A331163 a(n) is the number of occurrences of the most frequently seen difference between adjacent digits in the concatenation of a(0) to a(n-1), with a(0) = 0, a(1) = 0.

0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 12, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 17, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 19, 20, 21, 21, 21, 21, 22, 23, 24, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 27, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 29, 30, 31, 31, 31, 31, 31

Offset: 0

Author

Scott R. Shannon, Jan 11 2020

Keywords

• nonn
• base

Comments

In the first 10 million terms the most frequently seen digit difference is 1, which leads other digit difference counts for 2589052 terms. The least seen is 9, which only holds the lead for 990 terms and does not become the most frequently seen digit difference until a(23521), after a run of 1228 consecutive terms of 9090. The longest series of unchanging terms begins at a(7727945) = 7040480 which begins a run of 94553 consecutive terms of 7040480.

Examples

			a(2) = 1 as the concatenation of a(0) and a(1) = '00', and the only adjacent digit difference in '00' is 0, and that difference has occurred one time.
a(3) = 1 as the adjacent digit differences in '001' are 0 and 1, both of which have occurred one time.
a(4) = 2 as '0011' contains two pairs of adjacent digits which differ by 0.
a(22) = 12 as '001122334455667788991010' contains twelve pairs of adjacent digits which differ by 1.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A040115.

Programs

Maple

DC:= [0]: last:= 0:
Res:= 0,0:
Ct:= Array(0..9):
Ct[0]:= 1:
for n from 2 to 100 do
  v:= max(Ct);
  Res:= Res, v;
  L:= [last,op(ListTools:-Reverse(convert(v,base,10)))];
  DL:= map(abs,L[2..-1]-L[1..-2]);
  last:= L[-1];
  for i from 1 to nops(DL) do
    Ct[DL[i]]:= Ct[DL[i]]+1
  od;
od:
Res; # Robert Israel, Jan 16 2020