A130520 a(n) = Sum_{k=0..n} floor(k/5). (Partial sums of A002266.)

0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 7, 9, 11, 13, 15, 18, 21, 24, 27, 30, 34, 38, 42, 46, 50, 55, 60, 65, 70, 75, 81, 87, 93, 99, 105, 112, 119, 126, 133, 140, 148, 156, 164, 172, 180, 189, 198, 207, 216, 225, 235, 245, 255, 265, 275, 286, 297, 308, 319, 330, 342, 354, 366

Offset

0,7

Comments

Complementary with A130483 regarding triangular numbers, in that A130483(n) + 5*a(n) = n*(n+1)/2 = A000217(n).

Given a sequence b(n) defined by variables b(0) to b(5) and recursion b(n) = -(b(n-6) * a(n-2) * (b(n-4) * b(n-2)^3 - b(n-3)^3 * b(n-1)) - b(n-5) * b(n-3) * b(n-1) * (b(n-5) * b(n-2)^2 - b(n-4)^2 * b(n-1)))/(b(n-4) * (b(n-5) * b(n-3)^3 - b(n-4)^3 * b(n-2))). The denominator of b(n+1) has a factor of (b(1) * b(3)^3 - b(2)^3 * b(4))^a(n+1). For example, if b(0) = 2, b(1) = b(2) = b(3) = 1, b(4) = 1+x, b(5) = 4, then the denominator of b(n+1) is x^a(n+1). - Michael Somos, Nov 15 2023

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Formula

a(n) = floor(n/5)*(2*n - 3 - 5*floor(n/5))/2.

a(n) = A002266(n)*(2*n - 3 - 5*A002266(n))/2.

a(n) = A002266(n)*(n -3 +A010874(n))/2.

G.f.: x^5/((1-x^5)*(1-x)^2) = x^5/( (1+x+x^2+x^3+x^4)*(1-x)^3 ).

a(n) = floor((n-1)*(n-2)/10). - Mitch Harris, Sep 08 2008

a(n) = round(n*(n-3)/10) = ceiling((n+1)*(n-4)/10) = round((n^2 - 3*n - 1)/10). - Mircea Merca, Nov 28 2010

a(n) = A008732(n-5), n > 4. - R. J. Mathar, Nov 22 2008

a(n) = a(n-5) + n - 4, n > 4. - Mircea Merca, Nov 28 2010

a(5n) = A000566(n), a(5n+1) = A005476(n), a(5n+2) = A005475(n), a(5n+3) = A147875(n), a(5n+4) = A028895(n). - Philippe Deléham, Mar 26 2013

From Amiram Eldar, Sep 17 2022: (Start)

Sum_{n>=5} 1/a(n) = 518/45 - 2*sqrt(2*(sqrt(5)+5))*Pi/3.

Sum_{n>=5} (-1)^(n+1)/a(n) = 8*sqrt(5)*arccoth(3/sqrt(5))/3 + 92*log(2)/15 - 418/45. (End)

Maple

seq(floor((n-1)*(n-2)/10), n=0..70); # G. C. Greubel, Aug 31 2019

Mathematica

Accumulate[Floor[Range[0, 70]/5]] (* Harvey P. Dale, May 25 2016 *)

Prog

(Magma) [Round(n*(n-3)/10): n in [0..70]]; // Vincenzo Librandi, Jun 25 2011
(PARI) a(n) = sum(k=0, n, k\5); \\ Michel Marcus, May 13 2016
(Sage) [floor((n-1)*(n-2)/10) for n in (0..70)] # G. C. Greubel, Aug 31 2019
(GAP) List([0..70], n-> Int((n-1)*(n-2)/10)); # G. C. Greubel, Aug 31 2019

Crossrefs

Cf. A002264, A002265, A004526, A010872, A010873, A010874, A130481, A130482.

Sequence in context: A262249 A248421 A008732 * A005706 A173345 A226091

Adjacent sequences: A130517 A130518 A130519 * A130521 A130522 A130523

Keyword

nonn,easy

Author

Hieronymus Fischer, Jun 01 2007

Status

approved