A187326 -id:A187326 - cp's OEIS frontend

A187333 a(n) = floor(n/5) + floor(2n/5) + floor(3n/5) + floor(4n/5).

0, 0, 2, 4, 6, 10, 10, 12, 14, 16, 20, 20, 22, 24, 26, 30, 30, 32, 34, 36, 40, 40, 42, 44, 46, 50, 50, 52, 54, 56, 60, 60, 62, 64, 66, 70, 70, 72, 74, 76, 80, 80, 82, 84, 86, 90, 90, 92, 94, 96, 100, 100, 102, 104, 106, 110, 110, 112, 114, 116, 120, 120, 122, 124, 126, 130, 130, 132, 134, 136, 140, 140, 142, 144, 146, 150

Offset: 0

Clark Kimberling, Mar 08 2011

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

Cf. A187326.

Table[Floor[n/5]+Floor[2n/5]+Floor[3n/5]+Floor[4n/5], {n,0,120}]
Table[Total[Floor/@((Range[4]n)/5)],{n,0,80}] (* or *) LinearRecurrence[ {1,0,0,0,1,-1},{0,0,2,4,6,10},80] (* Harvey P. Dale, Oct 19 2018 *)

a(n) = floor(n/5)+floor(2n/5)+floor(3n/5)+floor(4n/5).

G.f.: 2*x^2*(1+x+x^2+2*x^3) / ( (x^4+x^3+x^2+x+1)*(x-1)^2 ). - R. J. Mathar, Mar 08 2011

A187331 a(n) = Sum_{k=1..4} floor(k*n/4).

Original entry on oeis.org

0, 1, 4, 6, 10, 11, 14, 16, 20, 21, 24, 26, 30, 31, 34, 36, 40, 41, 44, 46, 50, 51, 54, 56, 60, 61, 64, 66, 70, 71, 74, 76, 80, 81, 84, 86, 90, 91, 94, 96, 100, 101, 104, 106, 110, 111, 114, 116, 120, 121, 124, 126, 130, 131, 134, 136, 140, 141, 144, 146, 150, 151, 154, 156, 160, 161, 164

Offset: 0

Clark Kimberling, Mar 08 2011

nonn

M. F. Hasler, Table of n, a(n) for n = 0..1000

Cf. A187332 (complement), A187326 (= a(n) - n), A187333.

t=Table[Sum[Floor[k*n/4], {k,1,4}],{n,0,200}]  (* A187331 *)
Complement[Range[Length[t]], t]                (* A187332 *)

A187331(n)=n\4+n\2+3*n\4+n \\ M. F. Hasler, Nov 21 2017

From Chai Wah Wu, Jun 07 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 4.
G.f.: x*(4*x^3 + 2*x^2 + 3*x + 1)/(x^5 - x^4 - x + 1). (End)
a(n) = n + A187326(n), where A187326(n) = [n/4] + [n/2] + [3n/4], [.] = floor. - M. F. Hasler, Nov 21 2017

A244590 a(n) = sum( floor(k*n/8), k=1..7 ).

Original entry on oeis.org

0, 0, 4, 7, 12, 14, 18, 21, 28, 28, 32, 35, 40, 42, 46, 49, 56, 56, 60, 63, 68, 70, 74, 77, 84, 84, 88, 91, 96, 98, 102, 105, 112, 112, 116, 119, 124, 126, 130, 133, 140, 140, 144, 147, 152, 154, 158, 161, 168, 168

Offset: 0

Gary Detlefs, Jun 30 2014

This sequence is G(n,8) where G(n,m) = sum(floor(k*n/m), k=1..m-1). This function is referenced in A109004 and is used in the following formula for gcd(n,m): gcd(n,m) = n+m-n*m+2*G(n,m).
Listed sequences of this form are:
G(n,2) ... A004526;
G(3,n) ... A130481;
G(n,4) ... A187326;
G(n,5) ... A187333;
G(n,6) ... A187336;
G(n,7) ... A187337;
G(n,k*n)/k = n*(n-1)/2  = G(n,n+k)-G(n,k).
It is of interest to note that this alternate form of gcd(n,m) will be undefined if m is a function having a zero in it. For example, gcd(n, n mod 4) would be undefined but gcd(n mod 4, n) would be defined.

Cf. A109004.

[&+[Floor(k*n/8): k in [1..7]]: n in [0..50]]; // Bruno Berselli, Jul 01 2014

G:=(n,m)-> sum(floor(k*n/m), k=1..m-1): seq(G(n,8), n = 0..60);

Table[Sum[Floor[k n/8], {k, 1, 7}], {n, 0, 50}] (* Bruno Berselli, Jul 01 2014 *)

[sum(floor(k*n/8) for k in (1..7)) for n in (0..50)] # Bruno Berselli, Jul 01 2014

a(n) = sum( floor(k*n/8), k=1..7 ).
a(n) = ( gcd(n,8) - (n+8) + n*8 )/2.
G.f.: x^2*(4 + 3*x + 5*x^2 + 2*x^3 + 4*x^4 + 3*x^5 + 7*x^6)/((1 + x)*(1 - x)^2*(1 + x^2)*(1 + x^4)). [Bruno Berselli, Jul 01 2014]

Some terms corrected by Bruno Berselli, Jul 01 2014

cp's OEIS Frontend

A187333 a(n) = floor(n/5) + floor(2n/5) + floor(3n/5) + floor(4n/5).

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A187331 a(n) = Sum_{k=1..4} floor(k*n/4).

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A244590 a(n) = sum( floor(k*n/8), k=1..7 ).

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