Moser–de Bruijn sequence
In number theory, the Moser–de Bruijn sequence is an integer sequence named after Leo Moser and Nicolaas Govert de Bruijn, consisting of the sums of distinct powers of 4, or equivalently the numbers whose binary representations are nonzero only in even positions. These numbers grow in proportion to the square numbers, and are the squares for a modified form of arithmetic without carrying. No two doubled sequence members differ by a square, and every non-negative integer has a unique representation as the sum of a sequence member and a doubled sequence member. This decomposition  into sums can be used to define a bijection between the integers and pairs of integers, to define coordinates for the Z-order curve, and to construct inverse pairs of transcendental numbers with simple decimal representations. A simple recurrence relation allows values of the Moser–de Bruijn sequence to be calculated from earlier values, and can be used to prove that the Moser–de Bruijn sequence is a 2-regular sequence. Examples. The Moser–de Bruijn sequence begins For instance, 69 belongs to this sequence because it equals 64 + 4 + 1, a sum of three distinct powers of  4. Binary and related representations. Another definition of the Moser–de Bruijn sequence is that it is the ordered sequence of numbers whose binary representation has nonzero digits only in the even positions. For instance, 69 belongs to the sequence, because its binary representation 1000101 has nonzero digits in the positions for 2, 2, and 2, all of which have even exponents. The numbers in the sequence can also be described as the numbers whose base-4 representation uses only the digits 0 or 1. For a number in this sequence, the base-4 representation can be found from the binary representation  by skipping the binary digits in odd positions, which should all be zero. The hexadecimal representation of these numbers contains only the digits 0, 1, 4, 5. For instance, 69 = 1011 = 45. Equivalently, they are the numbers whose binary and negabinary representations are equal. Because there are no two consecutive nonzeros in their binary representations, the Moser–de Bruijn sequence forms a subsequence of the fibbinary numbers. It follows from either the binary or base-4 definitions of these numbers that they grow roughly in proportion to the square numbers. The number of elements in the Moser–de Bruijn  sequence that are below any given threshold formula_1 is proportional to formula_2, a fact which is also true of the square numbers. In fact the numbers in the Moser–de Bruijn sequence are the squares for a version of arithmetic without carrying on binary numbers, in which the addition and multiplication of single bits are respectively the exclusive or and logical conjunction operations. In connection with the Furstenberg–Sárközy theorem on sequences of numbers with no square difference, Imre Z. Ruzsa found a construction for large square-difference-free sets that, like the binary definition of the Moser–de Bruijn  sequence, restricts the digits in alternating positions in the base-formula_3 numbers. When applied to the base formula_4, Ruzsa's construction generates the Moser–de Bruijn sequence multiplied by two, a set that is again square-difference-free. However, this set is too sparse to provide nontrivial lower bounds for the Furstenberg–Sárközy theorem. Unique representation as sums. The Moser–de Bruijn sequence obeys a property similar to that of a Sidon sequence: the sums formula_5, where formula_6 and formula_7 both belong to the Moser–de Bruijn sequence, are all unique. No two of these sums have the same value . Moreover, every integer formula_1 can be represented as a sum formula_5, where formula_6 and formula_7 both belong to the Moser–de Bruijn sequence. To find the sum that represents formula_1, compute formula_13, the bitwise Boolean and of formula_1 with a binary value (expressed here in hexadecimal) that has ones in all of its even positions, and The Moser–de Bruijn sequence is the only sequence with this property, that all integers have a unique expression as formula_5. It is for this reason that the sequence was originally studied by . Extending the property, found infinitely many other linear expressions like formula_5  that, when formula_6 and formula_7 both belong to the Moser–de Bruijn sequence, uniquely represent all integers. Z-order curve and successor formula. Decomposing a number formula_1 into formula_20, and then applying to formula_6 and formula_7 an order-preserving map from the Moser–de Bruijn sequence to the integers (by replacing the powers of four in each number by the corresponding powers of two) gives a bijection from non-negative integers to ordered pairs of non-negative integers. The inverse of this bijection gives a linear ordering on the points in the plane with non-negative integer coordinates, which may  be used to define the Z-order curve. In connection with this application, it is convenient to have a formula to generate each successive element of the Moser–de Bruijn sequence from its predecessor. This can be done as follows. If formula_6 is an element of the sequence, then the next member after formula_6 can be obtained by filling in the bits in odd positions of the binary representation of formula_6 by ones, adding one to the result, and then masking off the filled-in bits. Filling the bits and adding one can be combined into a single addition operation . That is, the next member is the number given by the formula formula_26 The two hexadecimal constants appearing in this formula can be interpreted as the 2-adic numbers formula_27 and formula_28, respectively. Subtraction game. investigated a subtraction game, analogous to subtract a square, based on this sequence. In Golomb's game, two players take turns removing coins from a pile of formula_1 coins. In each move, a player may remove any number of coins that belongs to the Moser–de Bruijn sequence. Removing any other number of coins is not allowed. The winner is the player who removes the  last coin. As Golomb observes, the "cold" positions of this game (the ones in which the player who is about to move is losing) are exactly the positions of the form formula_30 where formula_7 belongs to the Moser–de Bruijn sequence. A winning strategy for playing this game is to decompose the current number of coins, formula_1, into formula_5 where formula_6 and formula_7 both belong to the Moser–de Bruijn sequence, and then (if formula_6 is nonzero) to remove formula_6 coins, leaving a cold position to the other player. If formula_6 is zero, this strategy is not possible, and there  is no winning move. Decimal reciprocals. The Moser–de Bruijn sequence forms the basis of an example of an irrational number formula_6 with the unusual property that the decimal representations of formula_6 and formula_41 can both be written simply and explicitly. Let formula_42 denote the Moser–de Bruijn sequence itself. Then for formula_43 a decimal number whose nonzero digits are in the positions given by the Moser–de Bruijn sequence, it follows that the nonzero digits of its reciprocal are located in positions given by doubling the numbers in formula_42 and adding one to all of them: formula_45 Alternatively, one  can write: formula_46 Similar examples also work in other bases. For instance, the two binary numbers whose nonzero bits are in the same positions as the nonzero digits of the two decimal numbers above are also irrational reciprocals. These binary and decimal numbers, and the numbers defined in the same way for any other base by repeating a single nonzero digit in the positions given by the Moser–de Bruijn sequence, are transcendental numbers. Their transcendence can be proven from the fact that the long strings of zeros in their digits allow them to be approximated more accurately by rational  numbers than would be allowed by Roth's theorem if they were algebraic numbers, having irrationality measure no less than 3. Generating function. The generating function formula_47 whose exponents in the expanded form are given by the Moser–de Bruijn sequence, obeys the functional equations formula_48 and formula_49 For example, this function can be used to describe the two decimal reciprocals given above: one is formula_50 and the other is formula_51. The fact that they are reciprocals is an instance of the first of the two functional equations. The partial products of the product form of the generating function can  be used to generate the convergents of the continued fraction expansion of these numbers themselves, as well as multiples of them. Recurrence and regularity. The Moser–de Bruijn sequence obeys a recurrence relation that allows the th value of the sequence, formula_52 (starting at formula_53) to be determined from the value at formula_54 Iterating this recurrence allows any subsequence of the form formula_55 to be expressed as a linear function of the original sequence, meaning that the Moser–de Bruijn sequence is a 2-regular sequence.