Contribution to the OEIS - A395743

 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

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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

a(n) = Sum_{i=1..k} C(d_i), where d_1, d_2, ..., d_k are the digits of n, and C(d_i) is the number of times digit d_i has appeared in the concatenation of all integers from 1 to n-1 plus the digits of n to the left of d_i.

a(n) = A343644(n) - A055642(n).

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

Cf. A007953, A343644, A055642.

Sequence in context: A152481 A157276 A010486 * A298185 A329082 A111423

Adjacent sequences: A395738 A395740 A395741 * A395744 A395745 A395746

KEYWORD

nonn,base,easy,new

AUTHOR

Vincenzo Manto and James C. McMahon, May 05 2026

STATUS

approved