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%I #38 Sep 08 2022 08:46:18
%S 7,799,78407,7683199,752875207,73774087199,7229107670407,
%T 708378777612799,69413891098384007,6801852948864019999,
%U 666512175097575576007,65311391306613542428799,6399849835873029582446407,627119972524250285537319199,61451357457540654953074835207
%N Numbers m such that T(m) + 2*T(m+1) is a square, where T = A000217.
%C It is well known that T(m) + k*T(m+1) is always a square for k=1. For k=3, the nonnegative values of m are the terms of A278310.
%C Square roots of T(m) + 2*T(m+1) are listed by A168520 (after 0).
%C Negative values of m for which T(m) + 2*T(m+1) is a square: -1, -2, -82, -7922, -776162, ...
%H Colin Barker, <a href="/A278438/b278438.txt">Table of n, a(n) for n = 1..500</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (99,-99,1).
%F O.g.f.: x*(7 + 106*x - x^2)/((1 - x)*(1 - 98*x + x^2)).
%F E.g.f.: (exp((5-2*sqrt(6))^2*x) + exp((5+2*sqrt(6))^2*x) - 14*exp(x))/12 + 1.
%F a(n) = 99*a(n-1) - 99*a(n-2) + a(n-3) for n>3.
%F a(n) = 98*a(n-1) - a(n-2) + 112 for n>2.
%F a(n) = a(-n) = ((5 + 2*sqrt(6))^(2*n) + (5 - 2*sqrt(6))^(2*n))/12 - 7/6.
%F a(n) = A001079(2*n)/6 - 7/6.
%F a(n) = 2*A001078(n)^2 - 1 = A122652(n)^2/2 - 1.
%F a(n) = -A278620(n+1) + 106*A278620(n) + 7*A278620(n-1).
%F Lim_{n -> infinity} a(n)/a(n-1) = (5 + 2*sqrt(6))^2.
%p P:=proc(q) local n; for n from 1 to q do if type(sqrt((3*n^2+7*n+4)/2),integer) then print(n); fi; od; end: P(10^9); # _Paolo P. Lava_, Nov 25 2016
%t Table[((5 + 2 Sqrt[6])^(2 n) + (5 - 2 Sqrt[6])^(2 n))/12 - 7/6, {n, 1, 20}]
%t RecurrenceTable[{a[1] == 7, a[2] == 799, a[n] == 98 a[n - 1] - a[n - 2] + 112}, a, {n, 1, 20}]
%o (Sage)
%o def A278438():
%o a, b = 7, 799
%o yield a
%o while True:
%o yield b
%o a, b = b, 98*b - a + 112
%o a = A278438(); print([next(a) for _ in range(15)]) # _Peter Luschny_, Nov 24 2016
%o (Magma) Iv:=[7, 799]; [n le 2 select Iv[n] else 98*Self(n-1)-Self(n-2)+112: n in [1..20]];
%o (PARI) Vec(x*(7 + 106*x - x^2)/((1 - x)*(1 - 98*x + x^2)) + O(x^20)) \\ _Colin Barker_, Nov 27 2016
%Y Subsequence of A056220.
%Y Cf. A000217, A001078, A001079, A122652, A168520, A278620.
%Y Cf. A278310: numbers m such that T(m) + 3*T(m+1) is a square.
%K nonn,easy
%O 1,1
%A _Bruno Berselli_, Nov 23 2016
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