A173653 Partial sums of floor(n^2/10) (A056865).

0, 0, 0, 0, 1, 3, 6, 10, 16, 24, 34, 46, 60, 76, 95, 117, 142, 170, 202, 238, 278, 322, 370, 422, 479, 541, 608, 680, 758, 842, 932, 1028, 1130, 1238, 1353, 1475, 1604, 1740, 1884, 2036, 2196, 2364, 2540, 2724, 2917, 3119, 3330, 3550, 3780, 4020, 4270, 4530, 4800

Offset: 0

Author

Mircea Merca, Nov 24 2010

Keywords

• nonn
• easy

Examples

			a(9) = 0+0+0+0+1+2+3+4+6+8 = 24.
		

Links

• Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,0,0,0,0,1,-3,3,-1).

Crossrefs

Cf. A056865

Programs

Mathematica

Accumulate[Floor[Range[0,50]^2/10]] (* Harvey P. Dale, May 31 2012 *)

PARI

a(n)=(2*n^3+3*n^2-26*n)\/60 - ((n+3)%10<2) \\ Charles R Greathouse IV, Jun 02 2025

Formula

a(n) = Sum_{k=0..n} floor(k^2/10).

a(n) = a(n-10)+(n-5)^2+n-1 , n>9.

From R. J. Mathar, Nov 24 2010: (Start)

G.f.: x^4*(1+x^4) / ( (1+x)*(x^4+x^3+x^2+x+1)*(x^4-x^3+x^2-x+1)*(x-1)^4 ).

a(n)= +3*a(n-1) -3*a(n-2) +a(n-3) +a(n-10) -3*a(n-11) +3*a(n-12) -a(n-13). (End)

a(n) = floor((2*n^3+3*n^2-26*n+24)/60) + floor((n+1)/10) - floor((n+3)/10). - Hoang Xuan Thanh, Jun 02 2025