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A161599
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The list of the B values in the common solutions to the 2 equations 15*k + 1 = A^2, 19*k + 1 = B^2.
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3
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1, 18, 305, 5167, 87534, 1482911, 25121953, 425590290, 7209912977, 122142930319, 2069219902446, 35054595411263, 593858902089025, 10060546740102162, 170435435679647729, 2887341859813909231, 48914376181156809198, 828657053219851847135, 14038255528556324592097
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OFFSET
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1,2
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COMMENTS
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The case C=15 of finding k such that C*k+1 and (C+4)*k+2 are both perfect squares (A160682).
The 2 equations are equivalent to the Pell equation x^2 - 285*y^2 = 1, with x = (285*k+17)/2 and y = A*B/2.
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LINKS
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Table of n, a(n) for n=1..19.
Andersen, K., Carbone, L. and Penta, D., Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
Index entries for linear recurrences with constant coefficients, signature (17, -1).
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FORMULA
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B(t+2) = 17*B(t+1) - B(t).
B(t) = ((285+19*w)*((17+w)/2)^(t-1)+(285-19*w)*((17-w)/2)^(t-1))/570 where w=sqrt(285).
G.f.: (1+x)*x/(1-17*x+x^2).
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MAPLE
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t:=0: for b from 1 to 1000000 do a:=sqrt((15*b^2+4)/19):
if (trunc(a)=a) then t:=t+1: n:=(b^2-1)/19: print(t, a, b, n): end if: end do:
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PROG
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(Sage) [(lucas_number2(n, 17, 1)-lucas_number2(n-1, 17, 1))/15 for n in range(1, 20)] # Zerinvary Lajos, Nov 10 2009
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CROSSREFS
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Cf. A160682, A161595 (sequence of A), A161583 (sequence of k).
Sequence in context: A228606 A228605 A193317 * A273434 A083451 A162804
Adjacent sequences: A161596 A161597 A161598 * A161600 A161601 A161602
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KEYWORD
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nonn
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AUTHOR
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Paul Weisenhorn, Jun 14 2009
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EXTENSIONS
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Edited, extended by R. J. Mathar, Sep 02 2009
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STATUS
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approved
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