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A264847 Pluritriangular numbers: a(0) = 0; a(n+1) = a(n) + the number of digits in terms a(0)..a(n). 4
0, 1, 3, 6, 10, 16, 24, 34, 46, 60, 76, 94, 114, 137, 163, 192, 224, 259, 297, 338, 382, 429, 479, 532, 588, 647, 709, 774, 842, 913, 987, 1064, 1145, 1230, 1319, 1412, 1509, 1610, 1715, 1824, 1937, 2054, 2175, 2300, 2429, 2562, 2699, 2840, 2985, 3134, 3287, 3444, 3605, 3770, 3939 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Due to its generation rule, a(n+1) is the sum of floor(log_10(a(n)))+1 terms of A000217 (triangular numbers), as the name suggests.

This is easy to verify by observing the following table:

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

|  n |  Tn | Tn'| Tn"|...| a(n)|

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

|  1 |   1 |    |    |   |   1 |

|  2 |   3 |    |    |   |   3 |

|  3 |   6 |    |    |   |   6 |

|  4 |  10 |    |    |   |  10 |

|  5 |  15 |  1 |    |   |  16 |

|  6 |  21 |  3 |    |   |  24 |

|  7 |  28 |  6 |    |   |  34 |

|  8 |  36 | 10 |    |   |  46 |

|  9 |  45 | 15 |    |   |  60 |

| 10 |  55 | 21 |    |   |  76 |

| 11 |  66 | 28 |    |   |  94 |

| 12 |  78 | 36 |    |   | 114 |

| 13 |  91 | 45 |  1 |   | 137 |

| 14 | 105 | 55 |  3 |   | 163 |

| 15 | 120 | 66 |  6 |   | 192 |

.

It is evident that each new Tn sequence starts after each a(k) terms of A265108, corresponding to the n (number of digits) change, as also pointed out in A265108 (see also Formula).

LINKS

Francesco Di Matteo, Table of n, a(n) for n = 0..100

FORMULA

a(n) = T(n) + T(n-k(1)) + T(n-(k(1)+ k(2))) + T(n-(k(1)+ k(2) + k(3))) + ... + T(n - Sum_{j=1..i} k(j)) with (n - Sum_{j=1..i} k(j)) > 0, where T are the triangular numbers and where k(j) is A265108(j).

E.g., a(25) = T(25) + T(25 - 4) + T(25 - 4 - 8) = 325 + 231 + 91 = 647.

G.f.: (1-x)^(-3) * Sum_{k>=1} x^(b(k)+1) where b(k) is the first m such that a(m) has k decimal digits (including b(1)=0). - Robert Israel, Dec 14 2015

a(n+1) = 2*a(n) - a(n-1) + floor(log_10(a(n))) + 1. - Danny Rorabaugh, Jan 20 2016

EXAMPLE

a(1) = 1 = 0 + 1 because a(0) = 0 and 0 has 1 digit.

...

a(6) = 24 = 16 + 8 because a(5) = 16 and 0, 1, 3, 6, 10, 16 have 8 digits.

a(7) = 34 = 24 + 10 because a(6) = 24 and 0, 1, 3, 6, 10, 16, 24 have 10 digits.

MAPLE

a[0]:= 0: d[0]:= 1;

for n from 1 to 300 do

  a[n]:= a[n-1] + d[n-1];

  d[n]:= d[n-1] + ilog10(a[n])+1;

od:

seq(a[i], i=0..300); # Robert Israel, Dec 14 2015

MATHEMATICA

a = {0}; Do[AppendTo[a, a[[n - 1]] + Length@ Flatten@ Map[IntegerDigits, a]], {n, 2, 68}]; a (* Michael De Vlieger, Nov 27 2015 *)

PROG

(Python)

a, b = 0, 0

print a

for k in range(1, 101):

   b = b + len(str(a))

   a = a + b

   print a

(PARI) lista(nn) = {v = vector(nn); for (i=2, nn, v[i] = v[i-1] + sum(k=1, i-1, #Str(v[k])); ); v; } \\ Michel Marcus, Dec 05 2015

CROSSREFS

Cf. A000217, A064223, A088235, A102685, A265108.

Sequence in context: A256528 A066377 A259823 * A173653 A122046 A078663

Adjacent sequences:  A264844 A264845 A264846 * A264848 A264849 A264850

KEYWORD

nonn,base

AUTHOR

Francesco Di Matteo, Nov 26 2015

STATUS

approved

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Last modified April 27 01:03 EDT 2018. Contains 303149 sequences. (Running on oeis4.)