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A276670 Numerator of (n-1)*n*(n+1)/4.

%I A276670 #57 Aug 14 2022 02:40:09
%S A276670 0,0,3,6,15,30,105,84,126,180,495,330,429,546,1365,840,1020,1224,2907,
%T A276670 1710,1995,2310,5313,3036,3450,3900,8775,4914,5481,6090,13485,7440,
%U A276670 8184,8976,19635,10710,11655,12654,27417,14820,15990,17220,37023,19866,21285
%N A276670 Numerator of (n-1)*n*(n+1)/4.
%C A276670 Consider the sequence [2/(n+1), autosequence of the second kind] (see A003506), and its successive differences:
%C A276670     2,    1,  2/3,   1/2,   2/5,   1/3,   2/7,   1/4,   2/9, ... (see A026741)
%C A276670    -1, -1/3, -1/6, -1/10, -1/15, -1/21, -1/28, -1/36, -1/45, ... (see A000217)
%C A276670   2/3,  1/6, 1/15,  1/30, 2/105,  1/84, 1/126, 1/180, 2/495, ...
%C A276670   ...
%C A276670 Each fraction in the third row is essentially the reciprocal of (n-1)*n*(n+1)/4 (3/2, 6, 15, 30, 105/2, ... ).
%C A276670 The numbers (= 3*A138190) are divisible by
%C A276670 1) -1, 1, 1, 1, 3, 2, 5, 3, 7, ... hence f(n) = 0, 0, 3, 6, 5, 15, 21, 28, 18, ...
%C A276670 2)  1, 1, 3, 3, 5, 5, 7, 7, 9, ... hence g(n) = 0, 0, 1, 2, 3, 6, 15, 12, 14, ...
%H A276670 Colin Barker, <a href="/A276670/b276670.txt">Table of n, a(n) for n = 0..1000</a>
%H A276670 <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,4,0,0,0,-6,0,0,0,4,0,0,0,-1).
%F A276670 a(n) = 3*A138190(n), for n>=1.
%F A276670 a(n) = 4*a(n-4) - 6*a(n-8) + 4*a(n-12) - a(n-16).
%F A276670 a(n) = A007531(n+1)/2 if n == 2 (mod 4), otherwise a(n) = A007531(n+1)/4. - _Robert Israel_, Oct 05 2016
%F A276670 G.f.: 3*x^2*(1 +2*x +5*x^2 +10*x^3 +31*x^4 +20*x^5 +22*x^6 +20*x^7 +31*x^8 +10*x^9 +5*x^10 +2*x^11 +x^12) / ((1 -x)^4*(1 +x)^4*(1 +x^2)^4). - _Colin Barker_, Oct 09 2016
%F A276670 Sum_{n>=2} 1/a(n) = 1 - log(2)/2. - _Amiram Eldar_, Aug 13 2022
%p A276670 seq(numer((n^3-n)/4), n=0..100); # _Robert Israel_, Oct 05 2016
%t A276670 f[n_] := Numerator[(n - 1) n (n + 1)/4]; Array[f, 40, 0] (* _Robert G. Wilson v_, Oct 05 2016 *)
%o A276670 (PARI) concat(vector(2), Vec(3*x^2*(1 +2*x +5*x^2 +10*x^3 +31*x^4 +20*x^5 +22*x^6 +20*x^7 +31*x^8 +10*x^9 +5*x^10 +2*x^11 +x^12) / ((1 -x)^4*(1 +x)^4*(1 +x^2)^4) + O(x^30))) \\ _Colin Barker_, Oct 09 2016
%Y A276670 Cf. A000217, A003506, A007531, A027641, A138190, A109613, A236398 (denominators).
%K A276670 nonn,easy
%O A276670 0,3
%A A276670 _Paul Curtz_, Oct 05 2016
%E A276670 More terms from _Robert G. Wilson v_, Oct 05 2016