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A333385
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a(n) = 3^n + 2 * 17^n for n >= 0.
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2
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3, 37, 587, 9853, 167123, 2839957, 48275867, 820679533, 13951521443, 237175772677, 4031987859947, 68543792792413, 1165244474990963, 19809156067406197, 336755653123584827, 5724846103033980493, 97322383751376783683, 1654480523772802668517
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refs;
listen;
history;
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OFFSET
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0,1
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COMMENTS
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This sequence was the subject of the 1st problem of the 27th British Mathematical Olympiad in 1991 (see the link BMO).
Proposition: a(n) is never a perfect square.
Proof (by induction): the unit digits of a(n) follow the pattern 3773, 3773, ...
Generalization: Steve Dinh proves that for nonnegative integers k, m, u and v, the numbers (10^k*u + 3)^n + 2*(10^m*v + 7)^n are never a perfect square for n >= 0 (see reference). - Bernard Schott, Dec 27 2021
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REFERENCES
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S. Dinh, The Hard Mathematical Olympiad Problems And Their Solutions, AuthorHouse, 2011, Problem 1 of British Mathematical Olympiad 1991, page 186.
A. Gardiner, The Mathematical Olympiad Handbook: An Introduction to Problem Solving, Oxford University Press, 1997, reprinted 2011, Problem 1 pp. 57 and 115 (1991).
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LINKS
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British Mathematical Olympiad, Problem 1, 1991.
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FORMULA
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G.f.: (3 - 23*x) / ((1 - 3*x)*(1 - 17*x)).
a(n) = 20*a(n-1) - 51*a(n-2) for n>1.
(End)
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EXAMPLE
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a(4) = 3^4 + 2 * 17^4 = 167123 = 7 * 19 * 1031 is not a perfect square.
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MAPLE
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S:=seq(3^n+2*17^n, n=0..40);
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MATHEMATICA
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a[n_] := 3^n + 2 * 17^n ; Array[a, 18, 0] (* Amiram Eldar, Mar 18 2020 *)
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PROG
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(PARI) Vec((3 - 23*x) / ((1 - 3*x)*(1 - 17*x)) + O(x^20)) \\ Colin Barker, Mar 18 2020
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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