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%I #16 Oct 03 2021 07:44:00
%S 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,
%T 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,
%U 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
%N 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).
%C The sequences a_k(n) = T(n,k) are self-inverse permutations of the natural numbers:
%C T(n,1) = A054429(n);
%C T(n,2) = A115303(n-1) for n>1;
%C T(n,3) = A115304(n-2) for n>2;
%C T(n,4) = A115305(n-3) for n>3;
%C T(n,5) = A115306(n-4) for n>4;
%C T(n,6) = A115307(n-5) for n>5;
%C T(n,7) = A115308(n-6) for n>6;
%C T(n,8) = A115309(n-7) for n>7;
%C T(n,9) = A106649(n-8) for n>8;
%C T(n,k) = n-k+1 for k > n/2, especially T(n,n)=1 and T(2*n-1,n)=n.
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e Triangle begins:
%e 1;
%e 3, 1;
%e 2, 2, 1;
%e 7, 5, 2, 1;
%e 6, 4, 3, 2, 1;
%e 5, 3, 7, 3, 2, 1;
%e 4, 8, 6, 4, 3, 2, 1;
%e ...
%t 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]];
%t Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Oct 03 2021 *)
%Y Cf. A054429, A115303, A115304, A115305, A115306, A115307, A115308, A115309, A106649.
%K nonn,tabl
%O 1,2
%A _Reinhard Zumkeller_, Jan 20 2006
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