cp's OEIS Frontend

A226736 Triangular numbers such that the binary representation is of the form either Tbt or Tt, where T and t are binary representations of triangular numbers with equal binary length, and b is a binary digit: b = 1 or 0.

%I A226736 #13 Jan 03 2016 04:38:48
%S A226736 3,6,15,28,105,351,496,1830,7140,30876,129795,508536,2029105,8260080,
%T A226736 32817151,33550336,133179360,533811475,2139135936,8564460003,
%U A226736 34292793216,137220150385,549219598080,2197036652910,8791800675840,35164716131980,140703139101696,562926884985640
%N A226736 Triangular numbers such that the binary representation is of the form either Tbt or Tt, where T and t are binary representations of triangular numbers with equal binary length, and b is a binary digit: b = 1 or 0.
%C A226736 The binary representation of the triangular number t (least significant bits of a(n)) is allowed to have leading zeros.
%C A226736 The sequence of indices of a(n) begins: 2, 3, 5, 7, 14, 26, 31, 60, 119, 248, 509, 1008, 2014, 4064, 8101, 8191, 16320, 32674, 65408, 130877, 261888, 523870, 1048064, 2096204, 4193280, 8386264, 16775168, 33553744, ...
%e A226736 105 = (1101001)_2 = (110)_2//(1)_2//(001)_2 is in the sequence, where 105, (110)_2 = 6 and (001)_2=1 are each triangular numbers.
%p A226736 A000217 := proc(n)
%p A226736     n*(n+1) /2 ;
%p A226736 end proc:
%p A226736 isA000217 := proc(n)
%p A226736     local t1;
%p A226736     t1:=floor(sqrt(2*n));
%p A226736     if n = t1*(t1+1)/2 then
%p A226736         true
%p A226736     else
%p A226736         false;
%p A226736     end if;
%p A226736 end proc:
%p A226736 for n from 2 do
%p A226736     tmain := A000217(n) ;
%p A226736     dgs := convert(tmain,base,2) ;
%p A226736     ndgs := floor(nops(dgs)/2) ;
%p A226736     tlo := [op(1..ndgs,dgs)] ;
%p A226736     if type(nops(dgs),'even') then
%p A226736         thi := [op(ndgs+1..2*ndgs,dgs)] ;
%p A226736     else
%p A226736         thi := [op(ndgs+2..2*ndgs+1,dgs)] ;
%p A226736     end if;
%p A226736     tlo := add(op(i,tlo)*2^(i-1),i=1..nops(tlo)) ;
%p A226736     if isA000217(tlo) then
%p A226736         thi := add(op(i,thi)*2^(i-1),i=1..nops(thi)) ;
%p A226736         if isA000217(thi) then
%p A226736             printf("%d,\n",tmain) ;
%p A226736         end if;
%p A226736     end if;
%p A226736 end do: # _R. J. Mathar_, Jun 18 2013
%o A226736 (C)
%o A226736 #include <stdio.h>
%o A226736 #include <math.h>
%o A226736 typedef unsigned long long U64;
%o A226736 U64 isTriangular(U64 a) {  // ! Must be a < (1<<63)
%o A226736     U64 s = sqrt(a*2);
%o A226736     return (s*(s+1) == a*2);
%o A226736 }
%o A226736 int main() {
%o A226736   U64 i, j, n, tn, t, T, prev=0;
%o A226736   for (n = tn = 3; tn > prev; prev = tn, tn += n, ++n) {
%o A226736     for (i = 64, j = tn; j < (1ULL<<63); j += j)
%o A226736       --i;  // binary length of tn
%o A226736     j = i >> 1;  //  TOt or Tt, binary length of t is j
%o A226736     t = tn & ((1ULL<<j)-1);
%o A226736     T = tn >> (j+(i&1));
%o A226736     if (isTriangular(t) && isTriangular(T))
%o A226736         printf("%20llu %10llu %10llu\n", tn, T, t);
%o A226736   }
%o A226736   return 0;
%o A226736 }
%Y A226736 Cf. A000217, A070939, A226836.
%K A226736 nonn,base,less
%O A226736 1,1
%A A226736 _Alex Ratushnyak_, Jun 16 2013