A130652 a(n) = 11^n - 2.

9, 119, 1329, 14639, 161049, 1771559, 19487169, 214358879, 2357947689, 25937424599, 285311670609, 3138428376719, 34522712143929, 379749833583239, 4177248169415649, 45949729863572159, 505447028499293769, 5559917313492231479, 61159090448414546289, 672749994932560009199

Offset: 1

Author

Alexander Adamchuk, Jun 20 2007

Keywords

• nonn
• easy

Comments

There are only two known primes in a(n): a(4) = 14639 and a(6) = 1771559 (see A128472 = smallest prime of the form (2n-1)^k - 2 for k > (2n-1), or 0 if no such number exists). 3 divides a(2k-1). 7 divides a(3k-1). 13 divides a(12k-5). 17 divides a(16k-14).

Final digit of a(n) is 9.

Final two digits of a(n) are periodic with period 10: a(n) mod 100 = {09, 19, 29, 39, 49, 59, 69, 79, 89, 99}.

Final three digits of a(n) are periodic with period 50: a(n) mod 1000 = {009, 119, 329, 639, 049, 559, 169, 879, 689, 599, 609, 719, 929, 239, 649, 159, 769, 479, 289, 199, 209, 319, 529, 839, 249, 759, 369, 079, 889, 799, 809, 919, 129, 439, 849, 359, 969, 679, 489, 399, 409, 519, 729, 039, 449, 959, 569, 279, 089, 999}.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..300
• Index entries for linear recurrences with constant coefficients, signature (12,-11).

Crossrefs

Cf. A001020, A024127, A034524. Cf. A104096 = Largest prime <= 11^n. Cf. A084714 = smallest prime of the form (2n-1)^k - 2, or 0 if no such number exists. Cf. A128472 = smallest prime of the form (2n-1)^k - 2 for k>(2n-1), or 0 if no such number exists. Cf. A014224, A109080, A090669, A128455, A128457, A128458, A128459, A128460, A128461.

Programs

Magma

[11^n - 2: n in [1..50]]; // Vincenzo Librandi, Jun 08 2011

Mathematica

LinearRecurrence[{12, -11},{9, 119},17] (* Ray Chandler, Aug 26 2015 *)

Formula

a(n) = 11*a(n-1) + 20; a(1)=9. - Vincenzo Librandi, Jun 08 2011

From Elmo R. Oliveira, Jun 16 2025: (Start)

G.f.: x*(11*x+9)/((11*x-1)*(x-1)).

E.g.f.: 1 + exp(x)*(exp(10*x) - 2).

a(n) = 12*a(n-1) - 11*a(n-2) for n > 2. (End)