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L.g.f.: log(1 + Sum_{n>=1} x^(n^2) + x^(3*n^2) ).
2

%I #32 Jun 03 2015 22:39:42

%S 1,-1,4,-1,1,-4,1,-1,13,-11,12,-16,14,-15,19,-1,1,-13,1,-11,25,-12,24,

%T -40,26,-14,40,-15,1,-29,1,-1,48,-35,36,-61,38,-39,56,-11,1,-39,1,-12,

%U 73,-24,48,-88,50,-36,55,-14,1,-40,12,-15,61,-59,60,-101,62,-63,97,-1,14,-48,1,-35,96,-60,72,-157,74,-38,119,-39,12,-56,1,-11,121,-83,84,-135,86,-87,91,-12,1,-83,14,-24,97,-48,96,-184,98,-64,156,-36,1,-89,1,-14,180,-107,108,-196,110,-132,152,-15,1,-99,24,-59,182,-60,120,-245,133

%N L.g.f.: log(1 + Sum_{n>=1} x^(n^2) + x^(3*n^2) ).

%H Paul D. Hanna, <a href="/A258328/b258328.txt">Table of n, a(n) for n = 1..1024</a>

%F a(n) = -1 iff n = 2^k for k>=1 [conjecture].

%F a(p) = +1 for primes p such that 3 is not a square mod p (A003630), and a(n) = +1 nowhere else except at n=0 [conjecture].

%e L.g.f.: L(x) = x - x^2/2 + 4*x^3/3 - x^4/4 + x^5/5 - 4*x^6/6 + x^7/7 - x^8/8 + 13*x^9/9 - 11*x^10/10 + 12*x^11/11 - 16*x^12/12 + 14*x^13/13 - 15*x^14/14 + 19*x^15/15 - x^16/16 +...+ a(n)*x^n/n +...

%e where

%e exp(L(x)) = 1 + x + x^3 + x^4 + x^9 + x^12 + x^16 + x^25 + x^27 + x^36 + x^48 + x^49 + x^64 + x^75 + x^81 + x^100 + x^108 +...+ x^(n^2) + x^(3*n^2) +...

%e Note that for n>1, a(n) = +1 at positions:

%e [5, 7, 17, 19, 29, 31, 41, 43, 53, 67, 79, 89, 101, 103, 113, 127, ...];

%e which appears to be A003630 (primes p such that 3 is not a square mod p).

%o (PARI) {a(n) = local(L=x); L = log(1 + sum(k=1,sqrtint(n+1), x^(k^2) + x^(3*k^2)) +x*O(x^n)); n*polcoeff(L,n)}

%o for(n=1,121, print1(a(n),", "))

%Y Cf. A256357, A003630, A038875.

%K sign

%O 1,3

%A _Paul D. Hanna_, Jun 03 2015