A319170 Triangular numbers of the form 2..21..1; n_times 2 followed with n_times 1; n >= 1.

21, 2211, 222111, 22221111, 2222211111, 222222111111, 22222221111111, 2222222211111111, 222222222111111111, 22222222221111111111, 2222222222211111111111, 222222222222111111111111, 22222222222221111111111111, 2222222222222211111111111111, 222222222222222111111111111111, 22222222222222221111111111111111

Offset: 1

Author

Ctibor O. Zizka, Sep 12 2018

Keywords

• nonn
• base

Comments

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

Examples

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

Links

• Colin Barker, Table of n, a(n) for n = 1..500
• Jiri Sedlacek, Trojuhelnikova cisla, In: Jiří Sedláček (author): Faktoriály a kombinační čísla. (Czech). Praha: Mladá fronta, 1964. pp. 60-71.
• Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Crossrefs

Cf. A000217, A002280, A002283, A199682.

Programs

Mathematica

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 *)

PARI

Vec(3*x*(7 - 40*x) / ((1 - x)*(1 - 10*x)*(1 - 100*x)) + O(x^20)) \\ Colin Barker, Sep 13 2018

Formula

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

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

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

From Colin Barker, Sep 13 2018: (Start)

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

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

(End)