OFFSET
0,3
COMMENTS
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).
REFERENCES
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).
LINKS
Colin Barker, Table of n, a(n) for n = 0..300
H. E. G. P. and E. P. Starke, Elementary problem No. E716, Professor Umbugio's Prediction, Solution by E. P. Starke, American Math. Monthly 54:1 (1947), pp. 43-44.
Index entries for linear recurrences with constant coefficients, signature (7266,-19690571,23585007306,-10533473613720).
FORMULA
a(n) = 1492^n - 1770^n - 1863^n + 2141^n.
From Colin Barker, Nov 21 2015: (Start)
a(n) = 7266*a(n-1) - 19690571*a(n-2) + 23585007306*a(n-3) - 10533473613720*a(n-4) for n>3.
G.f: 103138*x^2*(2-3633*x) / ((1-1492*x)*(1-1770*x)*(1-1863*x)*(1-2141*x)).
(End)
MATHEMATICA
Table[1492^n - 1770^n - 1863^n + 2141^n, {n, 0, 11}] (* Michael De Vlieger, Nov 21 2015 *)
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 *)
LinearRecurrence[{7266, -19690571, 23585007306, -10533473613720}, {0, 0, 206276, 1124101062}, 20] (* Harvey P. Dale, Oct 18 2020 *)
PROG
(PARI) a(n)=1492^n-1770^n-1863^n+2141^n \\ Charles R Greathouse IV, Sep 16 2015
(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
(Magma) [1492^n-1770^n-1863^n+2141^n: n in [0..20]]; // Vincenzo Librandi, Nov 22 2015
(SageMath) [1492^n -1770^n -1863^n +2141^n for n in range(31)] # G. C. Greubel, Jan 21 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Apr 25 2003
STATUS
approved