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A212427 a(n) = 17*n + A000217(n-1).

%I A212427 #36 Dec 12 2024 17:16:56
%S A212427 0,17,35,54,74,95,117,140,164,189,215,242,270,299,329,360,392,425,459,
%T A212427 494,530,567,605,644,684,725,767,810,854,899,945,992,1040,1089,1139,
%U A212427 1190,1242,1295,1349,1404,1460,1517,1575,1634,1694,1755,1817,1880,1944,2009
%N A212427 a(n) = 17*n + A000217(n-1).
%C A212427 Generalization: T(n,i) = A000217(i-1+n) - A000217(i-1) = i*n + A000217(n-1); in this case is i=17. See also the comment in A212428.
%H A212427 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A212427 a(n) = (16+n)*(17+n)/2 - 16*17/2 = 17*n + (n-1)*n/2 = n*(n+33)/2.
%F A212427 G.f.: x*(17-16*x)/(1-x)^3. - _Bruno Berselli_, Jun 22 2012
%F A212427 a(n) = 17*n - floor(n/2) + floor(n^2/2). - _Wesley Ivan Hurt_, Jun 15 2013
%F A212427 From _Amiram Eldar_, Jan 11 2021: (Start)
%F A212427 Sum_{n>=1} 1/a(n) = 2*A001008(33)/(33*A002805(33)) = 53676090078349/216605329340400.
%F A212427 Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/33 - 14606816124167/340379803249200. (End)
%F A212427 From _Elmo R. Oliveira_, Dec 12 2024: (Start)
%F A212427 E.g.f.: exp(x)*x*(34 + x)/2.
%F A212427 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
%t A212427 Table[-17 (17 - 1)/2 + (17 + n) (16 + n)/2, {n, 0, 100}]
%o A212427 (Magma) [n*(n+33)/2: n in [0..49]]; // _Bruno Berselli_, Jun 22 2012
%o A212427 (PARI) a(n)=n*(n+33)/2 \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A212427 Cf. A000217, A000096, A001008, A002805, A055998-A056000, A056115, A056119, A056121, A056126, A051942, A101859, A132754-A132758, A212428.
%Y A212427 For n > 22, T(n,17) matches A074170 but with opposite sign.
%K A212427 nonn,easy
%O A212427 0,2
%A A212427 _Jesse Han_, May 16 2012