Hello Math-Fun, Today we want to duplicate only the last digit "d" of a(n): we just make it jump over "d" digits to the right and paste. As usual, we want S to be the lexicographically earliest sequence of distinct positive terms with this property: S = 1, 2, 11, 21, 3, 12, 4, 322, 5, 42, 6, 7, 251, 8, 161, 71, 9, 14, 83, 17, 43, 19, ... Explanation: See the commas of S as "duplicating catapults" to the right; – the last digit of a(1) = 1 is 1; this 1 is duplicated after jumping to the right over 1 digit; – the last digit of a(2) = 2 is 2; this 2 is duplicated after jumping to the right over 2 digits; – the last digit of a(3) = 11 is 1; this 1 is duplicated after jumping to the right over 1 digit; – the last digit of a(4) = 21 is 1; this 1 is duplicated after jumping to the right over 1 digit; etc. How was S built: We start with a(1) = 1: S = 1, . . . The first comma catapults to the right a copy of 1 over 1 free space: S = 1, . 1 . . . What is the smallest available integer obeying the constraints of this pattern? Obviously 2: S = 1, 2, 1 . . . The second comma catapults to the right a copy of 2 over 2 free spaces: S = 1, 2, 1 . 2 . What is the smallest available integer obeying the constraints? Not 1 (already in S), nor 10 as the zero of 10 would jump over 0 digits to the right and bump into the digit 2 already there: S = 1, 2, 10, 2 . As 10 is forbidden, we try 11 – which is alright: S = 1, 2, 11, 2 . The third comma now catapults to the right a copy of 1 over the digit 2 already there: S = 1, 2, 11, 2 1 The digits 2 and 1 at the end of S form 21 – the smallest available integer legally extending S: S = 1, 2, 11, 21, The fourth comma now catapults to the right a copy of 1 over 1 free space: S = 1, 2, 11, 21, . 1 . What is the smallest available integer obeying the constraints of this pattern? Obviously 3: S = 1, 2, 11, 21, 3, 1 . The fifth comma now catapults to the right a copy of 3 over 3 boxes: S = 1, 2, 11, 21, 3, 1 . . 3 . . . What is the smallest available integer obeying the constraints of this pattern? As 10 is for ever banned (like all terms ending in zero), as 11 is already in S, we try 12: S = 1, 2, 11, 21, 3, 12, . 3 . . . The sixth comma now catapults to the right a copy of this 2 over 2 boxes: S = 1, 2, 11, 21, 3, 12, . 3 2 . . What is the smallest available integer obeying the constraints of this pattern? Obviously 4: S = 1, 2, 11, 21, 3, 12, 4, 3 2 . . . The seventh comma now catapults to the right a copy of this 4 over 4 boxes: S = 1, 2, 11, 21, 3, 12, 4, 3 2 . . 4 . . WARNING What is the smallest available integer (after 4) obeying the constraints of this pattern? Not 3 (as 3 is already in S), not 32 (as a comma after 32 would duplicate the 2 on the 4 already there), not 320 (no terms ending in zero), not 321 (as the 1 would be duplicated on the 4) – but 322: S = 1, 2, 11, 21, 3, 12, 4, 322, . 4 . . The eighth comma now catapults smoothly to the right a copy of the last digit 2 over 2 boxes: S = 1, 2, 11, 21, 3, 12, 4, 322, . 4 2 . Etc. Best, É. P.-S. My personal webpage about that (with colors and pix) is below, but please concatenate the 3 parts of the address: htt p://cinquantesignes.blogspot.com/2020/06/commas-as- duplicators.html