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A167708 Numbers k such that (k^2 - 81)/19 is a square.

%I A167708 #18 Sep 05 2018 02:30:07
%S A167708 9,10,66,105,959,1530,2441,22335,35634,326050,520191,829930,7593834,
%T A167708 12115455,110856041,176863410,282173759,2581881225,4119219066,
%U A167708 37690727890,60133039209,95938248130,877832022666,1400522366985
%N A167708 Numbers k such that (k^2 - 81)/19 is a square.
%D A167708 A. H. Beiler, "Recreations in the theory of numbers": Ex. 38, p. 298 (Dover Publications, Inc., New York, 1966).
%H A167708 G. C. Greubel, <a href="/A167708/b167708.txt">Table of n, a(n) for n = 1..1000</a>
%H A167708 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,340,0,0,0,0,-1).
%F A167708 G.f.: (9 + 10*z + 66*z^2 + 105*z^3 + 959*z^4 + 1530*z^5 + 2441*z^6 + 22335*z^7 + 35634*z^8 + 326050*z^9 - 340*z^5*(9 + 10*z + 66*z^2 + 105*z^3 +959*z^4))/ (1 - 340*z^5 + z^10).
%F A167708 a(n+10) = 340*a(n+5) - a(n).
%F A167708 On every subsequence mod 5: a(n+2) = 340*a(n+1) - a(n).
%F A167708 On every subsequence mod 5: a(n+2) = 170*a(n) + 39*sqrt(19*a(n)^2 - 1539).
%F A167708 for n == 0 (mod 5): a(n) = (9/2)*(170 + 39*sqrt(19))^(n) + (9/2)*(170 - 39*sqrt(19))^(n); the subsequence is 9, 1530, 520191, 176863410, 60133039209, 20445056467650, 6951259065961791, 2363407637370541290, 803551645446918076809, ...
%F A167708 for n == 1 (mod 5): a(n) = (sqrt(19) + 10)/2*(170 + 39*sqrt(19))^(n) + (-sqrt(19) + 10)/2*(170 - 39*sqrt(19))^(n); the subsequence is 10, 2441, 829930, 282173759, 95938248130, 32618722190441, 11090269606501810, 3770659047488424959, 1282012985876457984250, ...
%F A167708 for n == 2 (mod 5): a(n) = (15*sqrt(19) + 66)/2*(170 + 39*sqrt(19))^(n) + (-15*sqrt(19) + 66)/2*(170 - 39*sqrt(19))^(n); the subsequence is 66, 22335, 7593834, 2581881225, 877832022666, 298460305825215, 101475626148550434, 34501414430201322345, 11730379430642301046866, ...
%F A167708 for n == 3 (mod 5): a(n) = (24*sqrt(19) + 105)/2*(170 + 39*sqrt(19))^(n) + (-24*sqrt(19) + 105)/2*(170 - 39*sqrt(19))^(n); the subsequence is 105, 35634, 12115455, 4119219066, 1400522366985, 476173485555834, 161897584566616575, 55044702579164079666, 18715036979331220469865, ...
%F A167708 for n == 4 (mod 5): a(n) = (220*sqrt(19) + 959)/2*(170 + 39*sqrt(19))^(n) + (-220*sqrt(19) + 959)/2*(170 - 39*sqrt(19))^(n); the subsequence is 959, 326050, 110856041, 37690727890, 12814736626559, 4356972762302170, 1481357924446111241, 503657337338915519770, 171242013337306830610559, ...
%e A167708 a(0)=9 because (9^2 - 81)/19=0 in N; a(1)=10 because (10^2 - 81)/19=1 in N.
%p A167708 a(0):=9:a(1):=10:a(2):=66:a(3):=105:a(4):=959:a(5):=1530:a(6):=2441:a(7):=22335:a(8):=35634:a(9):=326050:for n from 0 to 40 do a(n+10):=340*a(n+5)-a(n):od:seq(a(n),n=0..40);
%t A167708 Table[n /. {ToRules[Reduce[n > 0 && k >= 0 && n^2-81 == 19*k^2, n, Integers] /. C[1] -> c]} // Simplify, {c, 0, 5}] // Flatten // Union (* _Jean-François Alcover_, Dec 19 2013 *)
%t A167708 LinearRecurrence[{0, 0, 0, 0, 340, 0, 0, 0, 0, -1}, {9, 10, 66, 105, 959, 1530, 2441, 22335, 35634, 326050}, 100] (* _G. C. Greubel_, Jun 20 2016 *)
%Y A167708 Cf. A167709, A167774, A167775, A167778, A167779, A167780.
%K A167708 easy,nonn
%O A167708 1,1
%A A167708 _Richard Choulet_, Nov 10 2009, corrected Nov 12 2009