A137822 First differences of A137821.

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

1,2

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).

From M. F. Hasler, Mar 15 2008: (Start)

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.

The PARI code given here (function A137822(n)) allows one to calculate hundreds of terms of A107755 in a few microseconds. (End)

Links

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

Formula

a(m) = A137821(m)-A137821(m-1), A137821(m) = Sum_{j=1..m} a(j).

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).

Example

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), ...

Mathematica

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

Prog

(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<<L--); 1 \\ M. F. Hasler, Mar 15 2008

Crossrefs

Cf. A000108, A107755, A137821, A137823, A137824.

Cf. A122983 (record values of this).

Sequence in context: A393781 A326607 A071189 * A300845 A344766 A302714

Adjacent sequences: A137819 A137820 A137821 * A137823 A137824 A137825

Keyword

nonn

Author

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

Status

approved