A228443 - OEIS

%I #16 Dec 30 2023 03:25:10

%S 1,2,6,6,7,10,14,12,18,18,12,22,31,20,30,30,20,36,38,28,42,42,42,46,

%T 43,36,54,60,36,58,62,42,84,66,44,70,74,62,60,78,61,82,108,60,90,84,

%U 60,108,98,70,102,102,72,106,110,76,114,132,98,108,111,84,156

%N G.f.: Sum_{k>=0} (2*k + 1) * x^k / (1 + x^(2*k + 1)).

%H G. C. Greubel, <a href="/A228443/b228443.txt">Table of n, a(n) for n = 0..5000</a>

%F a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1 (mod 4), b(p^e) = (p^(e+1) + (-1)^e) / (p + 1) if p == 3 (mod 4), with a(0) = 1.

%F G.f.: Sum_{k>=0} (-1)^k * x^k * (1 + x^(2*k + 1)) / (1 - x^(2*k + 1))^2

%F a(2*n - 1) = 2 * A053091(n).

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = 0.915965... is Catalan's constant (A006752). - _Amiram Eldar_, Dec 30 2023

%e G.f. = 1 + 2*x + 6*x^2 + 6*x^3 + 7*x^4 + 10*x^5 + 14*x^6 + 12*x^7 + 18*x^8 + ...

%e G.f. = q + 2*q^3 + 6*q^5 + 6*q^7 + 7*q^9 + 10*q^11 + 14*q^13 + 12*q^15 + ...

%t a[ n_] := If[ n < 0, 0, With[{m = 2 n + 1}, (-1)^n Sum[ (-1)^Quotient[k, 2] k, {k, Divisors@m}]]];

%o (PARI) {a(n) = if( n<0, 0, polcoeff( sum( k=0, n, (2*k + 1) * x^k / (1 + x^(2*k + 1)), x * O(x^n)), n))};

%o (PARI) {a(n) = my(A, p, e); if( n<0, 0, n = 2*n + 1; A = factor( n); prod( k=1, matsize(A)[1], if(p = A[k,1], e = A[k,2]; if( p==2, 0, if( p%4==1, (p^(e+1) - 1) / (p - 1), (p^(e+1) + (-1)^e) / (p + 1))))))};

%Y Cf. A006752, A053091.

%K nonn,easy

%O 0,2

%A _Michael Somos_, Nov 03 2013