A264847 Pluritriangular numbers: a(0) = 0; a(n+1) = a(n) + the number of digits in terms a(0)..a(n).

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

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, end=', ')
for k in range(1, 101):
   b += len(str(a))
   a += b
   print(a, end=', ')
(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