A249009 -id:A249009 - cp's OEIS frontend

A248034 a(n+1) gives the number of occurrences of the last digit of a(n) so far, up to and including a(n), with a(0)=0.

0, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 2, 10, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 3, 10, 4, 4, 5, 4, 6, 4, 7, 4, 8, 4, 9, 4, 10, 5, 5, 6, 5, 7, 5, 8, 5, 9, 5, 10, 6, 6, 7, 6, 8, 6, 9, 6, 10, 7, 7, 8, 7, 9, 7, 10, 8, 8, 9, 8, 10, 9, 9, 10

Offset: 0

Eric Angelini and M. F. Hasler, Oct 11 2014

In other words, the number to the right of a comma gives the number of occurrences of the digit immediately to the left of the comma, counting from the beginning up to that digit or comma.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Eric Angelini, Digit-counters updating themselves, SeqFan list, Oct 11 2014.
Zak Seidov, Graph of 10^6 terms

Cf. A249068 (analogous sequence in base 8).
Cf. A249009 (analogous sequence which uses the first, not the last digit).
Cf. A249069, A249070, A249144, A249148, A364788.

a:= proc(n) option remember; `if`(n=0, 0,
      coeff(b(n-1), x, irem(a(n-1), 10)))
    end:
b:= proc(n) option remember; `if`(n=0, 1, b(n-1)+
      add(x^i, i=convert(a(n), base, 10)))
    end:
seq(a(n), n=0..120);  # Alois P. Heinz, Oct 18 2014

nn = 120; a[0] = j = 0; c[] := 0; Do[Map[c[#]++ &, IntegerDigits[j]]; a[n] = j = c[Mod[j, 10]], {n, nn}]; Array[a, nn, 0] (* _Michael De Vlieger, Aug 07 2023 *)

c=vector(10);print1(a=0);for(n=1,99,apply(d->c[d+1]++,if(a,digits(a)));print1(","a=c[1+a%10]))
(MIT/GNU Scheme)
;; An implementation of memoization-macro definec can be found for example from: http://oeis.org/wiki/Memoization
(definec (A248034 n) (if (zero? n) n (vector-ref (A248034aux_digit_counts (- n 1)) (modulo (A248034 (- n 1)) 10))))
(definec (A248034aux_digit_counts n) (cond ((zero? n) (vector 1 0 0 0 0 0 0 0 0 0)) (else (let loop ((digcounts-for-n (vector-copy (A248034aux_digit_counts (- n 1)))) (n (A248034 n))) (cond ((zero? n) digcounts-for-n) (else (vector-set! digcounts-for-n (modulo n 10) (+ 1 (vector-ref digcounts-for-n (modulo n 10)))) (loop digcounts-for-n (floor->exact (/ n 10)))))))))
;; Antti Karttunen, Oct 22 2014

from itertools import islice
def A248034_gen(): # generator of terms
    c, clist = 0, [1]+[0]*9
    while True:
        yield c
        c = clist[c%10]
        for d in str(c):
            clist[int(d)] += 1
A248034_list = list(islice(A248034_gen(),30)) # Chai Wah Wu, Dec 13 2022

A249069 a(n+1) gives the number of occurrences of the first digit of a(n) in factorial base (i.e., A099563(a(n))) so far amongst the factorial base representations of all the terms up to and including a(n), with a(0)=0.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 1, 7, 9, 12, 2, 13, 3, 16, 5, 6, 18, 1, 19, 2, 21, 3, 25, 27, 30, 32, 35, 38, 40, 41, 43, 45, 48, 13, 14, 15, 16, 18, 6, 53, 20, 7, 57, 21, 8, 64, 24, 65, 27, 69, 28, 72, 10, 73, 11, 76, 12, 33, 80, 13, 34, 85, 14, 37, 89, 15, 41, 94, 17, 46, 96, 1, 97, 2, 99, 3, 103, 4, 48, 49, 50

