A256100 In S = A007376 (read as a sequence) the digit S(n) appears a(n) times in the sequence S(1), ..., S(n).

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 4, 5, 2, 6, 2, 7, 2, 8, 2, 9, 2, 10, 2, 11, 2, 12, 2, 3, 2, 4, 13, 5, 6, 7, 3, 8, 3, 9, 3, 10, 3, 11, 3, 12, 3, 13, 3, 4, 3, 5, 14, 6, 14, 7, 8, 9, 4, 10, 4, 11, 4, 12, 4, 13, 4, 14, 4, 5, 4, 6, 15, 7, 15, 8, 15, 9, 10, 11, 5, 12, 5, 13, 5, 14, 5, 15, 5, 6

Offset

1,10

Comments

The motivation to consider this sequence came from the proposal A256379 by Anthony Sand.

This sequence can also be read as an irregular triangle (array) in which a(n, k) is the number of appearances of the k-th digit of n in the digits of 1, ... ,n-1 and the first k digits of n. See the example for the head of this array. The row length is A055842(n), n >= 1.

This can also be described as the ordinal transform of A007376. - Franklin T. Adams-Watters, Oct 10 2015

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Formula

a(n) gives the number of digits A007376(n) in the sequence starting with A007376(1) and ending with A007376(n).

Example

a(10) = 2 because A007376(10) = 1 and that sequence up to n=10 is 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, and 1 appears twice.
a(24) = 10 because A007376(24) = 1 and this is the tenth 1 in A007376 up to, and including, A007376(24).
Read as a tabf array a(n, k) with row length A055842(n) this begins:
   n\k  1  2  ...
   1:   1
   2:   1
   3:   1
   4:   1
   5:   1
   6:   1
   7:   1
   8:   1
   9:   1
  10:   2  1
  11:   3  4
  12:   5  2
  13:   6  2
  14:   7  2
  15:   8  2
  16:   9  2
  17:  10  2
  18:  11  2
  19:  12  2
  20:   3  2
  ...

Mathematica

lim = 120; s = Flatten[IntegerDigits /@ Range@ lim]; f[n_] := Block[{d = IntegerDigits /@ Take[s, n] // Flatten // FromDigits}, DigitCount[d][[If[ s[[n]] == 0, 10, s[[n]] ]] ] ]; Array[f, lim] (* Michael De Vlieger, Apr 08 2015, after Robert G. Wilson v at A007376 *)

Prog

(Haskell)
a256100 n = a256100_list !! (n-1)
a256100_list = f a007376_list $ take 10 $ repeat 1 where
   f (d:ds) counts = y : f ds (xs ++ (y + 1) : ys) where
                           (xs, y:ys) = splitAt d counts
-- Reinhard Zumkeller, Aug 13 2015

Crossrefs

Cf. A007376, A065648.

Sequence in context: A356294 A338091 A284584 * A340383 A117407 A232095

Adjacent sequences: A256097 A256098 A256099 * A256101 A256102 A256103

Keyword

nonn,base,easy,look

Author

Wolfdieter Lang, Apr 08 2015

Status

approved