A131242 Partial sums of A059995: a(n) = sum_{k=0..n} floor(k/10).

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 156, 162, 168, 174, 180, 186, 192, 198

Offset: 0

Author

Hieronymus Fischer, Jun 21 2007

Keywords

• nonn
• easy

Comments

Complementary with A130488 regarding triangular numbers, in that A130488(n)+10*a(n)=n(n+1)/2=A000217(n).

Examples

			As square array :
    0,   0,   0,   0,   0,   0,   0,   0,   0,    0
    1,   2,   3,   4,   5,   6,   7,   8,   9,   10
   12,  14,  16,  18,  20,  22,  24,  26,  28,   30
   33,  36,  39,  42,  45,  48,  51,  54,  57,   60
   64,  68,  72,  76,  80,  84,  88,  92,  96,  100
  105, 110, 115, 120, 125, 130, 135, 140, 145,  150
  156, 162, 168, 174, 180, 186, 192, 198, 204,  210
... - _Philippe Deléham_, Mar 27 2013
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,1,-2,1).

Crossrefs

Cf. A008728, A059995, A010879, A002266, A130488, A000217, A002620, A130518, A130519, A130520, A174709, A174738, A118729, A218470.

Programs

Mathematica

Table[(1/2)*Floor[n/10]*(2*n - 8 - 10*Floor[n/10]), {n,0,50}] (* G. C. Greubel, Dec 13 2016 *)
Accumulate[Table[FromDigits[Most[IntegerDigits[n]]],{n,0,110}]] (* or *) LinearRecurrence[{2,-1,0,0,0,0,0,0,0,1,-2,1},{0,0,0,0,0,0,0,0,0,0,1,2},120] (* Harvey P. Dale, Apr 06 2017 *)

PARI

for(n=0,50, print1((1/2)*floor(n/10)*(2n-8-10*floor(n/10)), ", ")) \\ G. C. Greubel, Dec 13 2016

PARI

a(n)=my(k=n\10); k*(n-5*k-4) \\ Charles R Greathouse IV, Dec 13 2016

Formula

a(n) = (1/2)*floor(n/10)*(2n-8-10*floor(n/10)).

a(n) = A059995(n)*(2n-8-10*A059995(n))/2.

a(n) = (1/2)*A059995(n)*(n-8+A010879(n)).

a(n) = (n-A010879(n))*(n+A010879(n)-8)/20.

G.f.: x^10/((1-x^10)(1-x)^2).

From Philippe Deléham, Mar 27 2013: (Start)

a(10n) = A051624(n).

a(10n+1) = A135706(n).

a(10n+2) = A147874(n+1).

a(10n+3) = 2*A005476(n).

a(10n+4) = A033429(n).

a(10n+5) = A202803(n).

a(10n+6) = A168668(n).

a(10n+7) = 2*A147875(n).

a(10n+8) = A135705(n).

a(10n+9) = A124080(n). (End)

a(n) = A008728(n-10) for n>= 10. - Georg Fischer, Nov 03 2018