A137822 First differences of A137821 (numbers such that sum( Catalan(k), k=1..2n) = 0 (mod 3)).

1, 3, 2, 7, 2, 3, 1, 21, 2, 3, 1, 8, 1, 3, 2, 61, 2, 3, 1, 8, 1, 3, 2, 21, 1, 3, 2, 7, 2, 3, 1, 183, 2, 3, 1, 8, 1, 3, 2, 21, 1, 3, 2, 7, 2, 3, 1, 62, 1, 3, 2, 7, 2, 3, 1, 21, 2, 3, 1, 8, 1, 3, 2, 547, 2, 3, 1, 8, 1, 3, 2, 21, 1, 3, 2, 7, 2, 3, 1, 62, 1, 3, 2, 7, 2, 3, 1, 21, 2, 3, 1, 8, 1, 3, 2, 183, 1, 3, 2

Offset: 1

Author

M. F. Hasler, Feb 25 2008, revised Mar 15 2008

Keywords

• nonn

Comments

For the initial term, we use A137821(0)=0 (cf. formula).

Sequence A122983 lists record values of this one, which occur at index 2^j (cf. formula). The fact that these values roughly grow by a factor 3 is explained by the fact that these values are given as the sum of all preceding terms (up to +1 or +2 according to the parity of j, cf. formula).

The only values occurring in this sequence are { 1, 2, 3, 7, 8, 21, 61, 62, 183, 547, 548, 1641,... } = A137823, consisting of the record values a(2^j) and, for every other one of these (i.e. for even j), its successor a(2^j)+1, occurring first as a(3*2^j).

The remarkably simple sequence A137824 (= 1,3,2, 4,12,8,...: pattern 1,3,2 multiplied by powers of 4) gives the index at which the value A137823(m) first occurs. - M. F. Hasler, Mar 15 2008

The PARI code given here (function A137822(n)) allows one to calculate hundreds of terms of A107755 in a few microseconds. - M. F. Hasler, Mar 15 2008

Examples

			Record values are a(1)=1, a(2)=3, a(4)=7, a(8)=21, a(16)=61, ...
Apart from these values, the only other values occurring in the sequence are:
2=a(1)+1=a(3*1), 8=a(4)+1=a(3*4), 62=a(16)+1=a(3*16), ...
		

Links

• M. F. Hasler, Table of n, a(n) for n = 1..499.

Crossrefs

Cf. A000108, A107755, A137821-A137824.

Cf. A122983 (record values of this).

Programs

Mathematica

Join[{1},Differences[Flatten[Position[Accumulate[CatalanNumber[Range[3000]]],?(Mod[#,3]==0&)]]/2]] (* _Harvey P. Dale, Jun 19 2025 *)

PARI

A137822 = D( A137821 ) /* where D(v)=vector(#v-1,i,v[i+1]-v[i]) or D(v)=vecextract(v, "^1")-vecextract(v,"^-1") */

PARI

n=0; A137822=vector(499,i,{ o=n; if( bitand(i,i-1), while(n++ && s+=binomial(4*n-2, 2*n-1)/(2*n)*(10*n-1)/(2*n+1),),s=Mod(0,3); n=2*n+1+log(i+.5)\log(2)%2 ); n-o})

PARI

A137822(n)= local( L=log(n+.5)\log(2) ); while( n>0 || ((n+=2^L) && L=log(n+.5)\log(2)), (n-=2^L) || return( 3^(L+1)\4+1 ); (n-=2^(L-1)) || return( 3^L\4+1+L%2 );n<0 && n+=2<M. F. Hasler, Mar 15 2008

Formula

a(m) = A137821(m)-A137821(m-1), A137821(m)=sum( a(j), j=1..m).

a(2^j) = A122983(j-1) = A137821(2^j-1) + 1 (resp. +2) for j even (resp. odd).

a(3*2^j) = a(2^j) (resp. = a(2^j)+1) for j odd (resp. j even).