A023110 Squares which remain squares when the last digit is removed.

0, 1, 4, 9, 16, 49, 169, 256, 361, 1444, 3249, 18496, 64009, 237169, 364816, 519841, 2079364, 4678569, 26666896, 92294449, 341991049, 526060096, 749609641, 2998438564, 6746486769, 38453641216, 133088524969, 493150849009, 758578289296, 1080936581761

Offset

1,3

Comments

This A023110 = A031149^2 is the base 10 version of A001541^2 = A055792 (base 2), A001075^2 = A055793 (base 3), A004275^2 = A055808 (base 4), A204520^2 = A055812 (base 5), A204518^2 = A055851 (base 6), A204516^2 = A055859 (base 7), A204514^2 = A055872 (base 8) and A204502^2 = A204503 (base 9). - M. F. Hasler, Sep 28 2014

For the first 4 terms the square has only one digit. It is understood that deleting this digit yields 0. - Colin Barker, Dec 31 2017

References

R. K. Guy, Neg and Reg, preprint, Jan 2012.

Links

Jon E. Schoenfield, Table of n, a(n) for n = 1..70 (terms 1..38 from David W. Wilson, terms 39..40 from Robert G. Wilson v, terms 41..67 from Dmitry Petukhov)

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

Joshua Stucky, Pell's Equation and Truncated Squares, Number Theory Seminar, Kansas State University, Feb 19 2018.

Index to sequences related to truncating digits of squares.

Formula

Appears to satisfy a(n)=1444*a(n-7)+a(n-14)-76*sqrt(a(n-7)*a(n-14)) for n >= 16. For n = 15, 14, 13, ... this would require a(1) = 16, a(0) = 49, a(-1) = 169, ... - Henry Bottomley, May 08 2001; edited by Robert Israel, Sep 28 2014

a(n) = A031149(n)^2. - M. F. Hasler, Sep 28 2014

Conjectures from Colin Barker, Dec 31 2017: (Start)

G.f.: x^2*(1 + 4*x + 9*x^2 + 16*x^3 + 49*x^4 + 169*x^5 + 256*x^6 - 1082*x^7 - 4328*x^8 - 9738*x^9 - 4592*x^10 - 6698*x^11 - 6698*x^12 - 4592*x^13 + 361*x^14 + 1444*x^15 + 3249*x^16 + 256*x^17 + 169*x^18 + 49*x^19 + 16*x^20) / ((1 - x)*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)*(1 - 1442*x^7 + x^14)).

a(n) = 1443*a(n-7) - 1443*a(n-14) + a(n-21) for n>22.

(End)

Maple

count:= 1: A[1]:= 0:
for n from 0 while count < 35 do
  for t in [1, 4, 6, 9] do
    if issqr(10*n^2+t) then
       count:= count+1;
       A[count]:= 10*n^2+t;
    fi
  od
od:
seq(A[i], i=1..count); # Robert Israel, Sep 28 2014

Mathematica

fQ[n_] := IntegerQ@ Sqrt@ Quotient[n^2, 10]; Select[ Range@ 1000000, fQ]^2 (* Robert G. Wilson v, Jan 15 2011 *)

Prog

(PARI) for(n=0, 1e7, issquare(n^2\10) & print1(n^2", ")) \\  M. F. Hasler, Jan 16 2012

Crossrefs

Cf. A023111.

Cf. A031150, A053784, A031149, A055792, A055793, A055808, A055812, A055851, A055859, A055872.

Cf. A001541, A001075, A004275, A204520, A204518, A204516, A204514, A204502, A204503.

Sequence in context: A059931 A027382 A164840 * A277699 A368891 A073723

Adjacent sequences: A023107 A023108 A023109 * A023111 A023112 A023113

Keyword

nonn,base

Author

David W. Wilson

Extensions

More terms from M. F. Hasler, Jan 16 2012

Status

approved