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A213630 a(t_1 t_2...t_n) = (t_1 + t_2)*t_1 t_2 + ... + (t_n-1 + t_n)*t_n-1 t_n.

%I A213630 #39 Aug 03 2016 03:41:34
%S A213630 1,4,9,16,25,36,49,64,81,10,22,36,52,70,90,112,136,162,190,40,63,88,
%T A213630 115,144,175,208,243,280,319,90,124,160,198,238,280,324,370,418,468,
%U A213630 160,205,252,301,352,405,460,517,576,637,250,306,364,424,486,550,616
%N A213630 a(t_1 t_2...t_n) = (t_1 + t_2)*t_1 t_2 + ... + (t_n-1 + t_n)*t_n-1 t_n.
%C A213630 Differs from A057147 first at n=100.
%C A213630 a(n)/n = {1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 6, 7, 8, 9, 10, 11, 12, 13, 14, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 9, 10, 11, 12, 13, 14, 15, ...}. - _Alexander R. Povolotsky_, Jun 19 2012
%H A213630 Felipe Bottega Diniz and T. D. Noe, <a href="/A213630/b213630.txt">Table of n, a(n) for n = 1..1000</a> (first 100 terms from Felipe Bottega Diniz)
%H A213630 F. B. Diniz, <a href="http://arxiv.org/abs/1607.06082">About a New Family of Sequences</a>, arXiv:1607.06082 [math.GM], 2016.
%F A213630 Let t_1t_2...t_n be a natural number.
%F A213630 a(t_1t_2...t_n) = (t_1 + t_2)*t_1t_2 + ... + (t_n-1 + t_n)*t_n-1t_n, if n even.
%F A213630 If n is odd, t_1 t_2...t_n <- 0t_1t_2...t_n and do a(0t_1 t_2...t_n).
%e A213630 a(123456) = (1+2)*12 + (3+4)*34 + (5+6)*56 = 890.
%e A213630 a(890) = (0+8)*08 + (9+0)*90 = 874.
%e A213630 a(15) = (1+5)*15 = 90.
%e A213630 a(7) = (0+7)*07 = 49.
%p A213630 read("transforms"):
%p A213630 A213630 := proc(n)
%p A213630     local dgs,a,i ;
%p A213630     if n < 10 then
%p A213630         n^2;
%p A213630     else
%p A213630         dgs := convert(n,base,10) ;
%p A213630         a := 0 ;
%p A213630         for i from 2 to nops(dgs) do
%p A213630             a := a+(op(i-1,dgs)+op(i,dgs))*digcat2(op(i,dgs),op(i-1,dgs)) ;
%p A213630         end do:
%p A213630         a;
%p A213630     end if;
%p A213630 end proc: # _R. J. Mathar_, Aug 10 2012
%t A213630 num[n_] := Module[{d = IntegerDigits[n], d2}, If[OddQ[Length[d]], d = Prepend[d, 0]]; d2 = Partition[d, 2]; Sum[(d2[[i, 1]] + d2[[i, 2]])*(10*d2[[i, 1]] + d2[[i, 2]]), {i, Length[d2]}]]; Table[num[n], {n, 100}] (* _T. D. Noe_, Jun 19 2012 *)
%K A213630 easy,nonn,base
%O A213630 1,2
%A A213630 _Felipe Bottega Diniz_, Jun 16 2012