A343638 a(n) = (Sum of decimal digits of 3*n) / 3.

0, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5, 3, 4, 5, 3, 4, 5, 6, 1, 2, 3, 1, 2, 3, 1, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5, 3, 4, 5, 3, 4, 5, 6, 4, 5, 6, 1, 2, 3, 1, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5, 3, 4, 5, 3, 4, 5, 6, 4, 5, 6, 4, 5, 6

Offset: 0

Author

M. F. Hasler, May 19 2021

Keywords

• nonn
• base
• easy

Comments

Not surprisingly, the sequence has a nice self-similar structure. It can be written as a table with rows of length 10, which are of the form [a,b,c,d; r,s,t; x,y,z], b = a+1 etc, where in all rows r = 0, 1, 2, 4, 5, 7, 8, ... with r not congruent to 3, 6 or 9 (mod 10), (r,s,t) = (x,y,z) = (b,c,d). When r == 3 (mod 10), then (r,s,t) = (x,y,z); when r == 6 (mod 10), then (r,s,t) = (b,c,d).

In rows 3, 13, 23, 43, 53, 63, ... one has r = x = a-2 (i.e., t = z = a),

in rows 6, 16, 26, 36, 46, 56, 76, ... one has r = b but x = a-2 (i.e., z = a),

in rows 33, 133, ... one has r = x = a-5,

in rows 66, 166, ... one has r = b but x = a-5.

The rows can also be partitioned in groups of 4+3+3 with the initial terms of the rows having exactly the same pattern as the rows, including exceptions. In particular, the first 4 X 4 or 3 X 4 block of these groups (of 4 X 10 resp. 3 X 10 terms) always have constant antidiagonals.

Examples

			Written in rows of 4+3+3 terms, grouped in the same pattern, the table reads:
.
  a(  0) =  0,  1,  2,  3,        1,  2,  3,      1,  2,  3,
  a( 10) =  1,  2,  3,  4,        2,  3,  4,      2,  3,  4,
  a( 20) =  2,  3,  4,  5,        3,  4,  5,      3,  4,  5,
  a( 30) =  3,  4,  5,  6,        1,  2,  3,      1,  2,  3,
.
  a( 40) =  1,  2,  3,  4,        2,  3,  4,      2,  3,  4,
  a( 50) =  2,  3,  4,  5,        3,  4,  5,      3,  4,  5,
  a( 60) =  3,  4,  5,  6,        4,  5,  6,      1,  2,  3,
.
  a( 70) =  1,  2,  3,  4,        2,  3,  4,      2,  3,  4,
  a( 80) =  2,  3,  4,  5,        3,  4,  5,      3,  4,  5,
  a( 90) =  3,  4,  5,  6,        4,  5,  6,      4,  5,  6,
.
  a(100) =  1,  2,  3,  4,        2,  3,  4,      2,  3,  4,
  a(110) =  2,  3,  4,  5,        3,  4,  5,      3,  4,  5,
  a(120) =  3,  4,  5,  6,        4,  5,  6,      4,  5,  6,
  a(130) =  4,  5,  6,  7,        2,  3,  4,      2,  3,  4,
.
  a(140) =  2,  3,  4,  5,        3,  4,  5,      3,  4,  5,
  a(150) =  3,  4,  5,  6,        4,  5,  6,      4,  5,  6,
  a(160) =  4,  5,  6,  7,        5,  6,  7,      2,  3,  4,
.
  a(170) =  2,  3,  4,  5,        3,  4,  5,      3,  4,  5,
  a(180) =  3,  4,  5,  6,        4,  5,  6,      4,  5,  6,
  a(190) =  4,  5,  6,  7,        5,  6,  7,      5,  6,  7,
(...)
  a(330) =  6,  7,  8,  9,        1,  2,  3,      1,  2,  3,
(...)
  a(660) =  6,  7,  8,  9,        7,  8,  9,      1,  2,  3,
etc.
		

Crossrefs

Cf. A007953 (sum of digits), A008585 (3n), A343639 (same for 9), A002264 ([n/3]).

Cf. A083822 (reverse(3n)/3).

Programs

Mathematica

a[n_] := Plus @@ IntegerDigits[3*n]/3; Array[a, 100, 0] (* Amiram Eldar, May 19 2021 *)

PARI

A343638(n)=sumdigits(3*n)/3

Formula

a(n) = A002264(A007953(A008585(n))), i.e., A343638 = A002264 o A007953 o A008585, by definition.

a(3n) = A343639(n)/3, or: A343638 o A008585 = A002264 o A343639.