A023111 Squares that remain square when the digit 1 is appended.

0, 36, 51984, 74960964, 108093658176, 155870980128900, 224765845252215696, 324112192982714904804, 467369557515229640511744, 673946577824768158903030116, 971830497853758169908528915600, 1401378903958541456239939793265156, 2020787407677718926139823273359439424

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• base
• easy

Comments

The terms of the sequence are the squares of the y-values in the solution to the Pellian equation x^2-10*y^2=1. - Colin Barker, Sep 28 2013

After 0, the sequence lists the numbers k for which A055437(k) is a perfect square. - Bruno Berselli, Jan 16 2018

Examples

			36 is a term because both 36 and 361 are squares.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..300
• Index entries for linear recurrences with constant coefficients, signature (1443,-1443,1).

Crossrefs

Cf. A023110.

Programs

Mathematica

LinearRecurrence[{1443,-1443,1},{0,36,51984},20] (* Harvey P. Dale, Dec 23 2013 *)

PARI

concat(0, Vec(36*x^2*(1 + x) / ((1 - x)*(1 - 1442*x + x^2)) + O(x^15))) \\ Colin Barker, Dec 29 2017

Formula

G.f.: 36*x^2*(1 + x) / ((1 - x)*(1 - 1442*x + x^2)). - Colin Barker, Jan 31 2013

a(0)=0, a(1)=36, a(2)=51984, a(n) = 1443*a(n-1)-1443*a(n-2)+a(n-3). - Harvey P. Dale, Dec 23 2013

a(n) = (721 + 228*sqrt(10))^(-n)*(721+228*sqrt(10) - 2*(721+228*sqrt(10))^n + (721-228*sqrt(10))*(721+228*sqrt(10))^(2*n)) / 40. - Colin Barker, Dec 29 2017