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A232713 Doubly pentagonal numbers: a(n) = n*(3*n-2)*(3*n-1)*(3*n+1)/8.

%I A232713 #19 Aug 25 2022 04:35:39
%S A232713 0,1,35,210,715,1820,3876,7315,12650,20475,31465,46376,66045,91390,
%T A232713 123410,163185,211876,270725,341055,424270,521855,635376,766480,
%U A232713 916895,1088430,1282975,1502501,1749060,2024785,2331890,2672670,3049501,3464840,3921225,4421275
%N A232713 Doubly pentagonal numbers: a(n) = n*(3*n-2)*(3*n-1)*(3*n+1)/8.
%H A232713 Bruno Berselli, <a href="/A232713/b232713.txt">Table of n, a(n) for n = 0..1000</a>
%H A232713 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A232713 G.f.: x*(1 + 30*x + 45*x^2 + 5*x^3) / (1 - x)^5.
%F A232713 a(n) = A000326(A000326(n)) = A000332(3n+1).
%F A232713 From _Amiram Eldar_, Aug 25 2022: (Start)
%F A232713 Sum_{n>=1} 1/a(n) = 4 + 2*Pi/sqrt(3) - 6*log(3).
%F A232713 Sum_{n>=1} (-1)^(n+1)/a(n) = 32*log(2)/3 - 4*Pi/(3*sqrt(3)) - 4. (End)
%t A232713 Table[n (3 n - 2) (3 n - 1) (3 n + 1)/8, {n, 0, 40}]
%o A232713 (Magma) [n*(3*n-2)*(3*n-1)*(3*n+1)/8: n in [0..40]];
%o A232713 (PARI) a(n)=n*(3*n-2)*(3*n-1)*(3*n+1)/8 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A232713 Cf. A000326, A000332.
%Y A232713 Cf. similar sequences: A000583 for A000290(A000290(n)); A002817 for A000217(A000217(n)); A063249 for A000384(A000384(n)).
%Y A232713 Cf. A236770, A260810.
%K A232713 nonn,easy
%O A232713 0,3
%A A232713 _Bruno Berselli_, Nov 28 2013