A194221 Partial sums of A194220.

0, 0, 1, 2, 3, 5, 8, 11, 14, 17, 21, 25, 29, 34, 40, 46, 52, 58, 65, 72, 79, 87, 96, 105, 114, 123, 133, 143, 153, 164, 176, 188, 200, 212, 225, 238, 251, 265, 280, 295, 310, 325, 341, 357, 373, 390, 408, 426, 444, 462, 481, 500, 519, 539, 560, 581, 602

Offset

1,4

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

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

Formula

From Chai Wah Wu, Jun 10 2020: (Start)

a(n) = 2*a(n-1) - a(n-2) + a(n-8) - 2*a(n-9) + a(n-10) for n > 10.

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

a(n) = 3/16*n^2 - n/8 + O(1). - Charles R Greathouse IV, May 28 2026

Maple

seq(coeff(series( x^3*(1+x^3+x^4)/((1-x)^2*(1-x^8)) , x, n+1), x, n), n = 1..60); # G. C. Greubel, Jun 10 2020

Mathematica

r = 1/4;
a[n_] := Floor[Sum[FractionalPart[k*r], {k, 1, n}]]
Table[a[n], {n, 1, 90}]    (* A194220 *)
s[n_] := Sum[a[k], {k, 1, n}]
Table[s[n], {n, 1, 100}]   (* A194221 *)

Prog

(PARI) a(n)=(3*n^2-2*n+[0, -1, -8, -5, -8, -17, -16, -5][n%8+1])/16 \\ Charles R Greathouse IV, May 28 2026

Crossrefs

Cf. A194220.

Sequence in context: A332071 A391538 A225087 * A054925 A194248 A375981

Adjacent sequences: A194218 A194219 A194220 * A194222 A194223 A194224

Keyword

nonn,easy

Author

Clark Kimberling, Aug 19 2011

Status

approved