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%I #28 Jun 03 2021 03:31:24
%S 0,1,2,3,4,5,6,7,8,9,10,11,1,2,3,4,5,6,7,8,9,10,11,12,2,3,4,5,6,7,8,9,
%T 10,11,12,13,3,4,5,6,7,8,9,10,11,12,13,14,4,5,6,7,8,9,10,11,12,13,14,
%U 15,5,6,7,8,9,10,11,12,13,14,15,16,6,7,8,9,10,11,12,13,14,15,16,17,7,8
%N Sum of digits of n written in base 12.
%C Also the fixed point of the morphism 0->{0,1,2,3,4,5,6,7,8,9,10,11}, 1->{1,2,3,4,5,6,7,8,9,10,11,12}, 2->{2,3,4,5,6,7,8,9,10,11,12,13}, etc. - _Robert G. Wilson v_, Jul 27 2006
%H Reinhard Zumkeller, <a href="/A053832/b053832.txt">Table of n, a(n) for n = 0..10000</a>
%H Jeffrey O. Shallit, <a href="http://www.jstor.org/stable/2322179">Problem 6450</a>, Advanced Problems, The American Mathematical Monthly, Vol. 91, No. 1 (1984), pp. 59-60; <a href="http://www.jstor.org/stable/2322523">Two series, solution to Problem 6450</a>, ibid., Vol. 92, No. 7 (1985), pp. 513-514.
%H Robert Walker, <a href="http://robertinventor.com/ftswiki/Self_Similar_Sloth_Canon_Number_Sequences">Self Similar Sloth Canon Number Sequences</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Duodecimal.html">Duodecimal</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DigitSum.html">Digit Sum</a>.
%F From _Benoit Cloitre_, Dec 19 2002: (Start)
%F a(0) = 0, a(12n+i) = a(n)+i for 0 <= i <= 11.
%F a(n) = n-11*(Sum_{k>0} floor(n/12^k)) = n-11*A064459(n). (End)
%F a(n) = A138530(n,12) for n > 11. - _Reinhard Zumkeller_, Mar 26 2008
%F Sum_{n>=1} a(n)/(n*(n+1)) = 12*log(12)/11 (Shallit, 1984). - _Amiram Eldar_, Jun 03 2021
%e a(20) = 1 + 8 = 9 because 20 is written as 18 base 12.
%t Table[Plus @@ IntegerDigits[n, 12], {n, 0, 85}] (* or *)
%t Nest[ Flatten[ #1 /. a_Integer -> Table[a + i, {i, 0, 11}]] &, {0}, 2] (* _Robert G. Wilson v_, Jul 27 2006 *)
%o (PARI) a(n)=if(n<1,0,if(n%12,a(n-1)+1,a(n/12)))
%o (Haskell)
%o a053832 n = q 0 $ divMod n 12 where
%o q r (0, d) = r + d
%o q r (m, d) = q (r + d) $ divMod m 12
%o -- _Reinhard Zumkeller_, May 15 2011
%Y Cf. A000120, A007953, A064459, A138530.
%K base,nonn
%O 0,3
%A _Henry Bottomley_, Mar 28 2000
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