1,2

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.

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).

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

Sequence in context: A110370 A249517 A353907 * A031044 A367306 A380856

Adjacent sequences: A363268 A363269 A363270 * A363272 A363273 A363274

nonn,base

Simon Plouffe, May 24 2023

approved