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%I #35 Oct 25 2023 18:26:28
%S 14,84,490,2856,16646,97020,565474,3295824,19209470,111960996,
%T 652556506,3803378040,22167711734,129202892364,753049642450,
%U 4389094962336,25581520131566,149100025827060,869018634830794,5065011783157704,29521052064115430,172061300601534876
%N a(n) is the second number in a triple consisting of 3 numbers, which when squared are part of a right diagonal of a magic square of squares.
%C The multiplying factor 6 appears to come from the ratio of a(1)/a(0) of the sequence. Each of the lines of tables (V vs VII) or (VI vs VIII) in oddwheel.com/ImaginaryB.html generates this factor.
%H Colin Barker, <a href="/A273182/b273182.txt">Table of n, a(n) for n = 0..1000</a>
%H Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Nemeth/nemeth7.html">Ellipse Chains and Associated Sequences</a>, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
%H E. Gutierrez, <a href="http://www.oddwheel.com/ImaginaryF6.html">Recursion Methods to Generate New Integer Sequences (Part VIF)</a>
%H E. Gutierrez, <a href="http://www.oddwheel.com/ImaginaryB.html">Table of Tuples and Use of Magic Ratio for Tuple Conversion (Part IB)</a>
%H E. Gutierrez, <a href="http://www.oddwheel.com/ImaginaryC.html">Table of Tuples for Square of Squares (Part IC)</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-1).
%F a(0)=14, a(1)= 84, a(n+1)= a(n)*6 - a(n-1).
%F G.f.: 14 / (1-6*x+x^2). - _Colin Barker_, May 18 2016
%F E.g.f.: 7*(3*sqrt(2)*sinh(2*sqrt(2)*x) + 4*cosh(2*sqrt(2)*x))*exp(3*x)/2. - _Ilya Gutkovskiy_, May 18 2016
%e a(2) = 84*6 -14 = 490; a(3) = 490*6 - 84 = 2856; a(4) = 2856*6 - 490 = 16646.
%t CoefficientList[Series[14/(1 - 6 x + x^2), {x, 0, 21}], x] (* _Michael De Vlieger_, May 18 2016 *)
%o (PARI) Vec(14/(1-6*x+x^2) + O(x^50)) \\ _Colin Barker_, May 18 2016
%Y Cf. A178218, A273187, A273189.
%K nonn,easy
%O 0,1
%A _Eddie Gutierrez_, May 17 2016
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