cp's OEIS Frontend

A068052 Start from 1, shift one left and sum mod 2 (bitwise-XOR) to get 3 (11 in binary), then shift two steps left and XOR to get 15 (1111 in binary), then three steps and XOR to get 119 (1110111 in binary), then four steps and so on.

%I A068052 #25 Mar 07 2024 16:45:45
%S A068052 1,3,15,119,1799,59367,3743271,481693095,123123509927,62989418816679,
%T A068052 64491023022979239,132015402419352060071,540829047855347718631591,
%U A068052 4430403202865824763042320551,72583450474242118015031400337575,2378466805556971511916001231449723047
%N A068052 Start from 1, shift one left and sum mod 2 (bitwise-XOR) to get 3 (11 in binary), then shift two steps left and XOR to get 15 (1111 in binary), then three steps and XOR to get 119 (1110111 in binary), then four steps and so on.
%C A068052 a(n) = each row of A053632 reduced mod 2 and interpreted as a binary number.
%H A068052 Antti Karttunen, <a href="/A068052/b068052.txt">Table of n, a(n) for n = 0..64</a>
%F A068052 a(0) = 1; for n > 0, a(n) = a(n-1) XOR (2^n)*a(n-1), where XOR is bitwise-XOR (A003987).
%F A068052 a(n) = A248663(A285101(n)) = A048675(A285102(n)).
%F A068052 A000120(a(n)) = A285103(n). [Number of ones in binary representation.]
%F A068052 A080791(a(n)) = A285105(n). [Number of nonleading zeros.]
%p A068052 with(gfun,seriestolist); [seq(foo(map(`mod`,seriestolist(series(mul(1+(z^i),i=1..n),z,binomial(n+1,2)+1)),2)), n=0..20)];
%p A068052 foo := proc(a) local i; add(a[i]*2^(i-1),i=1..nops(a)); end;
%p A068052 # second Maple program:
%p A068052 a:= proc(n) option remember; `if`(n=0, 1,
%p A068052       (t-> Bits[Xor](2^n*t, t))(a(n-1)))
%p A068052     end:
%p A068052 seq(a(n), n=0..16);  # _Alois P. Heinz_, Mar 07 2024
%t A068052 FoldList[BitXor[#, #*#2]&, 1, 2^Range[20]] (* _Paolo Xausa_, Mar 07 2024 *)
%o A068052 (Scheme, with memoization-macro definec)
%o A068052 (definec (A068052 n) (if (zero? n) 1 (A003987bi (A068052 (- n 1)) (* (A000079 n) (A068052 (- n 1)))))) ;; A003987bi implements bitwise-XOR (A003987).
%o A068052 (PARI) a(n) = if(n<1, 1, bitxor(a(n - 1), 2^n*a(n - 1))); \\ _Indranil Ghosh_, Apr 15 2017, after formula by _Antti Karttunen_
%Y A068052 Same sequence shown in binary: A068053.
%Y A068052 Cf. A000120, A003987, A028362 (using + instead of XOR), A048675, A053632, A080791, A248663, A285101, A285102, A285103, A285105.
%K A068052 nonn,base
%O A068052 0,2
%A A068052 _Antti Karttunen_, Feb 13 2002
%E A068052 Formulas added by _Antti Karttunen_, Apr 15 2017