A184969 a(n) = [Pi]+[2*Pi]+...+[n*Pi], where []=floor.

3, 9, 18, 30, 45, 63, 84, 109, 137, 168, 202, 239, 279, 322, 369, 419, 472, 528, 587, 649, 714, 783, 855, 930, 1008, 1089, 1173, 1260, 1351, 1445, 1542, 1642, 1745, 1851, 1960, 2073, 2189, 2308, 2430, 2555, 2683, 2814, 2949, 3087, 3228, 3372, 3519, 3669, 3822, 3979, 4139

Offset

1,1

Links

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

Michel Lagneau, Proof of convergence

Formula

Partial sums of A022844.

a(n)/n^2 ~ Pi/2.

a(n) = Pi/2 * n^2 + e(n) * n, where Pi/2-1 < e(n) < Pi/2. [Charles R Greathouse IV, May 27 2011]

Maple

with(numtheory):Digits:=100:s:=0:for n from 1 to 100 do: s:=s+floor(n*Pi):printf(`%d,   `, s):od:

Mathematica

Table[Sum[Floor[k*Pi], {k, 1, n}], {n, 1, 50}] (* G. C. Greubel, Sep 24 2017 *)
Accumulate[Floor[Pi Range[60]]] (* Much more efficient than the above program. *) (* Harvey P. Dale, Aug 27 2023 *)

Prog

(PARI) for(n=1, 50, print1(sum(k=1, n, floor(k*Pi)), ", ")) \\ G. C. Greubel, Sep 24 2017

Crossrefs

Cf. A000796, A022844.

Sequence in context: A193567 A045943 A134479 * A194113 A330010 A194114

Adjacent sequences: A184966 A184967 A184968 * A184970 A184971 A184972

Keyword

nonn

Author

Michel Lagneau, Mar 27 2011

Status

approved