cp's OEIS Frontend

A363271 Vertical sum of n in base 10.

%I A363271 #20 May 26 2023 08:51:49
%S A363271 1,2,3,4,5,6,7,8,10,1,2,3,4,5,6,7,8,9,11,2,3,4,5,6,7,8,9,10,12,3,4,5,
%T A363271 6,7,8,9,10,11,13,4,5,6,7,8,9,10,11,12,14,5,6,7,8,9,10,11,12,13,15,6,
%U A363271 7,8,9,10,11,12
%N A363271 Vertical sum of n in base 10.
%C A363271 The Sum_{n>=1} a(n)/10^n = 10/81, it is the vertical sum of each integer. The pattern is easy to see but apparently impossible for a program to find any closed form or recurrence. The sequence is generated by adding each integer with an offset of 1 at each step.
%C A363271 If you sum integers with each term divided by 10^n, at n = 9 there are 2 terms in the column 9 + 1 = 10 which is a(10).
%C A363271 Here is the actual sum:
%C A363271   .100000000000000000000
%C A363271   .020000000000000000000
%C A363271   .003000000000000000000
%C A363271   .000400000000000000000
%C A363271   .000050000000000000000
%C A363271   .000006000000000000000
%C A363271   .000000700000000000000
%C A363271   .000000080000000000000
%C A363271   .000000009000000000000
%C A363271   .000000001000000000000
%C A363271   .000000000110000000000
%C A363271   .000000000012000000000
%C A363271   .000000000001300000000
%C A363271   .000000000000140000000
%C A363271   .000000000000015000000
%C A363271   .000000000000001600000
%C A363271   .000000000000000170000
%C A363271   .000000000000000018000
%C A363271   .000000000000000001900
%C A363271   .000000000000000000200
%C A363271   .000000000000000000021
%C A363271   .000000000000000000002
%C A363271   ...
%C A363271 By adding each column we get a(n), which explains why a(9) = 10.
%e A363271 The original sequence is 1 2 3 4 5 6 7 8 9 10 11 12 ... but when we sum digit per digit (in base 10) the sequence is not a rational fraction.
%p A363271 p:=proc(v) local n, aa, nn, s, k, t;
%p A363271     aa := v;
%p A363271     nn := nops(aa);
%p A363271     s := [seq(1 + aa[k]/10^k,
%p A363271         k = 1 .. nops(aa))];
%p A363271     [seq(sum(trunc(10*frac(10^t*s[k])),
%p A363271         k = 1 .. nops(aa)),
%p A363271         t = 0 .. nops(aa))]
%p A363271 end;
%p A363271 # enter a sequence like a(n) = [1, 2, 3, 4, ...] it will return a sequence r such that sum(r(n)/10^n) is equal to sum(a(n)/10^n).
%Y A363271 Cf. A021085 (10/81), A089400 (binary analog).
%K A363271 nonn,base
%O A363271 1,2
%A A363271 _Simon Plouffe_, May 24 2023