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A262566
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Sequence U_n found on page 154 of Poulet (1934).
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1
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0, 1, 1, -1, -37, -79, 281, 3359, 5661, -45341, -320819, -273824, 5978275, 30439553, -9213737, -719524811, -2714115501, 5008636187, 81197252233
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OFFSET
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0,5
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COMMENTS
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Unfortunately not enough of Poulet (1934) was scanned to identify this sequence (U_n on page 154). There is a companion sequence (V_n) on page 155 which could be entered once this sequence is identified.
This sequence appears on page 154 of in Poulet (1934) as an example of a higher-order primogenic sequence (*suite primogène*), a termwise product of five sequences satisfying second-order recurrence relations (the product having integer terms although the factors do not); the sequence itself satisfies a recurrence of order 32 but it is more compactly described in Poulet's notation by the polynomial -z^5-z^4+2z^3-z^2-z+1.
So far as I am able to glean from the text, this particular sequence has no very special interest in itself; Poulet tabulates it to show one means of calculating the terms of his primogenic sequences. Poulet's goal was to find sequences enjoying arithmetical properties similar to those of the Fibonacci and Lucas sequences (A000045, A000032) but growing more slowly; he hoped to find enough such sequences, growing slowly enough, to provide a powerful primality test applying to the majority of positive integers.
In this hope he was disappointed, but he was able to exploit the divisibility properties of his sequences in such a way as to prove some numbers of 9 or 10 digits prime without exhaustive search, hence his term "primogenic" for the sequences.
The sequence here has a partner V_n, bearing roughly the same relation to it as the Lucas sequence bears to the Fibonacci sequence, beginning 32, 1, -37, -281, -153, 4061. Poulet tabulates this sequence as well in his book.
Given Poulet's polynomial (in this case -z^5-z^4+2z^3-z^2-z+1), one calculates the sequence as follows. Find the roots zk of the polynomial and write each in the form z=p^2/q. (There is some flexibility in the choice of p,q and it is not perfectly obvious to me which choices Poulet wishes to be made; I think they lead to fewer degrees of freedom in the final sequence than one might expect.) Now construct sequences uk,vk obeying the recurrence relation u''=pu'-qu; let u have initial terms 0,1 and v have initial terms 2,p. Finally, take the termwise product.
(End)
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LINKS
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Paul Poulet, La Chasse aux Nombres, Librairie du Sphinx, Brussels, 1934; [Annotated scanned copy of three pages only, with notes from N. J. A. Sloane]
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FORMULA
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Comment from Gareth McCaughan, Oct 23 2015: If I have correctly understood a comment on page 156 of the Poulet book, the sequences Un and Vn obey the order-32 recurrence with coefficients 1, -1, 19, 75, 125, 755, 1773, 2803, 7747, -3341, 25661, -43722, 39170, -130858, 115862, -91398, 237094, -01398, 115862, -130858, 39170, -43722, 25661, -3341, 7747, 2803, 1773, 755, 125, 75, 19, -1, 1.
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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