cp's OEIS Frontend

A265849 First differences of A006751.

%I A265849 #26 Jul 07 2022 20:56:41
%S A265849 10,1100,2000,129000,1112990000,310198100000,12900010100000,
%T A265849 1113122099909791900000,31130009089198002000100000,
%U A265849 132082082098921801009009900000,11131221131211000108018890978199979090100000,31131122211299991892189900998999891000999919009909900000
%N A265849 First differences of A006751.
%C A265849 Also first differences of A006715, A001140, A001141, A001143, A001145, A001151, A001154. - _Michel Marcus_, Dec 16 2015
%C A265849 Note that A005150 has really different first differences characteristic because of its initial term that is 1.
%F A265849 a(n) = A006751(n+1) - A006751(n).
%F A265849 a(n) mod 10^5 = 0, for n > 5.
%F A265849 a(2*n+2) - a(2*n) mod 10^6 = 0, for n > 3.
%F A265849 a(2*n+1) - a(2*n-1) mod 10^7 = 0, for n > 3.
%e A265849 a(1) = A006751(2) - A006751(1) = 12 - 2 = 10.
%e A265849 a(2) = A006751(3) - A006751(2) = 1112 - 12 = 1100.
%t A265849 f[n_, d_: 1] := NestList[Flatten[Reverse /@ Map[Function[k, Through[{First, Length}@ k]], Split@ #]] &, {d}, n - 1]; Differences@ Array[FromDigits@ f[#, 2][[#]] &, {13}] (* _Michael De Vlieger_, Jan 03 2016, after _Zerinvary Lajos_ at A006751 *)
%o A265849 (PARI) dpt(n) = {vd = []; d = digits(n); nbd = 0; old = -1; for (k=1, #d, if (d[k] == old, nbd ++, if (old != -1, vd = concat(vd, nbd); vd = concat(vd, old);); nbd = 1;); old = d[k];); vd = concat(vd, nbd); vd = concat(vd, old); subst(Pol(vd), x, 10);}
%o A265849 lista(nn, x=2) = {v = vector(nn); v[1] = x; for (n=2, nn, nx = dpt(x); v[n] = nx; x = nx;); vector(nn-1, n, v[n+1] - v[n]);} \\ 2nd param x can any value between 2 and 9 \\ _Michel Marcus_, Dec 16 2015
%Y A265849 Cf. A006751, A006715, A001140, A001141, A001143, A001145, A001151, A001154.
%K A265849 nonn,base,easy
%O A265849 1,1
%A A265849 _Altug Alkan_, Dec 16 2015