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a(0) = 1; thereafter a(n) is the concatenation of a(n-1), n, a(n-1).
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%I #27 Apr 30 2020 18:57:58

%S 1,111,1112111,111211131112111,1112111311121114111211131112111,

%T 111211131112111411121113111211151112111311121114111211131112111

%N a(0) = 1; thereafter a(n) is the concatenation of a(n-1), n, a(n-1).

%F a(1)=1 and a(n) = a(n - 1)*10^(2^n) + n *10^(2^n - 1) + a(n - 1).

%t concatPal[0] = 1; concatPal[n_] := FromDigits @ Join[RealDigits[concatPal[n - 1]][[1]], {n}, RealDigits[concatPal[n - 1]][[1]]]; Table[concatPal[n], {n, 0, 10}]

%t nxt[{n_,a_}]:={n+1,FromDigits[Flatten[Join[IntegerDigits[a], IntegerDigits[ n+1], IntegerDigits[ a]]]]}; NestList[nxt,{0,1},5][[All,2]] (* _Harvey P. Dale_, Apr 30 2020 *)

%o (PARI) main(size) = {s=vector(size); s[1]=1; for(t=2,size,s[t]= eval(Str(s[t-1],t-1,s[t-1]))); return(s);} \\ _Anders Hellström_, Jul 05 2015

%Y Cf. A001704, A007908.

%K nonn,base

%O 0,2

%A _José María Grau Ribas_, Jul 04 2015