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A023111 Numbers k such that k remains square when digit 1 is appended. 3
0, 36, 51984, 74960964, 108093658176, 155870980128900, 224765845252215696, 324112192982714904804, 467369557515229640511744, 673946577824768158903030116, 971830497853758169908528915600, 1401378903958541456239939793265156, 2020787407677718926139823273359439424 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

LINKS

Colin Barker, Table of n, a(n) for n = 1..300

Index entries for linear recurrences with constant coefficients, signature (1443, -1443, 1).

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

EXAMPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A023110.

Sequence in context: A159435 A123397 A185097 * A295927 A059493 A212327

Adjacent sequences:  A023108 A023109 A023110 * A023112 A023113 A023114

KEYWORD

nonn,base,easy

AUTHOR

David W. Wilson

STATUS

approved

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Last modified December 17 09:23 EST 2018. Contains 318193 sequences. (Running on oeis4.)