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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. 10

%I #25 Mar 07 2024 16:45:45

%S 1,3,15,119,1799,59367,3743271,481693095,123123509927,62989418816679,

%T 64491023022979239,132015402419352060071,540829047855347718631591,

%U 4430403202865824763042320551,72583450474242118015031400337575,2378466805556971511916001231449723047

%N 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 a(n) = each row of A053632 reduced mod 2 and interpreted as a binary number.

%H Antti Karttunen, <a href="/A068052/b068052.txt">Table of n, a(n) for n = 0..64</a>

%F a(0) = 1; for n > 0, a(n) = a(n-1) XOR (2^n)*a(n-1), where XOR is bitwise-XOR (A003987).

%F a(n) = A248663(A285101(n)) = A048675(A285102(n)).

%F A000120(a(n)) = A285103(n). [Number of ones in binary representation.]

%F A080791(a(n)) = A285105(n). [Number of nonleading zeros.]

%p 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 foo := proc(a) local i; add(a[i]*2^(i-1),i=1..nops(a)); end;

%p # second Maple program:

%p a:= proc(n) option remember; `if`(n=0, 1,

%p (t-> Bits[Xor](2^n*t, t))(a(n-1)))

%p end:

%p seq(a(n), n=0..16); # _Alois P. Heinz_, Mar 07 2024

%t FoldList[BitXor[#, #*#2]&, 1, 2^Range[20]] (* _Paolo Xausa_, Mar 07 2024 *)

%o (Scheme, with memoization-macro definec)

%o (definec (A068052 n) (if (zero? n) 1 (A003987bi (A068052 (- n 1)) (* (A000079 n) (A068052 (- n 1)))))) ;; A003987bi implements bitwise-XOR (A003987).

%o (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 Same sequence shown in binary: A068053.

%Y Cf. A000120, A003987, A028362 (using + instead of XOR), A048675, A053632, A080791, A248663, A285101, A285102, A285103, A285105.

%K nonn,base

%O 0,2

%A _Antti Karttunen_, Feb 13 2002

%E Formulas added by _Antti Karttunen_, Apr 15 2017

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Last modified March 28 14:38 EDT 2024. Contains 371254 sequences. (Running on oeis4.)