A346391 Number of permutations f of {1,...,n} with f(n) = n and f(n-1) > f(1) such that f(1)*f(2) + ... + f(n-1)*f(n) + f(n)*f(1) == 0 (mod n^2).

0, 0, 0, 2, 17, 16, 209, 3192

Offset

3,4

Comments

Conjecture: a(n) > 0 for all n > 5.

Links

Table of n, a(n) for n=3..10.

Example

a(6) = 2, and 2*4 + 4*1 + 1*3 + 3*5 + 5*6 + 6*2 = 3*5 + 5*1 + 1*2 + 2*4 + 4*6 + 6*3 = 2*6^2.
a(7) > 0 with 1*3 + 3*4 + 4*5 + 5*6 + 6*2 + 2*7 + 7*1 = 2*7^2.
a(8) > 0 with 1*5 + 5*3 + 3*6 + 6*4 + 4*7 + 7*2 + 2*8 + 8*1 = 2*8^2.
a(9) > 0 with 1*2 + 2*3 + 3*5 + 5*4 + 4*6 + 6*8 + 8*7 + 7*9 + 9*1 = 3*9^2.
a(10) > 0 with 1*2 + 2*3 + 3*6 + 6*8 + 8*4 + 4*9 + 9*7 + 7*5 + 5*10 + 10*1 = 3*10^2.
a(11) > 0 with 1*3 + 3*4 + 4*5 + 5*8 + 8*6 + 6*9 + 9*7 + 7*10 + 10*2 + 2*11 + 11*1 = 3*11^2.

Mathematica

(* A program to compute a(7): *)
VV[i_]:=VV[i]=Part[Permutations[{1, 2, 3, 4, 5, 6}], i];
n=0; Do[If[VV[i][[1]]<VV[i][[6]]&&Mod[Sum[VV[i][[k]]*VV[i][[k+1]], {k, 1, 5}]+VV[i][[6]]*7+7*VV[i][[1]], 7^2]==0, n=n+1], {i, 1, 6!}]; Print[n]

Crossrefs

Cf. A000290, A322070, A322099, A342965, A346387.

Sequence in context: A359437 A077311 A196732 * A153652 A198290 A198596

Adjacent sequences: A346388 A346389 A346390 * A346392 A346393 A346394

Keyword

nonn,more

Author

Zhi-Wei Sun, Jul 15 2021

Status

approved