A051942 a(n) = n*(n+1)/2 - 45.

0, 10, 21, 33, 46, 60, 75, 91, 108, 126, 145, 165, 186, 208, 231, 255, 280, 306, 333, 361, 390, 420, 451, 483, 516, 550, 585, 621, 658, 696, 735, 775, 816, 858, 901, 945, 990, 1036, 1083, 1131, 1180, 1230, 1281, 1333, 1386, 1440, 1495, 1551, 1608, 1666

Offset: 9

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 21 1999

Keywords

• easy
• nice
• nonn

Examples

			a(10) = 10 +  0 = 10;
a(11) = 11 + 10 = 21;
a(12) = 12 + 21 = 33.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 9..1000
• Project Euler, Problem 834: Add and Divide.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000096, A001008, A001477, A002805, A056121, A056126, A079664.

Programs

Magma

[(n-9)*(n+10)/2: n in [9..80]]; // G. C. Greubel, Jul 31 2022

Maple

A051942:=n->(n^2+n-90)/2: seq(A051942(n), n=9..80); # Wesley Ivan Hurt, Jan 28 2017

Mathematica

Table[n(n+1)/2 -45, {n, 9, 100}] (* Vladimir Joseph Stephan Orlovsky, Jun 15 2011 *)
#-45&/@Drop[Accumulate[Range[60]],8] (* Harvey P. Dale, Jul 24 2011 *)
LinearRecurrence[{3,-3,1},{0,10,21},60] (* Harvey P. Dale, Mar 25 2015 *)

PARI

a(n)=(n-9)*(n+10)/2;

SageMath

[(n-9)*(n+10)/2 for n in (9..80)] # G. C. Greubel, Jul 31 2022

Formula

a(n) = (n^2 + n - 90)/2 = (n-9)*(n+10)/2 = n*(n+1)/2 - 45.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n>=13.

a(n) = n + a(n-1) (with a(9) = 0). - Vincenzo Librandi, Aug 06 2010

G.f.: x^10*(10 - 9*x)/(1-x)^3.

From Amiram Eldar, Jan 10 2021: (Start)

Sum_{n>=10} 1/a(n) = 2*A001008(19)/(19*A002805(19)) = 275295799/737176440.

Sum_{n>=10} (-1)^n/a(n) = 4*log(2)/19 - 33464927/442305864. (End)

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

Extensions

More terms from Zerinvary Lajos, Oct 01 2006