A055793 Numbers n such that n and floor[n/3] are both squares; i.e., squares which remain squares when written in base 3 and last digit is removed.

0, 1, 4, 49, 676, 9409, 131044, 1825201, 25421764, 354079489, 4931691076, 68689595569, 956722646884, 13325427460801, 185599261804324, 2585064237799729, 36005300067391876, 501489136705686529, 6984842613812219524, 97286307456665386801

Offset

1,3

Comments

Or, squares of the form 3n^2+1.

See A023110, A204503, A204512, A204517, A204519, A055812, A055808 and A055792 for the analog in other bases.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..800

Tom C. Brown and Peter J Shiue, Squares of second-order linear recurrence sequences, Fib. Quart., 33 (1994), 352-356.

M. F. Hasler, Truncated squares, OEIS wiki, Jan 16 2012

Giovanni Lucca, Integer Sequences and Circle Chains Inside a Circular Segment, Forum Geometricorum, Vol. 18 (2018), 47-55.

Index to sequences related to truncating digits of squares.

Index entries for linear recurrences with constant coefficients, signature (15,-15,1).

Formula

a(n) = 3*A098301(n-2)+1. - R. J. Mathar, Jun 11 2009

a(n) = 14*a(n-1)-a(n-2)-6, with a(0)=1, a(1)=4. (See Brown and Shiue)

a(n) = (A001075(n-2))^2. - Johannes Boot Dec 16 2011, corrected by M. F. Hasler, Jan 15 2012

G.f.: x*(1 - 11*x + 4*x^2)/((1 - x)*(1 - 14*x + x^2)). - M. F. Hasler, Jan 15 2012

Example

a(3) = 49 because 49 = 7^2 = 1211 base 3 and 121 base 3 = 16 = 4^2.

Maple

A055793 := proc(n) coeftayl(x*(1-11*x+4*x^2)/((1-x)*(1-14*x+x^2)), x=0, n); end proc: seq(A055793(n), n=0..20); # Wesley Ivan Hurt, Sep 28 2014

Mathematica

CoefficientList[Series[x*(1 - 11*x + 4*x^2)/((1 - x)*(1 - 14*x + x^2)), {x, 0, 20}], x] (* Wesley Ivan Hurt, Sep 28 2014 *)
LinearRecurrence[{15, -15, 1}, {0, 1, 4, 49}, 40] (* Harvey P. Dale, Jun 19 2021 *)

Prog

(PARI) sq3nsqplus1(n) = { for(x=1, n, y = 3*x*x+1; \ print1(y" ") if(issquare(y), print1(y" ")) ) }
(Magma) I:=[0, 1, 4]; [n le 3 select I[n] else 14*Self(n-1) - Self(n-2) - 6: n in [1..30]]; // Vincenzo Librandi, Jan 27 2013

Crossrefs

Cf. A001075, A023110, A098301.

Cf. also A023110, A204503, A204512, A204517, A204519, A055812, A055808 and A055792 for the analog in other bases.

Sequence in context: A199028 A189146 A086094 * A202829 A204233 A144656

Adjacent sequences: A055790 A055791 A055792 * A055794 A055795 A055796

Keyword

base,nonn,easy

Author

Henry Bottomley, Jul 14 2000

Extensions

More terms from Cino Hilliard, Mar 01 2003

Status

approved