A256944 - OEIS

%I #86 Jan 31 2024 07:50:31

%S 0,1,4,9,16,36,49,64,100,144,196,256,289,324,400,484,576,676,784,900,

%T 1024,1156,1296,1444,1600,1681,1764,1936,2116,2304,2500,2704,2916,

%U 3136,3364,3600,3844,4096,4356,4624,4900,5184,5476,5776,6084,6400,6724,7056,7396,7744,8100,8464

%N Squares which are not the sums of two consecutive nonsquares.

%C The union of A008843, A055792, and A016742. [Corrected by _Charles R Greathouse IV_, May 07 2015]

%C Consists of the squares of all even numbers and odd numbers in A078057 = (1, 3, 7, 17, 41, 99, ...), see also A001333 = abs(A123335). See A257282 for the square roots and A257292 for their complement in the nonnegative integers A001477. - _M. F. Hasler_, May 08 2015

%H Charles R Greathouse IV, <a href="/A256944/b256944.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) ~ 4n^2. - _Charles R Greathouse IV_, May 07 2015

%F a(n) = A257282(n)^2. - _M. F. Hasler_, May 08 2015

%e 0, 1, 4, 9, 16, 36, are in this sequence because first 14 sums of two consecutive nonsquares are 5, 8, 11, 13, 15, 18, 21, 23, 25, 27, 29, 32, 35, 37.

%t lim = 15000; s = Plus @@@ (Partition[#, 2, 1] & @ Complement[Range@ lim, Range[Floor@ Sqrt[lim]]^2]); Select[Range@ Floor[Sqrt[lim]]^2, !MemberQ[s, #] &] (* _Michael De Vlieger_, Apr 29 2015 *)

%t lst=Partition[Select[Range[0,10^6],!IntegerQ[Sqrt[#]]&],2,1]/.{a_,b_}->  a+b;a256944=Complement[Table[n^2,{n,0,Sqrt[Last[lst]]}],lst] (* timing improved by _Ivan N. Ianakiev_, Apr 30 2015 *)

%t Union[#, Range[0, Max@ #, 2]] &@ Numerator[Convergents[Sqrt@ 2, 6]]^2 (* _Michael De Vlieger_, Aug 06 2016, after _Harvey P. Dale_ at A001333 *)

%o (PARI) is(n)=issquare(n) && (n%2==0 || issquare(n\2) || issquare(n\2+1)) \\ _Charles R Greathouse IV_, May 07 2015

%Y Cf. A000037, A000290, A008843, A016742, A056792, A257282.

%K nonn,easy

%O 1,3

%A _Juri-Stepan Gerasimov_, Apr 25 2015