cp's OEIS Frontend

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.

%I A352610 #18 Jul 01 2025 23:01:47
%S A352610 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,
%T A352610 36,25,6,21,51,61,94,10,133,112,130,230,129,175,184,206,155,231,312,
%U A352610 353,106,426,131,416,445,81,381,517,486,143,651,642,249,172,629
%N 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.
%C A352610 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.
%C A352610 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.
%C A352610 a(28844) = 2417583609 is the first pandigital term. Similar to A279818.
%H A352610 Michael De Vlieger, <a href="/A352610/b352610.txt">Table of n, a(n) for n = 0..10000</a>
%H A352610 Michael De Vlieger, <a href="/A352610/a352610.png">Log log scatterplot of a(n)</a>, n = 0..10^6, representing a(n) = 0 instead as a(n) = 1/2 for visibility.
%e A352610 a(1) = 0 because there are no terms prior to a(0)=0;
%e A352610 a(2) = 1 because 0 has occurred just twice, and 0,0 can be counted just once.
%e A352610 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.
%t A352610 Block[{a, c, d, nn}, 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 *)
%Y A352610 Cf. A000217, A279818.
%K A352610 nonn,base
%O A352610 0,5
%A A352610 _David James Sycamore_, Mar 16 2022