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%I #70 Sep 28 2023 08:17:39
%S 0,10,21,33,46,60,75,91,108,126,145,165,186,208,231,255,280,306,333,
%T 361,390,420,451,483,516,550,585,621,658,696,735,775,816,858,901,945,
%U 990,1036,1083,1131,1180,1230,1281,1333,1386,1440,1495,1551,1608,1666
%N a(n) = n*(n+1)/2 - 45.
%H G. C. Greubel, <a href="/A051942/b051942.txt">Table of n, a(n) for n = 9..1000</a>
%H Project Euler, <a href="https://projecteuler.net/problem=834">Problem 834: Add and Divide</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = (n^2 + n - 90)/2 = (n-9)*(n+10)/2 = n*(n+1)/2 - 45.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n>=13.
%F a(n) = n + a(n-1) (with a(9) = 0). - _Vincenzo Librandi_, Aug 06 2010
%F G.f.: x^10*(10 - 9*x)/(1-x)^3.
%F From _Amiram Eldar_, Jan 10 2021: (Start)
%F Sum_{n>=10} 1/a(n) = 2*A001008(19)/(19*A002805(19)) = 275295799/737176440.
%F Sum_{n>=10} (-1)^n/a(n) = 4*log(2)/19 - 33464927/442305864. (End)
%F E.g.f.: (1/8!)*(1814400 +1774080*x +846720*x^2 +262080*x^3 +58800*x^4 +10080*x^5 +1344*x^6 +136*x^7 +9*x^8 - (1814400 -40320*x -20160*x^2)*exp(x)). - _G. C. Greubel_, Jul 31 2022
%e a(10) = 10 + 0 = 10;
%e a(11) = 11 + 10 = 21;
%e a(12) = 12 + 21 = 33.
%p A051942:=n->(n^2+n-90)/2: seq(A051942(n), n=9..80); # _Wesley Ivan Hurt_, Jan 28 2017
%t Table[n(n+1)/2 -45, {n, 9, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 15 2011 *)
%t #-45&/@Drop[Accumulate[Range[60]],8] (* _Harvey P. Dale_, Jul 24 2011 *)
%t LinearRecurrence[{3,-3,1},{0,10,21},60] (* _Harvey P. Dale_, Mar 25 2015 *)
%o (PARI) a(n)=(n-9)*(n+10)/2;
%o (Magma) [(n-9)*(n+10)/2: n in [9..80]]; // _G. C. Greubel_, Jul 31 2022
%o (SageMath) [(n-9)*(n+10)/2 for n in (9..80)] # _G. C. Greubel_, Jul 31 2022
%Y Cf. A000096, A001008, A001477, A002805, A056121, A056126, A079664.
%K easy,nice,nonn
%O 9,2
%A Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 21 1999
%E More terms from _Zerinvary Lajos_, Oct 01 2006
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