A270048 a(1) = 0; a(n+1) = a(n) + n * the number of digits of a(n).

0, 1, 3, 6, 10, 20, 32, 46, 62, 80, 100, 133, 169, 208, 250, 295, 343, 394, 448, 505, 565, 628, 694, 763, 835, 910, 988, 1069, 1181, 1297, 1417, 1541, 1669, 1801, 1937, 2077, 2221, 2369, 2521, 2677, 2837, 3001, 3169, 3341, 3517, 3697, 3881, 4069, 4261, 4457, 4657, 4861, 5069, 5281, 5497, 5717, 5941

Offset

1,3

Comments

In this sequence each a(n) term is the sum of k-terms, where k is the number of digits of a(n-1).

This is easy to verify by observing the following table:

+----+---------+---------+---------+--+-----+

| n | A000217 | A056000 | A101859 |..| a(n)|

+----+---------+---------+---------+--+-----+

| 1 | 0 | . | . | .| 0 |

| 2 | 1 | . | . | .| 1 |

| 3 | 3 | . | . | .| 3 |

| 4 | 6 | . | . | .| 6 |

| 5 | 10 | 0 | . | .| 10 |

| 6 | 15 | 5 | . | .| 20 |

| 7 | 21 | 11 | . | .| 32 |

| 8 | 28 | 18 | . | .| 46 |

| 9 | 36 | 26 | . | .| 62 |

| 10 | 45 | 35 | . | .| 80 |

| 11 | 55 | 45 | 0 | .| 100 |

| 12 | 66 | 56 | 11 | .| 133 |

| 13 | 78 | 68 | 23 | .| 169 |

| 14 | 91 | 81 | 36 | .| 208 |

| 15 | 105 | 95 | 50 | .| 250 |

| 16 | 120 | 110 | 65 | .| 295 |

| 17 | 136 | 126 | 81 | .| 343 |

.

As we can see each of those terms is a term of a different subsequence, that is generated with the same construction rule, that is: a(n) = n + a(n-1) + Z.

In fact:

A000217 --> a(n) = n + a(n-1) + 0;

A056000 --> a(n) = n + a(n-1) + 4;

A101859 --> a(n) = n + a(n-1) + 10.

And so on, where the Z value is the n value of this sequence when the number of digits of a(n) is greater than that of a(n-1), or Z = Sum_{j=1..i} k(j) where k(j) is A270270(j).

Links

Francesco Di Matteo, Table of n, a(n) for n = 1..1000

Example

a(4) = 3 + 3*1 = 6;
a(5) = 6 + 4*1 = 10;
a(6) = 10 + 5*2 = 20.

Mathematica

a[1] = 0; a[n_] := a[n] = # + (n - 1) If[# == 0, 1, IntegerLength@ #] &@ a[n - 1]; Table[a@ n, {n, 57}] (* Michael De Vlieger, Mar 09 2016 *)

Prog

(Python)
b = 0
print(b, end=', ')
for g in range(1, 100):
   b += g*len(str(b))
   print(b, end=', ')
(PARI) a(n) = if (n==1, 0, prec = a(n-1); prec + (n-1)*#Str(prec)); \\ Michel Marcus, Apr 03 2016

Crossrefs

Cf. A000217, A056000, A101859, A264847, A270270.

Sequence in context: A065614 A183324 A058356 * A295719 A178850 A018171

Adjacent sequences: A270045 A270046 A270047 * A270049 A270050 A270051

Keyword

nonn,base

Author

Francesco Di Matteo, Mar 09 2016

Status

approved