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%I #26 Jan 21 2024 09:24:27
%S 0,0,206276,1124101062,4106026092896,12565214785548390,
%T 34787981278581970376,90353184628933414448862,
%U 224610989213093282203310816,541037084832262355204120965110,1272999064631803815296028401200376,2942001006486252167427671506502189262
%N Professor E. P. B. Umbugio's sequence.
%C The problem was to prove that 1946 divides every a(n). The proof uses 2141 - 1770 = 371 = 1863 - 1492 and 2141 - 1863 = 278 = 1770 - 1492, gcd(278,371) = 1, 278*371 = 53*1946 and the fact that x - y not 0 divides x^n - y^n for n>=0. See the Starke reference. The primes that divide every a(n) are 2, 7, 53, 139. Note the historical dates other than 2141 in the formula. This AMM problem was proposed in 1946 (with a reference to April 1).
%D C. A. Pickover, Die Mathematik und das Goettliche, Spektrum Akademischer Verlag, Heidelberg, Berlin, 1999, pp. 56-8, 398 (English: The Loom of God, Plenum, 1997).
%H Colin Barker, <a href="/A082176/b082176.txt">Table of n, a(n) for n = 0..300</a>
%H H. E. G. P. and E. P. Starke, <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.
%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: 103138*x^2*(2-3633*x) / ((1-1492*x)*(1-1770*x)*(1-1863*x)*(1-2141*x)).
%F (End)
%t Table[1492^n - 1770^n - 1863^n + 2141^n, {n, 0, 11}] (* _Michael De Vlieger_, Nov 21 2015 *)
%t CoefficientList[Series[103138 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},{0,0,206276,1124101062},20] (* _Harvey P. Dale_, Oct 18 2020 *)
%o (PARI) a(n)=1492^n-1770^n-1863^n+2141^n \\ _Charles R Greathouse IV_, Sep 16 2015
%o (PARI) concat(vector(2), Vec(103138*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: n in [0..20]]; // _Vincenzo Librandi_, Nov 22 2015
%o (SageMath) [1492^n -1770^n -1863^n +2141^n for n in range(31)] # _G. C. Greubel_, Jan 21 2024
%Y Cf. A082177, A082178.
%K nonn,easy
%O 0,3
%A _Wolfdieter Lang_, Apr 25 2003