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A168077 a(2n) = A129194(2n)/2; a(2n+1) = A129194(2n+1).

%I A168077 #72 Nov 28 2022 08:16:41
%S A168077 0,1,1,9,4,25,9,49,16,81,25,121,36,169,49,225,64,289,81,361,100,441,
%T A168077 121,529,144,625,169,729,196,841,225,961,256,1089,289,1225,324,1369,
%U A168077 361,1521,400,1681,441,1849,484,2025,529,2209,576,2401,625,2601
%N A168077 a(2n) = A129194(2n)/2; a(2n+1) = A129194(2n+1).
%C A168077 From _Paul Curtz_, Mar 26 2011: (Start)
%C A168077 Successive A026741(n) * A026741(n+p):
%C A168077   p=0:  0, 1,  1,  9,  4, 25,  9,     a(n),
%C A168077   p=1:  0, 1,  3,  6, 10, 15, 21,  A000217,
%C A168077   p=2:  0, 3,  2, 15,  6, 35, 12,  A142705,
%C A168077   p=3:  0, 2,  5,  9, 14, 20, 27,  A000096,
%C A168077   p=4:  0, 5,  3, 21,  8, 45, 15,  A171621,
%C A168077   p=5:  0, 3,  7, 12, 18, 25, 33,  A055998,
%C A168077   p=6:  0, 7,  4, 27, 10, 55, 18,
%C A168077   p=7:  0, 4,  9, 15, 22, 30, 39,  A055999,
%C A168077   p=8:  0, 9,  5, 33, 12, 65, 21,  (see A061041),
%C A168077   p=9:  0, 5, 11, 18, 26, 35, 45,  A056000. (End)
%C A168077 The moment generating function of p(x, m=2, n=1, mu=2) = 4*x*E(x, 2, 1), see A163931 and A274181, is given by M(a) = (-4 * log(1-a) - 4 * polylog(2, a))/a^2. The series expansion of M(a) leads to the sequence given above. - _Johannes W. Meijer_, Jul 03 2016
%C A168077 Multiplicative because both A129194 and A040001 are. - _Andrew Howroyd_, Jul 26 2018
%H A168077 G. C. Greubel, <a href="/A168077/b168077.txt">Table of n, a(n) for n = 0..1000</a>
%H A168077 John M. Campbell, <a href="http://arxiv.org/abs/1105.3399">An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences</a>, arXiv preprint arXiv:1105.3399 [math.GM], 2011.
%H A168077 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).
%F A168077 From _R. J. Mathar_, Jan 22 2011: (Start)
%F A168077 G.f.: x*(1 + x + 6*x^2 + x^3 + x^4) / ((1-x)^3*(1+x)^3).
%F A168077 a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6).
%F A168077 a(n) = n^2*(5 - 3*(-1)^n)/8. (End)
%F A168077 a(n) = A026741(n)^2.
%F A168077 a(2*n) = A000290(n); a(2*n+1) = A016754(n).
%F A168077 a(n) - a(n-4) = 4*A064680(n+2). - _Paul Curtz_, Mar 27 2011
%F A168077 4*a(n) = A061038(n) * A010121(n+2) = A109043(n)^2, n >= 2. - _Paul Curtz_, Apr 07 2011
%F A168077 a(n) = A129194(n) / A040001(n). - _Andrew Howroyd_, Jul 26 2018
%F A168077 From _Peter Bala_, Feb 19 2019: (Start)
%F A168077 a(n) = numerator(n^2/(n^2 + 4)) = n^2/(gcd(n^2,4)) = (n/gcd(n,2))^2.
%F A168077 a(n) = n^2/b(n), where b(n) = [1, 4, 1, 4, ...] is a purely periodic sequence of period 2. Thus a(n) is a quasi-polynomial in n.
%F A168077 O.g.f.: x*(1 + x)/(1 - x)^3 - 3*x^2*(1 + x^2)/(1 - x^2)^3.
%F A168077 Cf. A181318. (End)
%F A168077 From _Werner Schulte_, Aug 30 2020: (Start)
%F A168077 Multiplicative with a(2^e) = 2^(2*e-2) for e > 0, and a(p^e) = p^(2*e) for prime p > 2.
%F A168077 Dirichlet g.f.: zeta(s-2) * (1 - 3/2^s).
%F A168077 Dirichlet convolution with A259346 equals A000290.
%F A168077 Sum_{n>0} 1/a(n) = Pi^2 * 7 / 24. (End)
%F A168077 Sum_{k=1..n} a(k) ~ (5/24) * n^3. - _Amiram Eldar_, Nov 28 2022
%p A168077 a := proc(n): n^2*(5-3*(-1)^n)/8 end: seq(a(n), n=0..46); # _Johannes W. Meijer_, Jul 03 2016
%t A168077 LinearRecurrence[{0,3,0,-3,0,1},{0,1,1,9,4,25},60] (* _Harvey P. Dale_, May 14 2011 *)
%t A168077 f[n_] := Numerator[(n/2)^2]; Array[f, 60, 0] (* _Robert G. Wilson v_, Dec 18 2012 *)
%t A168077 CoefficientList[Series[x(1+x+6x^2+x^3+x^4)/((1-x)^3(1+x)^3), {x,0,60}], x] (* _Vincenzo Librandi_, Jul 10 2016 *)
%o A168077 (PARI) concat(0, Vec(x*(1+x+6*x^2+x^3+x^4)/((1-x)^3*(1+x)^3) + O(x^60))) \\ _Altug Alkan_, Jul 04 2016
%o A168077 (PARI) a(n) = lcm(4, n^2)/4; \\ _Andrew Howroyd_, Jul 26 2018
%o A168077 (Magma) I:=[0,1,1,9,4,25]; [n le 6 select I[n] else 3*Self(n-2)-3*Self(n-4)+Self(n-6): n in [1..60]]; // _Vincenzo Librandi_, Jul 10 2016
%o A168077 (Sage) (x*(1+x+6*x^2+x^3+x^4)/(1-x^2)^3).series(x, 60).coefficients(x, sparse=False) # _G. C. Greubel_, Feb 20 2019
%Y A168077 Cf. A040001, A168068, A181318.
%K A168077 nonn,easy,mult
%O A168077 0,4
%A A168077 _Paul Curtz_, Nov 18 2009