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A308422 a(n) = n^2 if n odd, 3*n^2/4 if n even.

%I A308422 #21 Jan 23 2024 04:05:50
%S A308422 0,1,3,9,12,25,27,49,48,81,75,121,108,169,147,225,192,289,243,361,300,
%T A308422 441,363,529,432,625,507,729,588,841,675,961,768,1089,867,1225,972,
%U A308422 1369,1083,1521,1200,1681,1323,1849,1452,2025,1587,2209,1728,2401,1875,2601,2028,2809,2187,3025
%N A308422 a(n) = n^2 if n odd, 3*n^2/4 if n even.
%C A308422 Moebius transform of A076577.
%H A308422 Amiram Eldar, <a href="/A308422/b308422.txt">Table of n, a(n) for n = 0..10000</a>
%H A308422 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).
%F A308422 G.f.: x*(1 + 3*x + 6*x^2 + 3*x^3 + x^4)/(1 - x^2)^3.
%F A308422 G.f.: Sum_{k>=1} J_2(k)*x^k/(1 - x^(2*k)), where J_2() is the Jordan function (A007434).
%F A308422 E.g.f.: x*((4 + 3*x)*cosh(x) + (3 + 4*x)*sinh(x))/4.
%F A308422 Dirichlet g.f.: zeta(s-2)*(1 - 1/2^s).
%F A308422 a(n) = (7 - (-1)^n)*n^2/8.
%F A308422 a(n) = Sum_{d|n, n/d odd} J_2(d).
%F A308422 a(2*k+1) = A016754(k), a(2*k) = A033428(k).
%F A308422 Sum_{n>=1} 1/a(n) = 13*Pi^2/72 = 1.7820119057522453061...
%F A308422 Sum_{n>=1} (-1)^(n+1)/a(n) = 5*Pi^2/72 = 0.68538919452009434853...
%F A308422 Multiplicative with a(2^e) = 3*2^(2*e-2), and a(p^e) = p^(2*e) for odd primes p. - _Amiram Eldar_, Oct 26 2020
%F A308422 For n >= 1, n*a(n) = A309337(n) = Sum_{d divides n} (-1)^(d+1) * J(3, n/d), where the Jordan totient function J_3(n) = A059376. - _Peter Bala_, Jan 21 2024
%t A308422 a[n_] := If[OddQ[n], n^2, 3 n^2/4]; Table[a[n], {n, 0, 55}]
%t A308422 nmax = 55; CoefficientList[Series[x (1 + 3 x + 6 x^2 + 3 x^3 + x^4)/(1 - x^2)^3, {x, 0, nmax}], x]
%t A308422 LinearRecurrence[{0, 3, 0, -3, 0, 1}, {0, 1, 3, 9, 12, 25}, 56]
%t A308422 Table[(7 - (-1)^n) n^2/8, {n, 0, 55}]
%Y A308422 Cf. A000290, A007434, A016754, A026741, A033428, A076577, A308418, A309337.
%K A308422 nonn,easy,mult
%O A308422 0,3
%A A308422 _Ilya Gutkovskiy_, May 26 2019