A319170 - OEIS

%I #29 Sep 16 2019 10:55:56

%S 21,2211,222111,22221111,2222211111,222222111111,22222221111111,

%T 2222222211111111,222222222111111111,22222222221111111111,

%U 2222222222211111111111,222222222222111111111111,22222222222221111111111111,2222222222222211111111111111,222222222222222111111111111111,22222222222222221111111111111111

%N Triangular numbers of the form 2..21..1; n_times 2 followed with n_times 1; n >= 1.

%C Triangular numbers of the  form (5^(2x)*2^(2x+1)-10^x-1)/9. - _Harvey P. Dale_, Sep 16 2019

%H Colin Barker, <a href="/A319170/b319170.txt">Table of n, a(n) for n = 1..500</a>

%H Jiri Sedlacek, <a href="https://dml.cz/bitstream/handle/10338.dmlcz/403521/SkolaMladychMatematiku_010-1964-1_9.pdf">Trojuhelnikova cisla</a>, In: Jiří Sedláček (author): Faktoriály a kombinační čísla. (Czech). Praha: Mladá fronta, 1964. pp. 60-71.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (111,-1110,1000).

%F For n >= 1, a(n) = 2..21..1; n_times 2 followed with n_times 1.

%F a(n) = A000217(n_times 6), that is a(n) = A000217(A002280(n)).

%F a(n) = 1/9 * (2*10^n + 1) * (10^n - 1), that is a(n) = 1/9 * A199682(n) * A002283(n).

%F From _Colin Barker_, Sep 13 2018: (Start)

%F G.f.: 3*x*(7 - 40*x) / ((1 - x)*(1 - 10*x)*(1 - 100*x)).

%F a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n>3

%F (End)

%e a(1) = A000217(6) = 21; a(2) = A000217(66) = 2211; a(3) = A000217(666) = 222111.

%t Select[Table[FromDigits[Join[PadRight[{},n,2],PadRight[{},n,1]]],{n,20}], OddQ[ Sqrt[8#+1]]&] (& or *) Select[Table[(5^(2x) 2^(2x+1)-10^x-1)/9,{x,20}],OddQ[Sqrt[8#+1]]&] (* _Harvey P. Dale_, Sep 16 2019 *)

%o (PARI) Vec(3*x*(7 - 40*x) / ((1 - x)*(1 - 10*x)*(1 - 100*x)) + O(x^20)) \\ _Colin Barker_, Sep 13 2018

%Y Cf. A000217, A002280, A002283, A199682.

%K nonn,base

%O 1,1

%A _Ctibor O. Zizka_, Sep 12 2018