A075412 Squares of A002277.

0, 9, 1089, 110889, 11108889, 1111088889, 111110888889, 11111108888889, 1111111088888889, 111111110888888889, 11111111108888888889, 1111111111088888888889, 111111111110888888888889, 11111111111108888888888889, 1111111111111088888888888889, 111111111111110888888888888889

Offset: 0

Author

Michael Taylor (michael.taylor(AT)vf.vodafone.co.uk), Sep 14 2002

Keywords

• nonn
• easy

Comments

A transformation of the Wonderful Demlo numbers (A002477).

Examples

			a(2) = 33^2 = 1089.
Contribution from _Reinhard Zumkeller_, May 31 2010: (Start)
n=1: ...................... 9 = 9 * 1;
n=2: ................... 1089 = 99 * 11;
n=3: ................. 110889 = 999 * 111;
n=4: ............... 11108889 = 9999 * 1111;
n=5: ............. 1111088889 = 99999 * 11111;
n=6: ........... 111110888889 = 999999 * 111111;
n=7: ......... 11111108888889 = 9999999 * 1111111;
n=8: ....... 1111111088888889 = 99999999 * 11111111;
n=9: ..... 111111110888888889 = 999999999 * 111111111. (End)
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Gérard Villemin, Variations sur les carrés.
• Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Crossrefs

Cf. A075411, A075412, A075413, A075414, A075415, A075416, A075417, A002283, A178630, A178631, A178632, A178633, A178634, A178635.

Cf. A000042, A002277, A002275, A002282, A002283, A002477, A059988.

Programs

Mathematica

LinearRecurrence[{11, -10}, {0, 3}, 20]^2 (* Vincenzo Librandi, Mar 20 2014 *)
Table[FromDigits[PadRight[{},n,9]]FromDigits[PadRight[{},n,1]],{n,0,15}] (* Harvey P. Dale, Feb 12 2023 *)

Formula

a(n) = A002277(n)^2 = (3*A002275(n))^2 = 9*A002275(n)^2.

a(n) = {111111... (2n times)} - 2*{ 111... (n times)} a(n) = A000042(2*n) - 2*A000042(n). - Amarnath Murthy, Jul 21 2003

a(n) = {333... (n times)}^2 = {111...(n times)}{000... (n times)} - {111... (n times)}. For example, 333^2 = 111000 - 111 = 110889. - Kyle D. Balliet, Mar 07 2009

From Reinhard Zumkeller, May 31 2010: (Start)

a(n) = A002283(n)*A002275(n).

For n>0, a(n) = (A002275(n-1)*10^n + A002282(n-1))*10 + 9. (End)

a(n) = (10^(n+1)-10)^2/900. - José de Jesús Camacho Medina, Apr 01 2016

From Elmo R. Oliveira, Jul 27 2025: (Start)

G.f.: 9*x*(1+10*x)/((1-x)*(1-10*x)*(1-100*x)).

E.g.f.: exp(x)*(1 - 2*exp(9*x) + exp(99*x))/9.

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

a(n) = 9*A002477(n). (End)