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Flatten@NestList[#+1&,{1,0},8]

{1, 0, 2, 1, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, 8, 7, 9, 8}

In addition

Array[{#,#-1}&,9,1,Join]

and

Array[{#+1,#}&, 9,0,Join]==Array[{#,#-1}&, 9,1,Join]

Original Post

NestList[#+{1,1}&, {1,0},8]//Flatten

Update

From this by @azerbajdzan, which introduced me to SubstitutionSystem

SubstitutionSystem[{n_ :> n+1}, {1,0}, 8]//Flatten

(* {1,0,2,1,3,2,4,3,5,4,6,5,7,6,8,7,9,8} *) 

Original Answer

Flatten@NestList[#+1&,{1,0},8]

{1, 0, 2, 1, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, 8, 7, 9, 8}

In addition

Array[{#,#-1}&,9,1,Join]

and

Array[{#+1,#}&, 9,0,Join]==Array[{#,#-1}&, 9,1,Join]

Original Post

NestList[#+{1,1}&, {1,0},8]//Flatten

Flatten@NestList[#+1&,{1,0},8]

{1, 0, 2, 1, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, 8, 7, 9, 8}

Original Post

NestList[#+{1,1}&, {1,0},8]//Flatten

Flatten@NestList[#+1&,{1,0},8]

{1, 0, 2, 1, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, 8, 7, 9, 8}

In addition

Array[{#,#-1}&,9,1,Join]

and

Array[{#+1,#}&, 9,0,Join]==Array[{#,#-1}&, 9,1,Join]

Original Post

NestList[#+{1,1}&, {1,0},8]//Flatten

NestList[#+{1,1}&, {1,0},8]//Flatten

{1, 0, 2, 1, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, 8, 7, 9, 8}

Flatten@NestList[#+1&,{1,0},8]

{1, 0, 2, 1, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, 8, 7, 9, 8}

Original Post

NestList[#+{1,1}&, {1,0},8]//Flatten