This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A343638 #19 May 26 2021 02:40:32 %S A343638 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, %T A343638 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, %U A343638 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 %N A343638 a(n) = (Sum of decimal digits of 3*n) / 3. %C A343638 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). %C A343638 In rows 3, 13, 23, 43, 53, 63, ... one has r = x = a-2 (i.e., t = z = a), %C A343638 in rows 6, 16, 26, 36, 46, 56, 76, ... one has r = b but x = a-2 (i.e., z = a), %C A343638 in rows 33, 133, ... one has r = x = a-5, %C A343638 in rows 66, 166, ... one has r = b but x = a-5. %C A343638 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. %F A343638 a(n) = A002264(A007953(A008585(n))), i.e., A343638 = A002264 o A007953 o A008585, by definition. %F A343638 a(3n) = A343639(n)/3, or: A343638 o A008585 = A002264 o A343639. %e A343638 Written in rows of 4+3+3 terms, grouped in the same pattern, the table reads: %e A343638 . %e A343638 a( 0) = 0, 1, 2, 3, 1, 2, 3, 1, 2, 3, %e A343638 a( 10) = 1, 2, 3, 4, 2, 3, 4, 2, 3, 4, %e A343638 a( 20) = 2, 3, 4, 5, 3, 4, 5, 3, 4, 5, %e A343638 a( 30) = 3, 4, 5, 6, 1, 2, 3, 1, 2, 3, %e A343638 . %e A343638 a( 40) = 1, 2, 3, 4, 2, 3, 4, 2, 3, 4, %e A343638 a( 50) = 2, 3, 4, 5, 3, 4, 5, 3, 4, 5, %e A343638 a( 60) = 3, 4, 5, 6, 4, 5, 6, 1, 2, 3, %e A343638 . %e A343638 a( 70) = 1, 2, 3, 4, 2, 3, 4, 2, 3, 4, %e A343638 a( 80) = 2, 3, 4, 5, 3, 4, 5, 3, 4, 5, %e A343638 a( 90) = 3, 4, 5, 6, 4, 5, 6, 4, 5, 6, %e A343638 . %e A343638 a(100) = 1, 2, 3, 4, 2, 3, 4, 2, 3, 4, %e A343638 a(110) = 2, 3, 4, 5, 3, 4, 5, 3, 4, 5, %e A343638 a(120) = 3, 4, 5, 6, 4, 5, 6, 4, 5, 6, %e A343638 a(130) = 4, 5, 6, 7, 2, 3, 4, 2, 3, 4, %e A343638 . %e A343638 a(140) = 2, 3, 4, 5, 3, 4, 5, 3, 4, 5, %e A343638 a(150) = 3, 4, 5, 6, 4, 5, 6, 4, 5, 6, %e A343638 a(160) = 4, 5, 6, 7, 5, 6, 7, 2, 3, 4, %e A343638 . %e A343638 a(170) = 2, 3, 4, 5, 3, 4, 5, 3, 4, 5, %e A343638 a(180) = 3, 4, 5, 6, 4, 5, 6, 4, 5, 6, %e A343638 a(190) = 4, 5, 6, 7, 5, 6, 7, 5, 6, 7, %e A343638 (...) %e A343638 a(330) = 6, 7, 8, 9, 1, 2, 3, 1, 2, 3, %e A343638 (...) %e A343638 a(660) = 6, 7, 8, 9, 7, 8, 9, 1, 2, 3, %e A343638 etc. %t A343638 a[n_] := Plus @@ IntegerDigits[3*n]/3; Array[a, 100, 0] (* _Amiram Eldar_, May 19 2021 *) %o A343638 (PARI) A343638(n)=sumdigits(3*n)/3 %Y A343638 Cf. A007953 (sum of digits), A008585 (3n), A343639 (same for 9), A002264 ([n/3]). %Y A343638 Cf. A083822 (reverse(3n)/3). %K A343638 nonn,base,easy %O A343638 0,3 %A A343638 _M. F. Hasler_, May 19 2021