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Numbers that can't be represented as the sum of two squares, two triangular numbers, or a square and a triangular number.
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%I #13 May 19 2020 16:56:13

%S 23,33,47,62,63,86,118,134,138,143,158,167,188,195,203,204,209,223,

%T 230,243,248,275,283,294,318,323,348,368,383,385,395,398,408,411,413,

%U 418,419,426,437,440,448,454,467,473,476,489,492,503,508,518,523,558,563,566,572,608

%N Numbers that can't be represented as the sum of two squares, two triangular numbers, or a square and a triangular number.

%C Intersection of A014134, A020757, A022544.

%H Robert Israel, <a href="/A264101/b264101.txt">Table of n, a(n) for n = 1..10000</a>

%e Since 22 = 16+6, because 16 is a square and 6 is a triangular number, 22 is not a term.

%e 23 is a term because there is no representation as S+T or S1+S2 or T1+T2, where S, S1, S2 are squares, and T, T1, T2 are triangular numbers.

%p N:= 1000: # for terms <= N

%p S:= [seq(i^2,i=0..floor(sqrt(N)))]: nS:= nops(S):

%p T:= [seq(i*(i+1)/2, i=0..floor(sqrt(2*N)))]: nT:= nops(T):

%p sort(convert({$1..N} minus {seq(seq(S[i]+S[j], j=1..i),i=1..nS),

%p seq(seq(S[i]+T[j],i=1..nS),j=1..nT),

%p seq(seq(T[i]+T[j],j=1..i),i=1..nT)}, list)); # _Robert Israel_, May 19 2020

%t mx = 610; Complement[ Range@ mx, Union@ Flatten@ Table[{i^2 + j^2, i(i + 1)/2 + j^2, i(i + 1)/2 + j(j + 1)/2}, {i, 0, Sqrt[2 mx]}, {j, 0, Sqrt[2 mx]}]] (* _Robert G. Wilson v_, Nov 29 2015 *)

%Y Cf. A000217, A000290, A014134, A020757, A022544, A264118.

%K nonn

%O 1,1

%A _Alex Ratushnyak_, Nov 03 2015