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A276599 Values of n such that n^2 + 5 is a triangular number (A000217).

%I A276599 #21 Sep 08 2022 08:46:17
%S A276599 1,4,10,25,59,146,344,851,2005,4960,11686,28909,68111,168494,396980,
%T A276599 982055,2313769,5723836,13485634,33360961,78600035,194441930,
%U A276599 458114576,1133290619,2670087421,6605301784,15562409950,38498520085,90704372279,224385818726
%N A276599 Values of n such that n^2 + 5 is a triangular number (A000217).
%H A276599 Colin Barker, <a href="/A276599/b276599.txt">Table of n, a(n) for n = 1..1000</a>
%H A276599 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,6,0,-1).
%F A276599 a(n) = 6*a(n-2) - a(n-4) for n>4.
%F A276599 G.f.: x*(1+x)*(1+3*x+x^2) / ((1+2*x-x^2)*(1-2*x-x^2)).
%F A276599 a(n) = (1/4)*(P(n+2) + P(n+1) + (-1)^n*(3*P(n) - 7*P(n-1))), where P(n) = A000129(n). - _G. C. Greubel_, Sep 15 2021
%e A276599 4 is in the sequence because 4^2 + 5 = 21, which is a triangular number.
%t A276599 CoefficientList[Series[(1+x)*(1+3*x+x^2)/((1+2*x-x^2)*(1-2*x-x^2)), {x, 0, 30}], x] (* _Wesley Ivan Hurt_, Sep 07 2016 *)
%t A276599 LinearRecurrence[{0,6,0,-1},{1,4,10,25},30] (* _Harvey P. Dale_, Feb 13 2017 *)
%o A276599 (PARI) Vec(x*(1+x)*(1+3*x+x^2)/((1+2*x-x^2)*(1-2*x-x^2)) + O(x^40))
%o A276599 (PARI) a(n)=([0,1;-1,6]^(n\2)*if(n%2,[1;10],[-1;4]))[1,1] \\ _Charles R Greathouse IV_, Sep 07 2016
%o A276599 (Magma) I:=[1,4,10,25]; [n le 4 select I[n] else 6*Self(n-2) - Self(n-4): n in [1..31]]; // _G. C. Greubel_, Sep 15 2021
%o A276599 (Sage)
%o A276599 def P(n): return lucas_number1(n, 2, -1)
%o A276599 [(1/4)*(P(n+2) + P(n+1) + (-1)^n*(3*P(n) - 7*P(n-1))) for n in (1..30)] # _G. C. Greubel_, Sep 15 2021
%Y A276599 Cf. A000129, A000217, A230044.
%Y A276599 Cf. A001109 (k=0), A106328 (k=1), A077241 (k=2), A276598 (k=3), A276600 (k=6), A276601 (k=9), A276602 (k=10), where k is the value added to n^2.
%K A276599 nonn,easy
%O A276599 1,2
%A A276599 _Colin Barker_, Sep 07 2016