Offset: 0

Antti Karttunen, Oct 20 2014

			   a(0) =  0 (by definition)
   a(1) =  1 ('1' in factorial base), as 0 has occurred once in all the preceding terms.
   a(2) =  1 as 1 has occurred once in all the preceding terms.
   a(3) =  2 ('10' in factorial base), as digit '1' has occurred two times in total in all the preceding terms.
   a(4) =  3 ('11' in factorial base), as '1' occurs once in each a(1) and a(2) and a(3).
   a(5) =  5 ('21' in factorial base), as '1' occurs once in each of a(1), a(2) and a(3) and twice at a(4).
   a(6) =  1 as '2' so far occurs only once at a(5)
   a(7) =  7 = '101'
   a(8) =  9 = '111'
   a(9) = 12 = '200'
  a(10) =  2 = '2'
  a(11) = 13 = '201'
  a(12) =  3 = '11'
  a(12) =  3 = '11'
  a(13) = 16 = '220'
  a(14) =  5 = '21'
  a(15) =  6 = '100'
  a(16) = 18 = '300'
  a(17) =  1 = '1'
  a(18) = 19 = '301'
  a(19) =  2 = '10'
  a(20) = 21 = '311'
  a(21) =  3 = '11'
  a(22) = 25 = '1001'
  a(23) = 27 = '1011'
  a(24) = 30 = '1100'
  a(25) = 32 = '1110'
  a(26) = 35 = '1121'
  a(27) = 38 = '1210' as the leftmost digit '1' has occurred 38 times in total in the factorial base expansions of the preceding terms a(0) - a(26).
etc.

Antti Karttunen, Table of n, a(n) for n = 0..10080

Cf. A249009 (analogous sequence in base-10).
Differs from a variant A249070 for the first time at n=27, where a(27) = 38, while A249070(27) = 7.
Cf. also A007623, A099563, A246359.

A358851 a(n+1) is the number of occurrences of the largest digit of a(n) among all the digits of [a(0), a(1), ..., a(n)], with a(0)=0.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, 11, 13, 2, 2, 3, 3, 4, 2, 4, 3, 5, 2, 5, 3, 6, 2, 6, 3, 7, 2, 7, 3, 8, 2, 8, 3, 9, 2, 9, 3, 10, 15, 4, 4, 5, 5, 6, 4, 6, 5, 7, 4, 7, 5, 8, 4, 8, 5, 9, 4, 9, 5, 10, 17, 6, 6, 7, 7, 8, 6

Offset: 0

Bence Bernáth, Dec 08 2022

Up to a(19)=10, the terms are identical to A248034. The branches (distinct lines of terms indicating the largest digit of the preceding term) can be labeled by the counter digit (shown in the scatter plot). From 9 to 1 the branches gradually get fragmented. Below digit 5 it is harder to disentangle the branches in some places. A repeating pattern also appears (shown in the inset of the scatter plot).

Bence Bernáth, Table of n, a(n) for n = 0..10000
Eric Angelini, Digit-counters updating themselves
Bence Bernáth, Scatter plot for n=0..500000
Michael De Vlieger, Log log scatterplot of a(n) n = 1..10^6.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..10^5, with a color function indicating largest digit of preceding term, where black = 0, red = 1, orange = 2, ..., magenta = 9.

Cf. A248034, A249009, A356348, A336514.

length_seq=150;
sequence(1)=0;
seq_for_digits=(num2str(sequence(1))-'0');
for i1=1:1:length_seq
     sequence(i1+1)=sum(seq_for_digits==max((num2str(sequence(i1))-'0'))');
     seq_for_digits=[seq_for_digits, num2str(sequence(i1+1))-'0'];
end

nn = 79; s[] = 0; a[0] = 0; Do[(Set[d, Max[#]]; Map[s[#1] += #2 & @@ # &, Tally[#] ]) &@ IntegerDigits[a[n - 1]]; a[n] = s[d], {n, nn}]; Array[a, nn + 1, 0] (* _Michael De Vlieger, Dec 28 2022 *)

sequence=[0]
length=150
seq_for_digits=list(map(int, list(str(sequence[0]))))
for ii in range(length):
     sequence.append(seq_for_digits.count(max(list(map(int,list(str(sequence[-1])))))))
     seq_for_digits.extend(list(map(int, list(str(sequence[-1])))))

