A330135 a(n) = ((10^(n+1))^4 - 1)/9999 for n >= 0.

1, 10001, 100010001, 1000100010001, 10001000100010001, 100010001000100010001, 1000100010001000100010001, 10001000100010001000100010001, 100010001000100010001000100010001

Offset

0,2

Comments

This sequence was the subject of the 6th problem of the 15th British Mathematical Olympiad in 1979 (see the link BMO).

There are no prime numbers in this infinite sequence. Why?

a(0) = 1 and a(1) = 10001 = 73 * 137;

if n even = 2*k, k >= 1, then A094028(n) divides a(n);

if n odd = 2*k+1, k >= 1, then a(k) divides a(n).

References

A. Gardiner, The Mathematical Olympiad Handbook: An Introduction to Problem Solving, Oxford University Press, 1997, reprinted 2011, Pb 6 pp. 68 and 201 (1979).

Links

Colin Barker, Table of n, a(n) for n = 0..200

British Mathematical Olympiad, 1979 - Problem 6.

Index entries for linear recurrences with constant coefficients, signature (10001,-10000).

Formula

a(n) = (10^(4*n+4) - 1)/9999 for n >= 0.

G.f.: 1 / ((1 - x)*(1 - 10000*x)). - Colin Barker, Dec 05 2019

Example

a(2) = ((10^3)^4 - 1)/9999 = 100010001 = 10101 * 9901 where 10101 = A094028(2).
a(3) = ((10^4)^4 - 1)/9999 = 1000100010001 = 10001 * 100000001 where 10001 = a(1).
From Omar E. Pol, Dec 04 2019: (Start)
Illustration of initial terms:
                  1;
                10001;
              100010001;
            1000100010001;
          10001000100010001;
        100010001000100010001;
      1000100010001000100010001;
    10001000100010001000100010001;
  100010001000100010001000100010001;
...
(End)

Maple

A: = seq((10^(4*n+4)-1)/9999, n=1..4);

Mathematica

Table[((10^(n+1))^4 - 1)/9999, {n, 0, 8}] (* Amiram Eldar, Dec 04 2019 *)

Prog

(PARI) Vec(1 / ((1 - x)*(1 - 10000*x)) + O(x^11)) \\ Colin Barker, Dec 05 2019

Crossrefs

Cf. A000533 (1000...0001), A094028 (10101...101), A261544 (1001001...1001).

Cf. A131865 (similar, with 2^(n+1)).

Sequence in context: A082567 A364050 A360574 * A031598 A210760 A164968

Adjacent sequences: A330132 A330133 A330134 * A330136 A330137 A330138

Keyword

nonn,easy

Author

Bernard Schott, Dec 02 2019

Status

approved