This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A271675 #61 Dec 22 2024 09:05:51
%S A271675 0,4,7,15,20,32,39,55,64,84,95,119,132,160,175,207,224,260,279,319,
%T A271675 340,384,407,455,480,532,559,615,644,704,735,799,832,900,935,1007,
%U A271675 1044,1120,1159,1239,1280,1364,1407,1495,1540,1632,1679,1775,1824,1924,1975,2079,2132,2240,2295,2407
%N A271675 Numbers m such that 3*m + 4 is a square.
%C A271675 7 is the unique prime in this sequence. If m is in this sequence, then 3*m + 4 = k^2 for k is nonzero integer, that is, m = (k^2 - 4)/3 = (k-2)*(k+2)/3. So m can be only prime if one of divisors is prime and another one is 1. Otherwise there should be more than 1 prime divisors, that is n must be composite. - _Altug Alkan_, Apr 12 2016
%C A271675 From _Ray Chandler_, Apr 12 2016: (Start)
%C A271675 Square roots of resulting squares gives A001651 (with a different starting point).
%C A271675 Sequence is the union of (positive terms) in A140676 and A270710. (End)
%C A271675 The sequence terms are the exponents in the expansion of Sum_{n >= 0} q^n*(1 - q)*(1 - q^3)*...*(1 - q^(2*n+1)) = 1 - q^4 - q^7 + q^15 + q^20 - q^32 - q^50 + + - - .... - _Peter Bala_, Dec 19 2024
%H A271675 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A271675 O.g.f.: x^2*(4 + 3*x - x^3)/((1 + x)^2*(1 - x)^3).
%F A271675 E.g.f.: 1 + (1 - 2*x)*exp(-x)/8 - 3*(3 - 4*x - 2*x^2)*exp(x)/8.
%F A271675 a(n) = A001082(n+1) - 1 = (6*n*(n+1) + (2*n + 1)*(-1)^n - 1)/8 - 1. Therefore: a(2*k+1) = k*(3*k+4), a(2*k) = (k+1)*(3*k-1).
%F A271675 Sum_{n>=2} 1/a(n) = 19/16 - Pi/(4*sqrt(3)). - _Amiram Eldar_, Jul 26 2024
%e A271675 a(4) = 32 because 3*32 + 4 = 100 = 10*10.
%t A271675 Select[Range[0,2500], IntegerQ@ Sqrt[3 # + 4] &] (* _Michael De Vlieger_, Apr 12 2016 *)
%t A271675 LinearRecurrence[{1,2,-2,-1,1},{0,4,7,15,20},60] (* _Harvey P. Dale_, Dec 09 2016 *)
%o A271675 (Magma) [n: n in [0..4000] | IsSquare(3*n+4)];
%o A271675 (Python)
%o A271675 from gmpy2 import is_square
%o A271675 for n in range(0,10**5):
%o A271675 if(is_square(3*n+4)):print(n)
%o A271675 # _Soumil Mandal_, Apr 12 2016
%Y A271675 Cf. A001651, A140676, A270710.
%Y A271675 Cf. numbers n such that 3*n + k is a square: A120328 (k=-6), A271713 (k=-5), A056107 (k=-3), A257083 (k=-2), A033428 (k=0), A001082 (k=1), A080663 (k=3), this sequence (k=4), A100536 (k=6).
%K A271675 nonn,easy
%O A271675 1,2
%A A271675 _Juri-Stepan Gerasimov_, Apr 12 2016
%E A271675 Edited and extended by _Bruno Berselli_, Apr 12 2016