cp's OEIS Frontend

Also the fixed point of the morphism 0->{0,1,2,3,4,5,6,7,8}, 1->{1,2,3,4,5,6,7,8,9}, 2->{2,3,4,5,6,7,8,9,10}, etc. - Robert G. Wilson v, Jul 27 2006

Also: a(n) = A053824(5^k+n-1) in the limit k->infinity, where k plays the role of a row index in A053824. (See the comment by M. F. Hasler for the proof.)

This means: if A053824 is regarded as a triangle then the rows converge to this sequence.

See conjecture in the entry A000120, and the case of base 2 in A063787.

From R. J. Mathar, Dec 09 2010: (Start)

In base b=5, A053824 starts counting up from 1 each time the index wraps around a power of b: A053824(b^k)=1.

Obvious recurrences are A053824(m*b^k+i) = m+A053824(i), 1 <= m < b-1, 0 <= i < b^(k-1).

So A053824 can be decomposed into a triangle T(k,n) = A053824(b^k+n-1), assuming that column indices start at n=1; row lengths are (b-1)*b^k.

There is a self-similarity in these sequences; a sawtooth structure of periodicity b is added algebraically on top of a sawtooth structure of periodicity b^2, on top of a periodicity b^3 etc. This leads to some "fake" finitely periodic substructures in the early parts of each row of T(.,.): often, but not always, a(n+b)=1+a(n). Often, but not always, a(n+b^2)=1+a(n) etc.

The common part of the rows T(.,.) grows with the power of b as shown in the recurrence above, and defines a(n) in the limit of large row indices k. (End)

The two definitions agree because the first 5^r terms in each row correspond to numbers 5^r, 5^r+1,...,5^r+(5^r-1), which are written in base 5 as a leading 1 plus the digits of 0,...,5^r-1. - M. F. Hasler, Dec 09 2010

From Omar E. Pol, Dec 10 2010: (Start)

In the scatter plots of these sequences, the basic structure is an element with b^2 points, where b is the associated base. (Scatter plots are created with the "graph" button of a sequence.) Sketches of these structures look as follows, the horizontal axis a squeezed version of the index n, b consecutive points packed vertically, and the vertical axis a(n):

........................................................

................................................ * .....

............................................... ** .....

..................................... * ...... *** .....

.................................... ** ..... **** .....

.......................... * ...... *** .... ***** .....

......................... ** ..... **** ... ****** .....

............... * ...... *** .... ***** ... ***** ......

.............. ** ..... **** .... **** .... **** .......

.... * ...... *** ..... *** ..... *** ..... *** ........

... ** ...... ** ...... ** ...... ** ...... ** .........

... * ....... * ....... * ....... * ....... * ..........

........................................................

... b=2 ..... b=3 ..... b=4 ..... b=5 ..... b=6 ........

. A000120 . A053735 . A053737 . A053824 . A053827 ......

. A063787 . A173523 . A173524 . A173525 . A173526 ......

........................................................

............................................. * ........

............................................ ** ........

........................... * ............. *** ........

.......................... ** ............ **** ........

........... *............ *** ........... ***** ........

.......... ** .......... **** .......... ****** ........

......... ***.......... ***** ......... ******* ........

........ **** ........ ****** ........ ******** ........

....... ***** ....... ******* ....... ********* ........

...... ****** ...... ******** ....... ******** .........

..... ******* ...... ******* ........ ******* ..........

..... ****** ....... ****** ......... ****** ...........

..... ***** ........ ***** .......... ***** ............

..... **** ......... **** ........... **** .............

..... *** .......... *** ............ *** ..............

..... ** ........... ** ............. ** ...............

..... * ............ * .............. * ................

........................................................

..... b=7 .......... b=8 ............ b=9 ..............

... A053828 ...... A053829 ........ A053830 ............

... A173527 ...... A173528 ........ A173529 ............(End)

A053735 can be obtained as 0 followed by the first 2 terms of this sequence, followed by the first 6 terms, followed by the first 18 terms, ..., followed by the first 2*3^n terms, etc.

Similar observations are possible for: A063787 (base-2 case), and generic comments have been gathered in A173525 (base-5 case).

Fixed point of morphism 1->123, 2->234, 3->345 etc. (start with 1).

It appears that if A053737 is written as a triangle then the rows are initial segments of the present sequence; see the conjecture in A000120.

The comments in A173525 (base b=5 there) apply here with base b=4. The base b=3 is considered in A173523.

If A053829 is written as a triangle then the rows converge to this sequence; see the conjecture in A000120.

The sequence is the base b=8 case in a family of 8 sequences for base b=2 (A063787) and bases 3 to 9 (A173523 to A173529). Common aspects (recurrences etc.) of these are documented in A173525.

If A053827 is regarded as a triangle then the rows converge to this sequence, i.e., a(n) = A053827(6^k+n-1) in the limit k->infinity, where k plays the role of a row index in A053827.

See conjecture in the entry A000120.

This sequence is the base b=6 case equivalent to A063787 (b=2), A173523 (b=3), A173524 (b=4), A173525 (b=5). Generic comments concerning the various bases are in A173525.

If A053828 is regarded as a triangle then the rows converge to this sequence, i.e., a(n) = A053828(7^k+n-1) in the limit k->infinity, where k plays the role of a row index in A053828.

See conjecture in the entry A000120.

This is the case for base b=7 for the sum of digits. A063787 and A173523 to A173526 deal with the bases 2 to 6. A173525 contains generic remarks concerning these 8 sequences which look in equivalent ways at their sum of digits as a sequence with triangular structure.