A276986 Numbers n for which there is a permutation p of (1,2,3,...,n) such that k+p(k) is a Catalan number for 1<=k<=n.

0, 1, 3, 4, 9, 10, 12, 13, 28, 29, 31, 32, 37, 38, 40, 41, 90, 91, 93, 94, 99, 100, 102, 103, 118, 119, 121, 122, 127, 128, 130, 131, 297, 298, 300, 301, 306, 307, 309, 310, 325, 326, 328, 329, 334, 335, 337, 338, 387, 388, 390, 391, 396, 397, 399, 400, 415, 416

Offset: 1

Author

Gary E. Davis, Sep 24 2016

Keywords

• nonn

Comments

A001453 is a subsequence. - Altug Alkan, Sep 29 2016

n>=1 is in the sequence if and only if there is a Catalan number c such that c/2 <= n < c and c-n-1 is in the sequence. - Robert Israel, Nov 20 2016

Examples

			3 is in the sequence because the permutation (1,3,2) added termwise to (1,2,3) yields (2,5,5) and both 2 and 5 are Catalan numbers.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000108, A073364.

Programs

Maple

S:= {0}:
for i from 1 to 8 do
  c:= binomial(2*i,i)/(i+1);
  S:= S union map(t -> c - t - 1, S);
od:
sort(convert(S,list)); # Robert Israel, Nov 20 2016

Mathematica

CatalanTo[n0_] :=
Module[{n = n0}, k = 1; L = {};
  While[CatalanNumber[k] <= 2*n, L = {L, CatalanNumber[k]}; k++];
  L = Flatten[L]]
perms[n0_] := Module[{n = n0, S, func, T, T2},
  func[k_] := Cases[CatalanTo[n], x_ /; 1 <= x - k <= n] - k;
  T = Tuples[Table[func[k], {k, 1, n}]];
  T2 = Cases[T, x_ /; Length[Union[x]] == Length[x]];
  Length[T2]]
Select[Range[41], perms[#] > 0 &]

Formula

a(i) + a(2^n+1-i) = A000108(n+1)-1 for 1<=i<=2^n. - Robert Israel, Nov 20 2016

Extensions

More terms from Alois P. Heinz, Sep 28 2016

a(23)-a(58) from Robert Israel, Nov 18 2016