A023110 - OEIS

%I #83 Oct 28 2023 02:26:25

%S 0,1,4,9,16,49,169,256,361,1444,3249,18496,64009,237169,364816,519841,

%T 2079364,4678569,26666896,92294449,341991049,526060096,749609641,

%U 2998438564,6746486769,38453641216,133088524969,493150849009,758578289296,1080936581761

%N Squares which remain squares when the last digit is removed.

%C 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

%C 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

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

%H Jon E. Schoenfield, <a href="/A023110/b023110.txt">Table of n, a(n) for n = 1..70</a> (terms 1..38 from David W. Wilson, terms 39..40 from Robert G. Wilson v, terms 41..67 from Dmitry Petukhov)

%H M. F. Hasler, <a href="/wiki/M._F._Hasler/Truncated_squares">Truncated squares</a>, OEIS wiki, Jan 16 2012

%H Joshua Stucky, <a href="https://www.math.ksu.edu/~jstucky95/papers/Pell&#39;s%20Equation%20and%20Truncated%20Squares.pdf">Pell's Equation and Truncated Squares</a>, Number Theory Seminar, Kansas State University, Feb 19 2018.

%H <a href="/index/Sq#sqtrunc">Index to sequences related to truncating digits of squares</a>.

%F 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

%F a(n) = A031149(n)^2. - _M. F. Hasler_, Sep 28 2014

%F Conjectures from _Colin Barker_, Dec 31 2017: (Start)

%F 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)).

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

%F (End)

%p count:= 1: A[1]:= 0:

%p for n from 0 while count < 35 do

%p   for t in [1,4,6,9] do

%p     if issqr(10*n^2+t) then

%p        count:= count+1;

%p        A[count]:= 10*n^2+t;

%p     fi

%p   od

%p od:

%p seq(A[i],i=1..count); # _Robert Israel_, Sep 28 2014

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

%o (PARI) for(n=0,1e7, issquare(n^2\10) & print1(n^2",")) \\  _M. F. Hasler_, Jan 16 2012

%Y Cf. A023111.

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

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

%K nonn,base

%O 1,3

%A _David W. Wilson_

%E More terms from _M. F. Hasler_, Jan 16 2012