A220689 - OEIS

%I #33 Aug 07 2023 07:48:21

%S 1,21,21,105,105,105,105,946,946,666,1653,666,1378,946,1225,946,4005,

%T 1378,4005,1378,7381,1225,1378,1653,2485,4005,31125,4005,4005,4005,

%U 2485,13861,13861,5356,4005,7381,5356,5356,7381,4005,5356,29161,12561,12561,4186,4186,4186,4186

%N Triangular numbers generated in A224218. That is, the triangular numbers generated by the operation triangular(i) XOR triangular(i+1) along increasing i.

%F a(n) = A000217(A224218(n)) XOR A000217(A224218(n)+1).

%p read("transforms") ;

%p A220689 := proc(n)

%p     i := A224218(n) ;

%p     XORnos(A000217(i),A000217(i+1)) ;

%p end proc: # _R. J. Mathar_, Apr 23 2013

%t nmax = 100;

%t pmax = 2 nmax^2; (* increase coeff 2 if A224218 is too short *)

%t A224218 = Join[{0}, Flatten[Position[Partition[Accumulate[Range[pmax]], 2, 1], _?(OddQ[Sqrt[1 + 8 BitXor[#[[1]], #[[2]]]]]&), {1}, Heads -> False]]];

%t a[n_] := Module[{i}, i = A224218[[n]]; BitXor[PolygonalNumber[i], PolygonalNumber[i+1]]];

%t Table[a[n], {n, 1, nmax}] (* _Jean-François Alcover_, Aug 07 2023, after _Harvey P. Dale_ in A224218 *)

%o (Python)

%o def rootTriangular(a):

%o     sr = 1<<33

%o     while a < sr*(sr+1)//2:

%o       sr>>=1

%o     b = sr>>1

%o     while b:

%o         s = sr+b

%o         if a >= s*(s+1)//2:

%o           sr = s

%o         b>>=1

%o     return sr

%o for i in range(1<<12):

%o         s = (i*(i+1)//2) ^ ((i+1)*(i+2)//2)

%o         t = rootTriangular(s)

%o         if s == t*(t+1)//2:

%o             print(str(s), end=',')

%Y Cf. A000217, A224218, A224511.

%K nonn

%O 1,2

%A _Alex Ratushnyak_, Apr 13 2013