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.

1, 3, 15, 119, 1799, 59367, 3743271, 481693095, 123123509927, 62989418816679, 64491023022979239, 132015402419352060071, 540829047855347718631591, 4430403202865824763042320551, 72583450474242118015031400337575, 2378466805556971511916001231449723047

Offset

0,2

Comments

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

Links

Antti Karttunen, Table of n, a(n) for n = 0..64

Formula

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

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

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

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

Maple

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)];
foo := proc(a) local i; add(a[i]*2^(i-1), i=1..nops(a)); end;
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1,
      (t-> Bits[Xor](2^n*t, t))(a(n-1)))
    end:
seq(a(n), n=0..16);  # Alois P. Heinz, Mar 07 2024

Mathematica

FoldList[BitXor[#, #*#2]&, 1, 2^Range[20]] (* Paolo Xausa, Mar 07 2024 *)

Prog

(Scheme, with memoization-macro definec)
(definec (A068052 n) (if (zero? n) 1 (A003987bi (A068052 (- n 1)) (* (A000079 n) (A068052 (- n 1)))))) ;; A003987bi implements bitwise-XOR (A003987).
(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

Crossrefs

Same sequence shown in binary: A068053.

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

Sequence in context: A145161 A121422 A060639 * A379366 A379364 A068859

Adjacent sequences: A068049 A068050 A068051 * A068053 A068054 A068055

Keyword

nonn,base

Author

Antti Karttunen, Feb 13 2002

Extensions

Formulas added by Antti Karttunen, Apr 15 2017

Status

approved