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Moser–De Bruijn Sequence

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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.

transcendental numbers integer sequence recurrence relation

References

Sloane, N. J. A., ed. "Sequence A000695 (Moser–De Bruijn sequence)". OEIS Foundation. https://oeis.org/A000695.
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The constants in this formula are expressed in hexadecimal and based on a 32-bit word size. The same bit pattern should be extended or reduced in the obvious way to handle other word sizes.
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Moser–De Bruijn Sequence

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