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A270048 a(1) = 0; a(n+1) = a(n) + n * the number of digits of a(n). 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified November 28 13:23 EST 2021. Contains 349410 sequences. (Running on oeis4.)