A352610 a(0) = 0; a(n+1) = Sum_{d a distinct decimal digit in a(n)} binomial(c(d)-1,2), where c(d) is the number of occurrences of d in a(j), 0 <= j <= n; see comments.

0, 0, 1, 0, 3, 0, 6, 0, 10, 16, 4, 0, 21, 6, 3, 1, 10, 43, 4, 3, 6, 6, 10, 57, 0, 45, 7, 1, 28, 1, 36, 25, 6, 21, 51, 61, 94, 10, 133, 112, 130, 230, 129, 175, 184, 206, 155, 231, 312, 353, 106, 426, 131, 416, 445, 81, 381, 517, 486, 143, 651, 642, 249, 172, 629

Offset

0,5

Comments

In other words, a(n+1) is the sum, taken over each distinct d in a(n), of the number of pairs (d,d) counted from different occurrences of d in a(j), 0 <= j <= n.

Every term is the sum of one or more triangular numbers (A000217). If no digit in a(n) has occurred before, a(n+1) = 0 (there are 7 such terms). Some numbers (0,1,3,4,6,10,...) occur multiple times, while others (2,5,8,11,...) never occur.

a(28844) = 2417583609 is the first pandigital term. Similar to A279818.

Links

Table of n, a(n) for n=0..64.

Example

a(1) = 0 because there are no terms prior to a(0)=0;
a(2) = 1 because 0 has occurred just twice, and 0,0 can be counted just once.
To calculate a(39) given that a(38) = 133: c(1) = 14 so (1,1) can be counted 13*14/2=91 times, c(3) = 7 so (3,3) can be counted 6*7/2 = 21 times. Therefore a(39) = 91 + 21 = 112.

Mathematica

nn = 65; d[_] = -1; a[1] = d[0] = 0; Do[c = DigitCount[a[n - 1]]; MapIndexed[AddTo[d[First[#2]], #1] &, c]; a[n] = Total@ Map[# (# + 1)/2 &@ d[#] &, Position[c, _?(# > 0 &)][[All, 1]] ], {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Mar 23 2022 *)

Crossrefs

Cf. A000217, A279818.

Sequence in context: A328788 A007386 A007385 * A307644 A081978 A117784

Adjacent sequences: A352607 A352608 A352609 * A352611 A352612 A352613

Keyword

nonn,base

Author

David James Sycamore, Mar 16 2022

Status

approved