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%I #35 Mar 29 2020 15:58:01
%S 1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,1,3,3,3,3,3,3,3,3,2,1,4,4,4,4,4,
%T 4,4,3,2,1,5,5,5,5,5,5,4,3,2,1,6,6,6,6,6,5,4,3,2,1,7,7,7,7,6,5,4,3,2,
%U 1,8,8,8,7,6,5,4,3,2,1,9,9,8,7,6,5,4,3,2,1,10,9,8,7,6,5,4,3,2,1,3,4,5,6,7,8,9,10,11,10,5
%N n is the a(n)-th positive integer having its digitsum.
%C a(A051885(n)) = 1. - _Reinhard Zumkeller_, Oct 09 2015
%C Ordinal transform of A007953. - _Antti Karttunen_, May 20 2017
%H David A. Corneth, <a href="/A254524/b254524.txt">Table of n, a(n) for n = 1..10000</a>
%e 35 is the 4th positive integer having digitsum 8 (the others before are 8, 17 and 26) so a(35) = 4.
%t c[n_, k_] := If[n >= k, Binomial[n, k], 0]; b[s_, q_, n_] := (s1 = q; If[s <= q*(n - 1), s1 = s + q; Sum[(-1)^i*c[q, i]*c[s1 - 1 - n*i, q - 1], {i, 0, q - 1}], 0]); a[n_] := (r = 1; v = IntegerDigits[n]; l = v[[-1]]; For[i = Length[v] - 1, i >= 1, i--, For[j = 1, j <= v[[i]], j++, r += b[l + j, Length[v] - i, 10]]; l += v[[i]]]; r); Table[a[n], {n, 1, 110}] (* _Jean-François Alcover_, Nov 14 2016, adapted from PARI *)
%t With[{nn=400},#[[3]]&/@Sort[Flatten[Table[Flatten[#,1]&/@MapIndexed[ List,Select[ Table[{n,Total[IntegerDigits[n]]},{n,nn}],#[[2]]==k&]],{k,nn}],1]]](* _Harvey P. Dale_, Mar 29 2020 *)
%o (PARI)
%o \\This algorithm needs a modified binomial.
%o C(n, k)=if(n>=k, binomial(n, k), 0)
%o \\ways to roll s-q with q dice having sides 0 through n - 1.
%o b(s, q, n)=if(s<=q*(n-1), s+=q; sum(i=0, q-1, (-1)^i*C(q, i)*C(s-1-n*i, q-1)), 0)
%o \\main algorithm
%o a(n)={r = 1; v=digits(n); l=v[#v]; forstep(i = #v-1, 1, -1, for(j=1,v[i], r+=b(l+j, #v-i,10)); l+=v[i]);r}
%o (Haskell)
%o import Data.IntMap (empty, findWithDefault, insert)
%o a254524 n = a254524_list !! (n-1)
%o a254524_list = f 1 empty where
%o f x m = y : f (x + 1) (insert q (y + 1) m) where
%o y = findWithDefault 1 q m; q = a007953 x
%o -- _Reinhard Zumkeller_, Oct 09 2015
%Y Cf. A007953, A051885, A069877, A143164, A263017, A263109, A263110.
%Y Cf. A286478 (analogous sequence for factorial base).
%K nonn,base,look,nice
%O 1,10
%A _David A. Corneth_, Jan 31 2015
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