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%I #37 Aug 22 2023 08:01:42
%S 0,1,5,6,22,23,27,28,92,93,97,98,114,115,119,120,376,377,381,382,398,
%T 399,403,404,468,469,473,474,490,491,495,496,1520,1521,1525,1526,1542,
%U 1543,1547,1548,1612,1613,1617,1618,1634,1635,1639,1640,1896,1897,1901,1902,1918
%N a(n) = Sum_{k=0..n} n XOR k where XOR is the bitwise logical exclusive-or operator.
%H Michael De Vlieger, <a href="/A224915/b224915.txt">Table of n, a(n) for n = 0..10000</a>
%F a(n) = Sum_{j=1..n} 4^(v_2(j)), where v_2(j) is the exponent of highest power of 2 dividing j. - _Ridouane Oudra_, Jun 08 2019
%F a(n) = n + 3*Sum_{j=1..floor(log_2(n))} 4^(j-1)*floor(n/2^j), for n>=1. - _Ridouane Oudra_, Dec 09 2020
%F From _Kevin Ryde_, Dec 17 2021: (Start)
%F a(2*n+b) = 4*a(n) + n + b where b = 0 or 1.
%F a(n) = (A001196(n) - n)/2.
%F a(n) = A350093(n) - A222423(n), being XOR = OR - AND.
%F (End)
%e a(2) = (0 xor 2) + (1 xor 2) = 2 + 3 = 5.
%p read("transforms"):
%p A051933 := proc(n,k)
%p XORnos(n,k) ;
%p end proc:
%p A224915 := proc(n)
%p add(A051933(n,k),k=0..n) ;
%p end proc: # _R. J. Mathar_, Apr 26 2013
%p # second Maple program:
%p with(MmaTranslator[Mma]):
%p seq(add(BitXor(n,i),i=0..n),n=0..60); # _Ridouane Oudra_, Dec 09 2020
%t Array[Sum[BitXor[#, k], {k, 0, #}] &, 53, 0] (* _Michael De Vlieger_, Dec 09 2020 *)
%o (Python)
%o for n in range(59):
%o s = 0
%o for k in range(n): s += n ^ k
%o print(s, end=',')
%o (Python)
%o def A224915(n): return 3*int(bin(n)[2:],4)-n>>1 # _Chai Wah Wu_, Aug 21 2023
%o (PARI) a(n) = sum(k=0, n, bitxor(n, k)); \\ _Michel Marcus_, Jun 08 2019
%o (PARI) a(n) = (3*fromdigits(binary(n),4) - n) >>1; \\ _Kevin Ryde_, Dec 17 2021
%Y Cf. A001196 (bit doubling).
%Y Row sums of A051933.
%Y Other sums: A222423 (AND), A350093 (OR), A265736 (IMPL), A350094 (CNIMPL), A004125 (mod).
%K nonn,easy
%O 0,3
%A _Alex Ratushnyak_, Apr 19 2013
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