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A155563 Intersection of A001481 and A003136: N = a^2 + b^2 = c^2 + 3d^2 for some integers a,b,c,d.

%I A155563 #21 Jun 27 2022 11:13:43
%S A155563 0,1,4,9,13,16,25,36,37,49,52,61,64,73,81,97,100,109,117,121,144,148,
%T A155563 157,169,181,193,196,208,225,229,241,244,256,277,289,292,313,324,325,
%U A155563 333,337,349,361,373,388,397,400,409,421,433,436,441,457,468,481,484
%N A155563 Intersection of A001481 and A003136: N = a^2 + b^2 = c^2 + 3d^2 for some integers a,b,c,d.
%C A155563 Contains A155561 as a subsequence (obtained by restricting a,b,c,d to be nonzero). Also contains A000290 (squares) as subsequence.
%H A155563 Ron Lifshitz, <a href="https://doi.org/10.1103/RevModPhys.69.1181">Theory of color symmetry for periodic and quasiperiodic crystals</a>, Rev. Mod. Phys. 69, 1181 (1997). This sequence coincides with the row N = 12 of Table VII.
%o A155563 (PARI) isA155563(n,/* use optional 2nd arg to get other analogous sequences */c=[3,1]) = { for(i=1,#c, for(b=0,sqrtint(n\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
%o A155563 for( n=0,500, isA155563(n) & print1(n","))
%o A155563 (PARI) is(n)=(n==0) || (#bnfisintnorm(bnfinit(z^2+z+1), n) && #bnfisintnorm(bnfinit(z^2+1), n));
%o A155563 select(n->is(n), vector(1500,j,j-1)) \\ _Joerg Arndt_, Jan 11 2015
%o A155563 (Python)
%o A155563 from itertools import count, islice
%o A155563 from sympy import factorint
%o A155563 def A155563_gen(): # generator of terms
%o A155563     return filter(lambda n: all(e & 1 == 0 or (p & 3 != 3 and p % 3 < 2) for p, e in factorint(n).items()),count(0))
%o A155563 A155563_list = list(islice(A155563_gen(),30)) # _Chai Wah Wu_, Jun 27 2022
%Y A155563 Cf. A000290, A001481, A003136, A155561.
%K A155563 easy,nonn
%O A155563 1,3
%A A155563 _M. F. Hasler_, Jan 24 2009