OFFSET
0,3
COMMENTS
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
LINKS
Indranil Ghosh, Table of n, a(n) for n = 0..59049
Jeffrey O. Shallit, Problem 6450, Advanced Problems, The American Mathematical Monthly, Vol. 91, No. 1 (1984), pp. 59-60; Two series, solution to Problem 6450, ibid., Vol. 92, No. 7 (1985), pp. 513-514.
Robert Walker, Self Similar Sloth Canon Number Sequences.
Eric Weisstein's World of Mathematics, Digit Sum.
FORMULA
From Benoit Cloitre, Dec 19 2002: (Start)
a(0) = 0, a(9n+i) = a(n) + i for 0 <= i <= 8;
a(n) = n - 8*Sum_{k>=1} floor(n/9^k) = n - 8*A054898(n). (End)
a(n) = A138530(n,9) for n > 8. - Reinhard Zumkeller, Mar 26 2008
a(n) = Sum_{k>=0} A031087(n,k). - Philippe Deléham, Oct 21 2011
a(0) = 0; a(n) = a(n - 9^floor(log_9(n))) + 1. - Ilya Gutkovskiy, Aug 24 2019
Sum_{n>=1} a(n)/(n*(n+1)) = 9*log(9)/8 (Shallit, 1984). - Amiram Eldar, Jun 03 2021
EXAMPLE
a(20) = 2+2 = 4 because 20 is written as 22 base 9.
From Omar E. Pol, Feb 23 2010: (Start)
It appears that this can be written as a triangle (see the conjecture in the entry A000120):
0;
1,2,3,4,5,6,7,8;
1,2,3,4,5,6,7,8,9,2,3,4,5,6,7,8,9,10,3,4,5,6,7,8,9,10,11,4,5,6,7,8,9,10,11,...
where the rows converge to A173529. (End)
MATHEMATICA
Table[Plus @@ IntegerDigits[n, 9], {n, 0, 100}] (* or *)
Nest[ Flatten[ #1 /. a_Integer -> Table[a + i, {i, 0, 8}]] &, {0}, 3] (* Robert G. Wilson v, Jul 27 2006 *)
PROG
(PARI) a(n)=if(n<1, 0, if(n%9, a(n-1)+1, a(n/9)))
(Magma) [&+Intseq(n, 9):n in [0..100]]; // Marius A. Burtea, Aug 24 2019
CROSSREFS
KEYWORD
base,nonn
AUTHOR
Henry Bottomley, Mar 28 2000
STATUS
approved