A160717 - OEIS

%I #18 Nov 05 2024 18:05:11

%S 0,105,406,703,903,11026,13041,14028,15051,27028,36046,41041,43071,

%T 46056,61075,66066,75078,77028,83028,85078,93096,1110795,1130256,

%U 1160526,1180416,1250571,1290421,1330896,1350546,1360425,1380291

%N Cyclops triangular numbers.

%C Triangular numbers (A000217) that are also cyclops numbers (A134808).

%H Kenny Lau, <a href="/A160717/b160717.txt">Table of n, a(n) for n = 1..20001</a>

%e 105 is in the sequence since it is both a triangular number (105 = 1 + 2 + ... + 14) and a Cyclops number (number of digits is odd, and the only zero is the middle digit). - _Michael B. Porter_, Jul 08 2016

%p count:= 1: A[1]:= 0:

%p for d from 1 to 3 do

%p   for x from 0 to 9^d-1 do

%p     L:= convert(x+9^d,base,9);

%p     X:= add((L[i]+1)*10^(i-1),i=1..d);

%p     for y from 0 to 9^d-1 do

%p       L:= convert(y+9^d,base,9);

%p       Y:= add((L[i]+1)*10^(i-1),i=1..d);

%p       Z:= Y + 10^(d+1)*X;

%p       if issqr(1+8*Z) then

%p         count:= count+1;

%p         A[count]:= Z;

%p       fi

%p od od od:

%p seq(A[i],i=1..count); # _Robert Israel_, Jul 08 2016

%t cyclopsQ[n_] := Block[{id=IntegerDigits@n,lg=Floor[Log[10,n]+1]}, Count[id,0]==1 && OddQ@lg && id[[(lg+1)/2]]==0]; lst = {0}; Do[t = n (n + 1)/2; If[ cyclopsQ@t, AppendTo[lst, t]], {n, 0, 1670}]; lst (* _Robert G. Wilson v_, Jun 09 2009 *)

%t cyclpsQ[n_]:=With[{len=IntegerLength[n]},OddQ[len]&&DigitCount[n,10,0]==1&&IntegerDigits[n][[(len+1)/2]]==0]; Join[{0},Select[ Accumulate[ Range[2000]],cyclpsQ]] (* _Harvey P. Dale_, Nov 05 2024 *)

%Y Cf. A000217, A134808, A134809, A138131, A138148, A153806, A160711, A160712.

%K base,easy,nonn

%O 1,2

%A _Omar E. Pol_, Jun 08 2009

%E More terms from _Robert G. Wilson v_, Jun 09 2009

%E Offset and b-file changed by _N. J. A. Sloane_, Jul 27 2016