A115310 Triangle read by rows, 0<=k<=n: T(n,k) = if n < 2*k then n-k+1 else (k+1)*T(k-1+floor((n-k+1)/(k+1)),k) + k - (n-k+1) mod (k+1).

1, 3, 1, 2, 2, 1, 7, 5, 2, 1, 6, 4, 3, 2, 1, 5, 3, 7, 3, 2, 1, 4, 8, 6, 4, 3, 2, 1, 15, 7, 5, 9, 4, 3, 2, 1, 14, 6, 4, 8, 5, 4, 3, 2, 1, 13, 17, 11, 7, 11, 5, 4, 3, 2, 1, 12, 16, 10, 6, 10, 6, 5, 4, 3, 2, 1, 11, 15, 9, 5, 9, 13, 6, 5, 4, 3, 2, 1, 10, 14, 8, 14, 8, 12, 7, 6, 5, 4, 3, 2, 1, 9, 13, 15, 13, 7, 11, 15, 7, 6, 5, 4, 3, 2, 1

Offset

1,2

Comments

The sequences a_k(n) = T(n,k) are self-inverse permutations of the natural numbers:

T(n,1) = A054429(n);

T(n,2) = A115303(n-1) for n>1;

T(n,3) = A115304(n-2) for n>2;

T(n,4) = A115305(n-3) for n>3;

T(n,5) = A115306(n-4) for n>4;

T(n,6) = A115307(n-5) for n>5;

T(n,7) = A115308(n-6) for n>6;

T(n,8) = A115309(n-7) for n>7;

T(n,9) = A106649(n-8) for n>8;

T(n,k) = n-k+1 for k > n/2, especially T(n,n)=1 and T(2*n-1,n)=n.

Links

Table of n, a(n) for n=1..105.

Index entries for sequences that are permutations of the natural numbers

Example

Triangle begins:
  1;
  3, 1;
  2, 2, 1;
  7, 5, 2, 1;
  6, 4, 3, 2, 1;
  5, 3, 7, 3, 2, 1;
  4, 8, 6, 4, 3, 2, 1;
  ...

Mathematica

T[n_, k_] := T[n, k] = If[n < 2 k, n - k + 1, (k + 1)*T[k - 1 + Floor[(n - k + 1)/(k + 1)], k] + k - Mod[n - k + 1, k + 1]];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 03 2021 *)

Crossrefs

Cf. A054429, A115303, A115304, A115305, A115306, A115307, A115308, A115309, A106649.

Sequence in context: A016568 A327314 A021888 * A236172 A139381 A225849

Adjacent sequences: A115307 A115308 A115309 * A115311 A115312 A115313

Keyword

nonn,tabl

Author

Reinhard Zumkeller, Jan 20 2006

Status

approved