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A306380 Squares of the form 5*k^2 + 5.

%I A306380 #110 Feb 16 2025 08:33:55
%S A306380 25,7225,2325625,748843225,241125192025,77641562988025,
%T A306380 25000342156951225,8050032532975305625,2592085475275891459225,
%U A306380 834643473006304074564025,268752606222554636118156025,86537504560189586525971675225
%N A306380 Squares of the form 5*k^2 + 5.
%C A306380 All terms of this sequence are odd, hence they end with 5.
%H A306380 Stefano Spezia, <a href="/A306380/b306380.txt">Table of n, a(n) for n = 1..390</a>
%H A306380 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PellEquation.html">Pell Equation</a>.
%H A306380 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (323,-323,1).
%F A306380 O.g.f.: 25*x*(1 - 34*x + x^2)/((1 - x)*(1 - 322*x + x^2)).
%F A306380 E.g.f.: (5/4)*x*(2*exp(x) + (9 - 4*sqrt(5))*exp((9 - 4*sqrt(5))^2*x) + (9 + 4*sqrt(5))*exp((9 + 4*sqrt(5))^2*x)).
%F A306380 a(n) = 323*a(n-1) - 323*a(n-2) + a(n-3) for n > 3.
%F A306380 a(n) = (5/4)*(2 + (9 - 4*sqrt(5))^(2*n)*(9 + 4*sqrt(5)) + (9 - 4*sqrt(5))*(9 + 4*sqrt(5))^(2*n)).
%F A306380 a(n) = 5*A000290(A075796(n)) + 5.
%p A306380 a := n ->(5/4)*(2+(9-4*sqrt(5))^(2*n-2)*(9+sqrt(5))+(9+4*sqrt(5))^(2*n-2)*(9-sqrt(5))): op(map(simplify, [seq(a(n), n = 1 .. 20)]))
%t A306380 LinearRecurrence[{323, -323, 1}, {25, 7225, 2325625}, 30]
%o A306380 (GAP) a:=[25,7225,2325625];; for n in [4..20] do a[n]:=323*a[n-1]-323*a[n-2]+a[n-3]; od; a;
%o A306380 (Magma) I:=[25, 7225, 2325625]; [n le 3 select I[n] else 323*Self(n-1)-323*Self(n-2)+Self(n-3): n in [1..20]];
%o A306380 (Maxima) a[1]:25$ a[2]:7225$ a[3]:2325625$ a[n]:=323*a[n-1]-323*a[n-2]+a[n-3]$ create_list(a[n], n, 1, 20);
%o A306380 (PARI) Vec(25*x*(1-34*x+x^2)/((1-x)*(1-322*x+x^2)) + O(x^20))
%Y A306380 Cf. A000290, A075796 (associated k).
%K A306380 nonn,easy
%O A306380 1,1
%A A306380 _Stefano Spezia_, Feb 13 2019