A358967 a(n+1) gives the number of occurrences of the smallest digit of a(n) so far, up to and including a(n), with a(0)=0.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 2, 10, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 3, 10, 4, 4, 5, 4, 6, 4, 7, 4, 8, 4, 9, 4, 10, 5, 5, 6, 5, 7, 5, 8, 5, 9, 5, 10, 6, 6, 7, 6, 8, 6, 9, 6, 10, 7, 7, 8, 7, 9, 7, 10, 8, 8, 9, 8, 10, 9, 9, 10, 10, 11, 22, 12, 23, 14

Offset: 0

Bence Bernáth, Dec 08 2022

Up to a(103)=12, the terms are identical to A248034.

Bence Bernáth, Table of n, a(n) for n = 0..10000
Eric Angelini, Digit-counters updating themselves

Cf. A248034, A249009, A356348, A336514, A358851.

length_seq=150;
sequence(1)=0;
seq_for_digits=(num2str(sequence(1))-'0');
for i1=1:1:length_seq
     sequence(i1+1)=sum(seq_for_digits==min((num2str(sequence(i1))-'0'))');
     seq_for_digits=[seq_for_digits, num2str(sequence(i1+1))-'0'];
end

sequence=[0]
length=150
seq_for_digits=list(map(int, list(str(sequence[0]))))
for ii in range(length):
   sequence.append(seq_for_digits.count(min(list(map(int,list(str(sequence[-1])))))))
   seq_for_digits.extend(list(map(int, list(str(sequence[-1])))))

A359031 a(n+1) gives the number of occurrences of the mode of the digits of a(n) among all the digits of [a(0), a(1), ..., a(n)], with a(0)=0.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 2, 10, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 3, 10, 4, 4, 5, 4, 6, 4, 7, 4, 8, 4, 9, 4, 10, 5, 5, 6, 5, 7, 5, 8, 5, 9, 5, 10, 6, 6, 7, 6, 8, 6, 9, 6, 10, 7, 7, 8, 7, 9, 7, 10

Offset: 0

Bence Bernáth, Dec 12 2022

The mode is the most frequently occurring value among the digits of a(n). When there are multiple values occurring equally frequently, the mode is the smallest of those values.

Up to a(464)=110, the terms are identical to A358967.

Bence Bernáth, Table of n, a(n) for n = 0..20000
Eric Angelini, Digit-counters updating themselves

Cf. A248034, A249009, A356348, A336514, A358967, A358851, A322182.

length_seq=470;
sequence(1)=0;
seq_for_digits=(num2str(sequence(1))-'0');
for i1=1:1:length_seq
     sequence(i1+1)=sum(seq_for_digits==mode((num2str(sequence(i1))-'0'))');
     seq_for_digits=[seq_for_digits, num2str(sequence(i1+1))-'0'];
end

import statistics as stat
sequence=[0]
length=470
seq_for_digits=list(map(int, list(str(sequence[0]))))
for ii in range(length):
    sequence.append(seq_for_digits.count(stat.mode(list(map(int, list(str(sequence[-1])))))))
    seq_for_digits.extend(list(map(int, list(str(sequence[-1])))))

cp's OEIS Frontend

A248034 a(n+1) gives the number of occurrences of the last digit of a(n) so far, up to and including a(n), with a(0)=0.

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A249069 a(n+1) gives the number of occurrences of the first digit of a(n) in factorial base (i.e., A099563(a(n))) so far amongst the factorial base representations of all the terms up to and including a(n), with a(0)=0.

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A358851 a(n+1) is the number of occurrences of the largest digit of a(n) among all the digits of [a(0), a(1), ..., a(n)], with a(0)=0.

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A358967 a(n+1) gives the number of occurrences of the smallest digit of a(n) so far, up to and including a(n), with a(0)=0.

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MATLAB

Python

A359031 a(n+1) gives the number of occurrences of the mode of the digits of a(n) among all the digits of [a(0), a(1), ..., a(n)], with a(0)=0.

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Author

Keywords

Comments

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Crossrefs

Programs

MATLAB

Python