0,3

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.

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.

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.

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

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

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

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

Sequence in context: A117373 A132677 A010882 * A293207 A106590 A194074

Adjacent sequences: A343635 A343636 A343637 * A343639 A343640 A343641

nonn,base,easy

M. F. Hasler, May 19 2021

approved