A021085 -id:A021085 - cp's OEIS frontend

A115595 The sequence 11,0,1,3333,2,3,5555,4,5,7777,6,7,9999,9,0,2222,1,2,4444,3,4,6666,5,6,8888,7,9,11 has three subsequences that have interesting patterns inside it. Namely, 11,0,(1),3333,2,(3),5555,4,(5),7777,6,(7),9999,9,(0),2222,1,(2),4444,3,(4),6666,5,(6),8888,7,(9),11.

11, 0, 1, 3333, 2, 3, 5555, 4, 5, 7777, 6, 7, 9999, 9, 0, 2222, 1, 2, 4444, 3, 4, 6666, 5, 6, 8888, 7, 9, 11

Offset: 1

Satoshi Hashiba (fantasia_sato205(AT)kcc.zaq.ne.jp), Mar 10 2006

You can generate very interesting sequences by using periodic numbers and the square root.

Sqrt[991199991199991199991199991199991199991199991199991199]

You can get this sequence by Sqrt[991199991199991199991199991199991199991199991199991199], where Sqrt is the square root. Or (991199991199991199991199991199991199991199991199991199)/9.

A363271 Vertical sum of n in base 10.

Original entry on oeis.org

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, 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, 7, 8, 9, 10, 11, 12

Offset: 1

Simon Plouffe, May 24 2023

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.
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).
Here is the actual sum:
  .100000000000000000000
  .020000000000000000000
  .003000000000000000000
  .000400000000000000000
  .000050000000000000000
  .000006000000000000000
  .000000700000000000000
  .000000080000000000000
  .000000009000000000000
  .000000001000000000000
  .000000000110000000000
  .000000000012000000000
  .000000000001300000000
  .000000000000140000000
  .000000000000015000000
  .000000000000001600000
  .000000000000000170000
  .000000000000000018000
  .000000000000000001900
  .000000000000000000200
  .000000000000000000021
  .000000000000000000002
  ...
By adding each column we get a(n), which explains why a(9) = 10.

			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.

Cf. A021085 (10/81), A089400 (binary analog).

p:=proc(v) local n, aa, nn, s, k, t;
    aa := v;
    nn := nops(aa);
    s := [seq(1 + aa[k]/10^k,
        k = 1 .. nops(aa))];
    [seq(sum(trunc(10*frac(10^t*s[k])),
        k = 1 .. nops(aa)),
        t = 0 .. nops(aa))]
end;
# 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).

cp's OEIS Frontend

A115595 The sequence 11,0,1,3333,2,3,5555,4,5,7777,6,7,9999,9,0,2222,1,2,4444,3,4,6666,5,6,8888,7,9,11 has three subsequences that have interesting patterns inside it. Namely, 11,0,(1),3333,2,(3),5555,4,(5),7777,6,(7),9999,9,(0),2222,1,(2),4444,3,(4),6666,5,(6),8888,7,(9),11.

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