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a(n) = 18*n + A000217(n-1).
3

%I #51 Jun 09 2024 13:50:52

%S 0,18,37,57,78,100,123,147,172,198,225,253,282,312,343,375,408,442,

%T 477,513,550,588,627,667,708,750,793,837,882,928,975,1023,1072,1122,

%U 1173,1225,1278,1332,1387,1443,1500,1558,1617,1677,1738,1800,1863,1927,1992

%N a(n) = 18*n + A000217(n-1).

%C Generalization: T(n,i) = A000217(i-1+n) - A000217(i-1) = i*n + A000217(n-1) (corrected by _Zak Seidov_, Jun 21 2012); in this case is i=18.

%C For i = 11..16, _Milan Janjic_ observed that if we define f(n,b,i) = Sum_{k=0..n-b} binomial(n,k)*Stirling1(n-k,b)*Product_{j=0..k-1} (-i - j), then T(n-1,i) = -f(n,n-1,i) for n >= 1.

%H Vincenzo Librandi, <a href="/A212428/b212428.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = (17+n)*(18+n)/2 - 17*18/2 = 18*n + (n-1)*n/2 = n*(n+35)/2.

%F G.f.: x*(18-17*x)/(1-x)^3. - _Bruno Berselli_, Jun 21 2012

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Vincenzo Librandi_, Jul 10 2012

%F a(n) = 18n - floor(n/2) + floor(n^2/2). - _Wesley Ivan Hurt_, Jun 15 2013

%F From _Amiram Eldar_, Jan 11 2021: (Start)

%F Sum_{n>=1} 1/a(n) = 2*A001008(35)/(35*A002805(35)) = 54437269998109/229732925058000.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/35 - 102126365345729/2527062175638000. (End)

%t Table[-18 (18 - 1)/2 + (18 + n) (17 + n)/2, {n, 0, 100}]

%t LinearRecurrence[{3,-3,1},{0,18,37},60] (* _Harvey P. Dale_, Jun 09 2024 *)

%o (Magma) [n*(n+35)/2: n in [0..48]]; // _Bruno Berselli_, Jun 21 2012

%o (PARI) a(n)=n*(n+35)/2 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A000217, A000096, A001008, A002805, A055998-A056000, A056115, A056119, A056121, A056126, A051942, A101859, A132754-A132758, A212427.

%K nonn,easy

%O 0,2

%A _Jesse Han_, May 16 2012