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%I #29 Dec 24 2022 19:03:06
%S 1,4,25,100,289,676,1369,2500,4225,6724,10201,14884,21025,28900,38809,
%T 51076,66049,84100,105625,131044,160801,195364,235225,280900,332929,
%U 391876,458329,532900,616225,708964,811801,925444,1050625,1188100
%N Main diagonal of A082043: a(n) = n^4 + 2*n^2 + 1.
%C a(n) = longest side b of all integer-sided triangles with sides a <= b <= c and inradius n >= 1. Triangle has sides (n^2+2, n^4+2*n^2+1, n^4+3*n^2+1).
%H G. C. Greubel, <a href="/A082044/b082044.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = n^4 + 2*n^2 + 1.
%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - _Harvey P. Dale_, Feb 27 2015
%F a(n) = (2*(2* A000217(n-1)^2 +(A002061(n)))^2 / A082044(n-1). - _Bruce J. Nicholson_, Apr 17 2017
%F a(n) = A002522(n)^2 = (n^2 + 1)^2 = a(-n) for all n in Z. - _Michael Somos_, Apr 17 2017
%F G.f.: (1 - x + 15*x^2 + 5*x^3 + 4*x^4 ) / (1 - x)^5. - _Michael Somos_, Apr 17 2017
%F From _Amiram Eldar_, Nov 02 2021: (Start)
%F Sum_{n>=0} 1/a(n) = Pi^2*csch(Pi)^2/4 + Pi*coth(Pi)/4 + 1/2.
%F Sum_{n>=0} (-1)^n/a(n) = Pi^2*csch(Pi)*coth(Pi)/4 + Pi*csch(Pi)/4 + 1/2. (End)
%F E.g.f.: (1 + 3*x + 9*x^2 + 6*x^3 + x^4)*exp(x). - _G. C. Greubel_, Dec 24 2022
%e G.f. = 1 + 4*x + 25*x^2 + 100*x^3 + 289*x^4 + 676*x^5 + 1369*x^6 + ...
%p seq(fibonacci(3,n)^2,n=0..33); # _Zerinvary Lajos_, Apr 09 2008
%t Fibonacci[3,Range[0,40]]^2 (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,4,25,100,289},40] (* _Harvey P. Dale_, Feb 27 2015 *)
%o (PARI) a(n) = n^4+2*n^2+1; \\ _Michel Marcus_, Jan 22 2016
%o (Magma) [(n^2+1)^2: n in [0..40]]; // _G. C. Greubel_, Dec 24 2022
%o (SageMath) [(n^2+1)^2 for n in range(41)] # _G. C. Greubel_, Dec 24 2022
%Y Cf. A000217, A002061 A002522, A059826, A082043, A082044, A082047.
%Y See A120062 for sequences related to integer-sided triangles with integer inradius n.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Apr 03 2003
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