OFFSET
1,2
COMMENTS
Sporadic solutions (not consisting only of digits 0 and 1): a(4) = 149, a(11) = 14499, a(17) = 316261, a(209) = 4604367505011, a(715) = 10959977245460011, a(1015) = 110000500908955011, a(1665) = 10099510939154979751, ... Three infinite subsequences are given by numbers of the form 10...01, 10...011 and 110...01, but there are many others. - M. F. Hasler, Nov 14 2017
From Zhao Hui Du, Mar 12 2024: (Start)
Most terms have a special pattern in that they have only digits 0 and 1 and could be written as Sum_{h=0..t} 10^x(h), where 2x(h) and x(h1)+x(h2) are distinct and x(0)=0 for the nonzero ending constraint. The number of n-digit terms in the sequence in the special pattern is A143823(n) - 2*A143823(n-1) + A143823(n-2) for n >= 2.
Terms with only digits 0 and 1 but not in the special pattern exist as well. If we define f(x) = 1 + x^768 + x^960 + x^1008 + x^1020 + x^1028 + x^1040 + x^1088 + x^1280 + x^2048, f(x)^2 is a function with all nonzero coefficients 1,2,10 (the only coefficient of x^2048 is 10 and the coefficient of x^2049 is 0). So f(10) is in the sequence but not in the special pattern. (End)
LINKS
Zhao Hui Du, Table of n, a(n) for n = 1..4000 (first 1000 terms from Chai Wah Wu; 1001..1269 from M. F. Hasler)
Patrick De Geest, Index to related sequences.
Hisanori Mishima, Sporadic tridigital solutions.
FORMULA
a(n) = sqrt(A058412(n)). - Zak Seidov, Jul 01 2013
MAPLE
R[1]:= {1, 9};
for m from 2 to 10 do
R[m]:= select(t -> max(convert(t^2 mod 10^m, base, 10)) <= 2, map(s -> seq(s + i*10^(m-1), i=0..9), R[m-1]))
od:
Res:= {seq(op(select(t -> t >= 10^(m-1) and max(convert(t^2, base, 10)) <= 2, R[m])), m=1..10)}:
sort(convert(Res, list)); # Robert Israel, Feb 23 2016
MATHEMATICA
Select[Range[10^6], And[Total@ Take[RotateRight@ DigitCount@ #, -7] == 0, Mod[#, 10] != 0] &[#^2] &] (* Michael De Vlieger, Nov 14 2017 *)
PROG
(Python)
A058411_list = [i for i in range(10**6) if i % 10 and max(str(i**2)) < '3'] # Chai Wah Wu, Feb 23 2016
(PARI) isok(n)={ n%10 && vecmax(digits(n^2)) < 3 } \\ Michel Marcus, Feb 24 2016, edited by M. F. Hasler, Nov 14 2017
(Magma) [n: n in [1..2*10^8 by 2] | Set(Intseq(n^2)) subset [0, 1, 2]]; // Vincenzo Librandi, Feb 24 2016
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Patrick De Geest, Nov 15 2000
EXTENSIONS
b-file corrected by Zhao Hui Du, Mar 07 2024
STATUS
approved