A048716 Numbers n such that binary expansion matches ((0)*00(1?)1)*(0*).

0, 1, 2, 3, 4, 6, 8, 9, 12, 16, 17, 18, 19, 24, 25, 32, 33, 34, 35, 36, 38, 48, 49, 50, 51, 64, 65, 66, 67, 68, 70, 72, 73, 76, 96, 97, 98, 99, 100, 102, 128, 129, 130, 131, 132, 134, 136, 137, 140, 144, 145, 146, 147, 152, 153, 192, 193, 194, 195, 196, 198, 200, 201

Offset: 1

Author

Antti Karttunen, Mar 30 1999

Keywords

• nonn
• base
• easy

Comments

If bit i is 1, then bits i+-2 must be 0. All terms satisfy A048725(n) = 5*n.

It appears that n is in the sequence if and only if C(5n,n) is odd (cf. A003714). - Benoit Cloitre, Mar 09 2003

Yes, as remarked in A048715, "This is easily proved using the well-known result that the multiplicity with which a prime p divides C(n+m,n) is the number of carries when adding n+m in base p." - Jason Kimberley, Dec 21 2011

A116361(a(n)) <= 2. - Reinhard Zumkeller, Feb 04 2006

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Index entries for 2-automatic sequences.
• Index entries for sequences defined by congruent products between domains N and GF(2)[X]
• Index entries for sequences defined by congruent products under XOR

Crossrefs

Superset of A048715 and A048719. Union of A004742 and A003726.

Cf. A048729, A003714, A115845, A115847, A116360.

Programs

Mathematica

Reap[Do[If[OddQ[Binomial[5n, n]], Sow[n]], {n, 0, 400}]][[2, 1]]
(* Second program: *)
filterQ[n_] := With[{bb = IntegerDigits[n, 2]}, MatchQ[bb, {0}|{1}|{1, 1}|{_, 0, , 1, __}|{_ 1, , 0, __}] && !MatchQ[bb, {_, 1, , 1, __}]];
Select[Range[0, 201], filterQ] (* Jean-François Alcover, Dec 31 2020 *)

PARI

is(n)=!bitand(n,n>>2) \\ Charles R Greathouse IV, Oct 03 2016

PARI

list(lim)=my(v=List(),n,t); while(n<=lim, t=bitand(n,n>>2); if(t, n+=1<Charles R Greathouse IV, Oct 22 2021