A395743
Sum of the cumulative number of previous occurrences of the digits of n in the sequence 1..n excluding the current occurrence of each digit.
0
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 5, 5, 6, 7, 8, 9, 10, 11, 12, 3, 15, 9, 8, 9, 10, 11, 12, 13, 14, 5, 17, 18, 13, 11, 12, 13, 14, 15, 16, 7, 19, 20, 21, 17, 14, 15, 16, 17, 18, 9, 21, 22, 23, 24, 21, 17, 18, 19, 20, 11, 23, 24, 25, 26, 27, 25, 20, 21, 22, 13
OFFSET
1,11
COMMENTS
This sequence is the exclusive counterpart to A343644. While A343644 counts the occurrences of digits in the range [1,n], a(n) counts only the occurrences strictly preceding each digit of n. Formally a(n) = A343644(n) - A055642(n). Thus, this sequence represents the exclusive scan of digit occurrences, whereas A343644 is the inclusive scan. a(n) = 0 for all single-digit n as it measures the cumulative repetition of digits at the exact moment n is formed.
LINKS
Vincenzo Manto, Table of n, a(n) for n = 1..10000
FORMULA
EXAMPLE
For n = 11:
The digits are '1' and '1'.
- First '1': Appeared previously in {1, 10} -> count = 2.
- Second '1': Appeared previously in {1, 10} AND as the first digit of 11 -> count = 3.
a(11) = 2 + 3 = 5.
For n = 12:
The digits are '1' and '2'.
- Digit '1': Appeared in {1, 10, 11 (twice)} -> count = 4.
- Digit '2': Appeared in {2} -> count = 1.
a(12) = 4 + 1 = 5.
MATHEMATICA
a[n_]:=Module[{sm=0, id=IntegerDigits[n], d=IntegerDigits/@Range[n-1]//Flatten}, Do[sm=sm+Count[Join[d, Take[id, i-1]], id[[i]]], {i, IntegerLength[n]}]; sm]; Array[a, 70]
PROG
(Python)
def generate_sequence(limit):
seq = []
counts = [0] * 10
for n in range(1, limit + 1):
v = 0
s = str(n)
for char in s:
d = ord(char) - 48
v += counts[d]
counts[d] += 1
seq.append(v)
return seq
print(generate_sequence(70))
CROSSREFS
KEYWORD
nonn,base,easy,new
AUTHOR
Vincenzo Manto and James C. McMahon, May 05 2026
STATUS
approved
