|
%I #39 Jan 26 2021 10:18:01
%S 1,2,3,4,5,2,3,4,5,6,3,4,5,6,7,4,5,6,7,8,5,6,7,8,9,2,3,4,5,6,3,4,5,6,
%T 7,4,5,6,7,8,5,6,7,8,9,6,7,8,9,10,3,4,5,6,7,4,5,6,7,8,5,6,7,8,9,6,7,8,
%U 9,10,7,8,9,10,11,4,5,6,7,8,5,6,7,8,9,6,7,8,9,10,7,8,9,10,11,8,9,10,11,12
%N a(n) = 1 + A053824(n-1), where A053824 = sum of digits in base 5.
%C 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.)
%C This means: if A053824 is regarded as a triangle then the rows converge to this sequence.
%C See conjecture in the entry A000120, and the case of base 2 in A063787.
%C From _R. J. Mathar_, Dec 09 2010: (Start)
%C In base b=5, A053824 starts counting up from 1 each time the index wraps around a power of b: A053824(b^k)=1.
%C Obvious recurrences are A053824(m*b^k+i) = m+A053824(i), 1 <= m < b-1, 0 <= i < b^(k-1).
%C 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.
%C 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.
%C 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)
%C 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
%C From _Omar E. Pol_, Dec 10 2010: (Start)
%C 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):
%C ........................................................
%C ................................................ * .....
%C ............................................... ** .....
%C ..................................... * ...... *** .....
%C .................................... ** ..... **** .....
%C .......................... * ...... *** .... ***** .....
%C ......................... ** ..... **** ... ****** .....
%C ............... * ...... *** .... ***** ... ***** ......
%C .............. ** ..... **** .... **** .... **** .......
%C .... * ...... *** ..... *** ..... *** ..... *** ........
%C ... ** ...... ** ...... ** ...... ** ...... ** .........
%C ... * ....... * ....... * ....... * ....... * ..........
%C ........................................................
%C ... b=2 ..... b=3 ..... b=4 ..... b=5 ..... b=6 ........
%C . A000120 . A053735 . A053737 . A053824 . A053827 ......
%C . A063787 . A173523 . A173524 . A173525 . A173526 ......
%C ........................................................
%C ............................................. * ........
%C ............................................ ** ........
%C ........................... * ............. *** ........
%C .......................... ** ............ **** ........
%C ........... *............ *** ........... ***** ........
%C .......... ** .......... **** .......... ****** ........
%C ......... ***.......... ***** ......... ******* ........
%C ........ **** ........ ****** ........ ******** ........
%C ....... ***** ....... ******* ....... ********* ........
%C ...... ****** ...... ******** ....... ******** .........
%C ..... ******* ...... ******* ........ ******* ..........
%C ..... ****** ....... ****** ......... ****** ...........
%C ..... ***** ........ ***** .......... ***** ............
%C ..... **** ......... **** ........... **** .............
%C ..... *** .......... *** ............ *** ..............
%C ..... ** ........... ** ............. ** ...............
%C ..... * ............ * .............. * ................
%C ........................................................
%C ..... b=7 .......... b=8 ............ b=9 ..............
%C ... A053828 ...... A053829 ........ A053830 ............
%C ... A173527 ...... A173528 ........ A173529 ............(End)
%H Reinhard Zumkeller, <a href="/A173525/b173525.txt">Table of n, a(n) for n = 1..3126=5^5+1</a>
%H Robert Walker, <a href="http://robertinventor.com/ftswiki/Self_Similar_Sloth_Canon_Number_Sequences">Self Similar Sloth Canon Number Sequences</a>
%F a(n) = A053824(5^k + n - 1) where k >= ceiling(log_5(n/4)). - _R. J. Mathar_, Dec 09 2010
%p A053825 := proc(n) add(d, d=convert(n,base,5)) ; end proc:
%p A173525 := proc(n) local b,k; b := 5 ; if n < b then n; else k := n/(b-1); k := ceil(log(k)/log(b)) ; A053825(b^k+n-1) ; end if; end proc:
%p seq(A173525(n),n=1..100) ;
%t Total[IntegerDigits[#,5]]+1&/@Range[0,100] (* _Harvey P. Dale_, Jun 14 2015 *)
%o (PARI) A173525(n)={ my(s=1); n--; until(!n\=5, s+=n%5); s } \\ _M. F. Hasler_, Dec 09 2010
%o (PARI) A173525(n)={ my(s=1+(n=divrem(n-1,5))[2]); while((n=divrem(n[1],5))[1],s+=n[2]); s+n[2] } \\ _M. F. Hasler_, Dec 09 2010
%o (Haskell)
%o a173525 = (+ 1) . a053824 . (subtract 1) -- _Reinhard Zumkeller_, Jan 31 2014
%Y Cf. A000120, A053824, A063787, A173523, A173524, A173526, A173527, A173528, A173529.
%K nonn,base,look
%O 1,2
%A _Omar E. Pol_, Feb 20 2010
%E More terms from _Vincenzo Librandi_, Aug 02 2010
| 