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%I #24 Jan 22 2024 05:52:28
%S 0,0,2,10899,39810992,121829149155,337295480604452,876041659029003999,
%T 2177771424820078750832,5245758933004928883671595,
%U 12342677428608309403866939452,28524898742328260848840112339799,65009312944028099855926272111730472
%N Professor Umbugio's sequence A082176 divided by 2*7*53*139 = 103138.
%C For references and details see A082176.
%H Colin Barker, <a href="/A082178/b082178.txt">Table of n, a(n) for n = 0..300</a>
%H H. E. G. P., <a href="http://www.jstor.org/stable/2304934">Elementary problem No. E716, Professor Umbugio's Prediction</a>, Solution by E. P. Starke, American Math. Monthly 54:1 (1947), pp. 43-44.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (7266,-19690571,23585007306,-10533473613720).
%F a(n) = (1492^n - 1770^n - 1863^n + 2141^n)/103138 = A082176(n)/103138 = A082177(n)/53.
%F From _Colin Barker_, Nov 21 2015: (Start)
%F a(n) = 7266*a(n-1) - 19690571*a(n-2) + 23585007306*a(n-3) - 10533473613720*a(n-4) for n>3.
%F G.f.: x^2*(2-3633*x) / ((1-1492*x)*(1-1770*x)*(1-1863*x)*(1-2141*x)). (End)
%F E.g.f.: exp(1492*x)*(1 - exp(278*x) - exp(371*x) + exp(649*x))/103138. - _G. C. Greubel_, Jan 22 2024
%t Table[(1492^n - 1770^n - 1863^n + 2141^n)/103138, {n, 0, 12}] (* _Michael De Vlieger_, Nov 21 2015 *)
%t CoefficientList[Series[x^2 (2-3633*x)/((1-1492 x) (1-1770 x) (1-1863 x) (1-2141 x)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Nov 22 2015 *)
%t LinearRecurrence[{7266,-19690571,23585007306,-10533473613720},{0,0,2,10899},20] (* _Harvey P. Dale_, Jun 25 2017 *)
%o (PARI) concat(vector(2), Vec(x^2*(2-3633*x) / ((1-1492*x)*(1-1770*x)*(1-1863*x)*(1-2141*x)) + O(x^15))) \\ _Colin Barker_, Nov 21 2015
%o (Magma) [(1492^n-1770^n-1863^n+2141^n)/103138 : n in [0..20]]; // _Vincenzo Librandi_, Nov 22 2015
%o (SageMath) [(1492^n-1770^n-1863^n+2141^n)/103138 for n in range(21)] # _G. C. Greubel_, Jan 22 2024
%Y Cf. A082176, A082177.
%K nonn,easy
%O 0,3
%A _Wolfdieter Lang_, Apr 25 2003
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