A118112 -id:A118112 - cp's OEIS frontend

A118113 Even Fibbinary numbers + 1; also 2*Fibbinary(n) + 1.

1, 3, 5, 9, 11, 17, 19, 21, 33, 35, 37, 41, 43, 65, 67, 69, 73, 75, 81, 83, 85, 129, 131, 133, 137, 139, 145, 147, 149, 161, 163, 165, 169, 171, 257, 259, 261, 265, 267, 273, 275, 277, 289, 291, 293, 297, 299, 321, 323, 325, 329, 331, 337, 339, 341, 513, 515, 517

Offset: 0

Labos Elemer, Apr 13 2006

m for which binomial(3*m-2,m) (see A117671) is odd, since by Kummer's theorem that happens exactly when the binary expansions of m and 2*m-2 have no 1 bit at the same position in each, and so m odd and no 11 bit pairs except optionally the least significant 2 bits. - Kevin Ryde, Jun 14 2025

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000108, A118112, A022341.
Cf. A003714 (Fibbinary numbers), A022340 (even Fibbinary numbers).
Cf. A117671, A263190, A171791, A263075.

F:= combinat[fibonacci]:
b:= proc(n) local j;
      if n=0 then 0
    else for j from 2 while F(j+1)<=n do od;
         b(n-F(j))+2^(j-2)
      fi
    end:
a:= n-> 2*b(n)+1:
seq(a(n), n=0..70);  # Alois P. Heinz, Aug 03 2012

Select[Table[Mod[Binomial[3*k,k], k+1], {k,1200}], #>0&]

a(n) = A022340(n) + 1.
a(n) = 2*A003714(n) + 1.
Solutions to {x : binomial(3x,x) mod (x+1) != 0 } are given in A022341. The corresponding values of binomial(3x,x) mod (x+1) are given here.

New definition from T. D. Noe, Dec 19 2006

A118114 a(n) = binomial(3n,n) mod((n+1)(n+2)).

Original entry on oeis.org

3, 3, 4, 15, 21, 28, 0, 81, 55, 99, 0, 0, 84, 120, 0, 153, 171, 285, 0, 231, 253, 0, 360, 0, 0, 0, 0, 522, 0, 496, 0, 561, 833, 945, 0, 703, 741, 156, 0, 861, 903, 1419, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2016, 1664, 2145, 2211, 3417, 0, 2415, 2485, 2556, 0

Offset: 1

Labos Elemer, Apr 13 2006

nonn

Compared with A118112: larger nonzero value more often and in non-monotonic order.

			For n=5, binomial(15,5) = 3003 = (5+1)*(5+2)*71 + 21, a(5) = 21, the residue.
Interestingly, a very large zone of zeros occurs between about n=5460 and n=7800, uninterrupted by nonzero residues.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A000108, A118112, A118113.

seq(binomial(3*n,n) mod((n+1)*(n+2)),n=1..71); # Emeric Deutsch, Apr 15 2006

Table[Mod[Binomial[3*k, k], (k + 1)*(k + 2)], {k, 1, 1000}]

a(n) = binomial(3*n,n) % ((n+1)*(n+2)); \\ Michel Marcus, Jan 05 2017

cp's OEIS Frontend

A118113 Even Fibbinary numbers + 1; also 2*Fibbinary(n) + 1.

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