A030154 - OEIS

%I #26 Aug 06 2018 05:36:53

%S 0,1,4,9,16,25,36,49,81,256,529,729,1296,4761,5476,6561,9216,16129,

%T 32761,34969,87616,763876,5414929,5612161,7414729,7436529,7634169,

%U 14561856,21058921,34503876,43072969,43414921,45252529,69272329

%N Squares such that in n and sqrt(n) the parity of digits alternates.

%C The more digits there are in n, the lower the likelihood that the parity of n's digits will strictly alternate. Thus, the terms of the sequence become increasingly rare as n gets larger. - _Harvey P. Dale_, Aug 05 2018

%C For n > 3 the last digit of a(n) isn't 0 or 4. - _David A. Corneth_, Aug 05 2018

%H Andrew Howroyd, <a href="/A030154/b030154.txt">Table of n, a(n) for n = 1..1226</a> (first 101 terms from Harvey P. Dale, terms 102..223 from David A. Corneth)

%t pdaQ[n_]:=Module[{a=Mod[IntegerDigits[n],2],b=Mod[IntegerDigits[ Sqrt[ n]],2]},Length[ Split[a]] ==IntegerLength[n]&&Length[Split[b]]== IntegerLength[ Sqrt[n]]]; Join[{0},Select[Range[8500]^2,pdaQ]] (* _Harvey P. Dale_, Aug 05 2018 *)

%o (PARI) alternating(n)={my(v=digits(n)%2);0==#select(i->v[i]==v[i-1],[2..#v])}

%o { for(n=0, 10^5, if(alternating(n^2) && alternating(n), print1(n^2, ", "))) } \\ _Andrew Howroyd_, Aug 05 2018

%o (PARI) \\ for larger n: requires alternating function above

%o upto(n)={local(R=List([0])); my(recurse(s,b)=if(b<n, for(i=0, 9, if(b==1||(s\(b\10)-i)%2, my(k=i*b+s); if(k<=n&&(b==1||k^2\(b\10)*11\10%2), if(i>0&&alternating(k^2\b), listput(R, k)); self()(k, 10*b)))))); recurse(0,1); listsort(R); Vec(R)}

%o apply(n->n^2, upto(sqrtint(10^12))) \\ _Andrew Howroyd_, Aug 05 2018

%Y Cf. A030141, A030151, A030152, A030153.

%K nonn,base

%O 1,3

%A _Patrick De Geest_

%E Offset changed by _David A. Corneth_, Aug 05 2018