A204502 Numbers k such that floor(k^2 / 9) is a square.

0, 1, 2, 3, 4, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168, 171, 174, 177

Offset

1,3

Comments

Or, numbers k such that k^2, with its last base-9 digit dropped, is again a square. (Except maybe for the 3 initial terms whose square has only 1 digit in base 9.)

Links

Table of n, a(n) for n=1..63.

Sela Fried, Proof of a conjecture stated in A204502, 2025.

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

Index to sequences related to truncating digits of squares.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Formula

From Colin Barker, Nov 23 2012: (Start)

a(n) = 3*n - 12 for n > 5.

G.f.: x^2*(x^2+x+1)*(x^3-x+1)/(x-1)^2. (End)

Mathematica

Select[Range[0, 200], IntegerQ[Sqrt[Floor[#^2/9]]]&] (* Harvey P. Dale, May 05 2018 *)

Prog

(PARI) b=9; for(n=0, 200, issquare(n^2\b) & print1(n", "))

Crossrefs

The squares are in A204503, the squares with last base-9 digit dropped in A204504, and the square roots of the latter in A028310.

Cf. A031149=sqrt(A023110) (base 10), A204514=sqrt(A055872) (base 8), A204516=sqrt(A055859) (base 7), A204518=sqrt(A055851) (base 6), A204520=sqrt(A055812) (base 5), A004275=sqrt(A055808) (base 4), A001075=sqrt(A055793) (base 3), A001541=sqrt(A055792) (base 2).

Sequence in context: A128166 A240470 A112249 * A062437 A060729 A229169

Adjacent sequences: A204499 A204500 A204501 * A204503 A204504 A204505

Keyword

nonn,easy

Author

M. F. Hasler, Jan 15 2012

Status

approved