A082176 Professor E. P. B. Umbugio's sequence.

0, 0, 206276, 1124101062, 4106026092896, 12565214785548390, 34787981278581970376, 90353184628933414448862, 224610989213093282203310816, 541037084832262355204120965110, 1272999064631803815296028401200376, 2942001006486252167427671506502189262

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

Cf. A082177, A082178.

Sequence in context: A234061 A234060 A115946 * A178285 A101701 A092011

Adjacent sequences: A082173 A082174 A082175 * A082177 A082178 A082179

Keyword

nonn,easy

Author

Wolfdieter Lang, Apr 25 2003

Status

approved