Lucasn.
/ˈluːkəs/, U.S. /ˈlukəs/ Mathematics.
Used attributively to designate (a) the sequence of integers 1, 3, 4, 7,…, formed in the same way as the Fibonacci numbers; (b) the sequences generated by the recurrence relation un + 2 = Pun+1− Qun when u0 = 0, u1 = 1 (the Fibonacci numbers being a particular case corresponding to P = 1, Q = −1) and when u0 = 2, u1 = P, which are respectively defined by un = (an − bn)/(a − b) and un = an + bn(n = 0, 1, 2,..), where a and b are the roots of x2 − Px + Q = 0.
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the world > relative properties > number > mathematical number or quantity > numerical arrangement > [noun] > set > sequence
subsequence1847
sequence1882
word1936
Lucas1953
Cauchy sequence1955
1919 L. E. Dickson Hist. Theory Numbers I. xvii. 393 (heading) Recurring series; Lucas' un, vn.]
1953 Scripta Math. 19 278 (heading) Linear expressions for the powers of Fibonacci and Lucas numbers.
1953 Scripta Math. 19 278 The ith term of the Lucas sequence 1, 3, 4, 7...
1954 Duke Math. Jrnl. 21 608 If any term of (W) vanishes, (W) is essentially the well-known Lucas sequence Ln = (αn − βn)/(α− β).
1961 Pacific Jrnl. Math 11 385 It would be interesting to make a numerical study of several recurrences..to endeavor to find out whether the two Lucas sequences 0, 1, P, … and 2, P, P2 − 2Q, … and their translates are essentially the only ones for which a global characterization of the divisors is possible.
1966 C. S. Ogilvy & J. T. Anderson Excursions Number Theory 164 The Lucas numbers satisfy the same recursion relation as the Fibonacci numbers, but have starting values L1 = 1, L2 = 3.
1972 P. Ribenboim Algebraic Numbers i. 8 Prove: (a) Two consecutive Lucas numbers are relatively prime. (b) bn2 − bn − 1bn + 1 = (− 1)n.5 for every n≧ 1.
This entry has not yet been fully updated (first published 1976; most recently modified version published online March 2022).
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