p;

%p       p:= 1+`if`(n=1, 1, b(n-1));

%p       while irem(p, 4)<2 do p:= nextprime(p) od; p

%p     end:

%p T:= n-> (p-> seq(modp(-(-1)^k*ceil(k/2)^2, p), k=0..p-1))(b(n)):

%p seq(T(n), n=1..8);

%t 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 T[n_] := With[{p = b[n]}, Table[Mod[-(-1)^k*Ceiling[k/2]^2, p], {k, 0, p-1}]];

%t Table[T[n], {n, 1, 8}] // Flatten (* _Jean-François Alcover_, Oct 29 2021, after _Alois P. Heinz_ *)

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

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

%Y Row lengths give A045326.

%Y Row sums give A000217(A281664(n)).

%Y Cf. A000040, A230077, A331103.

%K nonn,look,tabf

%O 1,5

%A _Alois P. Heinz_, Jan 08 2020