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Professor Umbugio's sequence A082176 divided by 1946.
3

%I #20 Jan 22 2024 05:52:06

%S 0,0,106,577647,2109982576,6456944905215,17876660472035956,

%T 46430207928537211947,115421885515464173794096,

%U 278025223449261230834594535,654161903716240398404947790956,1511819633343397824988525954009347,3445493586033489292364092421921715016

%N Professor Umbugio's sequence A082176 divided by 1946.

%C For references and details see A082176.

%H Colin Barker, <a href="/A082177/b082177.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)/1946 = A082176(n)/1946.

%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.: 53*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))/1946. - _G. C. Greubel_, Jan 21 2024

%p A082177:=n->(1492^n - 1770^n - 1863^n + 2141^n)/1946: seq(A082177(n), n=0..15); # _Wesley Ivan Hurt_, Nov 21 2015

%t Table[(1492^n - 1770^n - 1863^n + 2141^n)/1946, {n,0,20}] (* _Michael De Vlieger_, Nov 21 2015 *)

%t CoefficientList[Series[53 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 *)

%o (PARI) concat(vector(2), Vec(53*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)/1946 : n in [0..15]]; // _Wesley Ivan Hurt_, Nov 21 2015

%o (Magma) I:=[0,0,106,577647]; [n le 4 select I[n] else 7266*Self(n-1)-19690571*Self(n-2)+23585007306*Self(n-3)- 10533473613720*Self(n-4): n in [1..20]]; // _Vincenzo Librandi_, Nov 22 2015

%o (SageMath) [(1492^n - 1770^n - 1863^n + 2141^n)/1946 for n in range(21)] # _G. C. Greubel_, Jan 21 2024

%Y Cf. A082176, A082178.

%K nonn,easy

%O 0,3

%A _Wolfdieter Lang_, Apr 25 2003