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A281381 a(n) = n*(n + 1)*(4*n + 5)/2.

%I A281381 #36 Sep 22 2022 08:59:19
%S A281381 0,9,39,102,210,375,609,924,1332,1845,2475,3234,4134,5187,6405,7800,
%T A281381 9384,11169,13167,15390,17850,20559,23529,26772,30300,34125,38259,
%U A281381 42714,47502,52635,58125,63984,70224,76857,83895,91350,99234,107559,116337,125580,135300,145509,156219,167442,179190,191475
%N A281381 a(n) = n*(n + 1)*(4*n + 5)/2.
%C A281381 Shares its digital root, zero together with period 9: repeat [3, 3, 3, 6, 6, 6, 9, 9, 9] with A027480.
%C A281381 Final digits cycle a length period 20: repeat [0, 9, 9, 2, 0, 5, 9, 4, 2, 5, 5, 4, 4, 7, 5, 0, 4, 9, 7, 0].
%H A281381 Colin Barker, <a href="/A281381/b281381.txt">Table of n, a(n) for n = 0..1000</a>
%H A281381 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A281381 a(n) = 2*n^3 + 9*n^2/2 + 5*n/2.
%F A281381 a(n) = 3*A016061(n).
%F A281381 a(n) = A006002(n+1)*(n) - A006002(n)*(n-1).
%F A281381 a(n) = A007742(n)*(n - 1)/2.
%F A281381 From _Colin Barker_, Jan 21 2017: (Start)
%F A281381 G.f.: 3*x*(3 + x) / (1 - x)^4.
%F A281381 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. (End)
%F A281381 From _Stefano Spezia_, Aug 30 2022: (Start)
%F A281381 E.g.f.: exp(x)*x*(18 + 21*x + 4*x^2)/2.
%F A281381 Sum_{n>0} 1/a(n) = 2*(20*log(8) + 10*Pi - 71)/25 = 0.1603805895595720759728288896228498341201... . (End)
%F A281381 Sum_{n>=1} (-1)^(n+1)/a(n) = 4*sqrt(2)*Pi/5 + 4*(3+sqrt(2))*log(2)/5 - 8*sqrt(2)*log(2-sqrt(2))/5 - 178/25. - _Amiram Eldar_, Sep 22 2022
%t A281381 Table[n (n + 1) (4 n + 5)/2, {n, 0, 45}] (* or *)
%t A281381 CoefficientList[Series[3 x (3 + x)/(1 - x)^4, {x, 0, 45}], x] (* _Michael De Vlieger_, Jan 21 2017 *)
%o A281381 (PARI) concat(0, Vec(3*x*(3 + x) / (1 - x)^4 + O(x^50))) \\ _Colin Barker_, Jan 21 2017
%o A281381 (PARI) a(n) = n*(n + 1)*(4*n + 5)/2 \\ _Charles R Greathouse IV_, Feb 01 2017
%o A281381 (Magma) [n*(n+1)*(4*n+5)/2 : n in [0..50]]; // _Wesley Ivan Hurt_, Aug 30 2022
%Y A281381 Partial sums of A195319.
%Y A281381 Cf. A006002, A007742, A016061, A027480, A281258.
%K A281381 nonn,easy
%O A281381 0,2
%A A281381 _Peter M. Chema_, Jan 21 2017