A183143 -id:A183143 - cp's OEIS frontend

A183144 [1/s]+[2/s]+...+[n/s], where s=(3+sqrt(3))/2, []=floor.

0, 0, 1, 2, 4, 6, 8, 11, 14, 18, 22, 27, 32, 37, 43, 49, 56, 63, 71, 79, 87, 96, 105, 115, 125, 135, 146, 157, 169, 181, 194, 207, 220, 234, 248, 263, 278, 294, 310, 326, 343, 360, 378, 396, 415, 434, 453, 473, 493, 514, 535, 556, 578, 600, 623

Offset: 1

Clark Kimberling, Dec 26 2010

nonn

A183143 + A183144 = A000217 (the triangular numbers).

Cf. A183142, A183143.

With[{c=(3+Sqrt[3])/2},Accumulate[Floor[Range[60]/c]]] (* Harvey P. Dale, Mar 30 2018 *)

[1/s]+[2/s]+...+[n/s], where s=(3+sqrt(3))/2, []=floor.

A362873 a(n) is the number of points with integer coordinates that are inside an equilateral triangle inscribed in a circle of radius n, the location of the triangle in the Oxy coordinate plane is described in the comments.

Original entry on oeis.org

1, 4, 12, 17, 33, 42, 64, 77, 105, 122, 158, 177, 219, 242, 292, 319, 375, 406, 470, 503, 573, 610, 688, 729, 813, 856, 948, 995, 1093, 1144, 1248, 1303, 1415, 1472, 1592, 1653, 1779, 1844, 1976, 2045, 2185, 2256, 2402, 2477, 2631, 2710, 2870, 2951, 3119, 3204, 3378, 3467, 3649

Offset: 1

Nicolay Avilov, May 07 2023

nonn

An equilateral triangle is located in the coordinate plane Oxy so that its center coincides with the origin O, one of the vertices lies on the Oy axis.

			a(3) = 4 + 2*4 = 12;
a(4) = 5 + 2*6 = 17.

Nicolay Avilov, Illustration of initial terms

Cf. A194106, A183143.

a[n_]:=(3n-2+Mod[n,2])/2+2Sum[Floor[(3n+Mod[n,2])/2-Sqrt[3]k],{k,Floor[Sqrt[3]n/2]}]; Array[a,53] (* Stefano Spezia, May 08 2023 *)

a(n) = (3*n - 2)/2 + 2*Sum_{k=1..floor(sqrt(3)*n/2)} floor(-sqrt(3)*k + 3*n/2) if n is even;

a(n) = (3*n - 1)/2 + 2*Sum_{k=1..floor(sqrt(3)*n/2)} floor(-sqrt(3)*k + (3*n + 1)/2) if n is odd.

A184977 a(n) = Sum_{k=1..n} floor(k*gamma) where gamma is Euler's constant (A001620).

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 13, 17, 22, 27, 33, 39, 46, 54, 62, 71, 80, 90, 100, 111, 123, 135, 148, 161, 175, 190, 205, 221, 237, 254, 271, 289, 308, 327, 347, 367, 388, 409, 431, 454, 477, 501, 525, 550, 575, 601, 628, 655, 683, 711, 740, 770, 800, 831, 862, 894, 926, 959, 993, 1027, 1062

Offset: 1

Michel Lagneau, Mar 27 2011

a(n) = A183143(n) for n = 1..96 where A183143(n) is the sequence floor(1/r) + floor(2/r) + ... + floor(n/r) and r = sqrt(3). It is interesting to note that a(n)/n^2 converges to gamma/2.
gamma = 0.57721566490153286060651209... (A002852)
1/sqrt(3) = 0.577350269189625764509148... (A020760)
Starts to differ from A183143 at a(97). - R. J. Mathar, Aug 28 2025

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A001620, A038128.

R:=RealField(100); [(&+[Floor(k*EulerGamma(R)): k in [1..n]]): n in [1..50]]; // G. C. Greubel, Aug 27 2018

with(numtheory):Digits:=500:s:=0:c:=evalf(gamma(0)):for n from 1 to 100 do:
  s:=s+floor(n*c):printf(`%d, `,s):od:

Table[Sum[Floor[k*EulerGamma], {k, 1, n}], {n, 50}] (* G. C. Greubel, Jun 02 2017 *)

a(n) = sum(k=1, n, floor(k*Euler)); \\ Michel Marcus, Apr 02 2017

Partial sums of A038128.

Name edited by Jon E. Schoenfield, Apr 02 2017

cp's OEIS Frontend

A183144 [1/s]+[2/s]+...+[n/s], where s=(3+sqrt(3))/2, []=floor.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A362873 a(n) is the number of points with integer coordinates that are inside an equilateral triangle inscribed in a circle of radius n, the location of the triangle in the Oxy coordinate plane is described in the comments.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A184977 a(n) = Sum_{k=1..n} floor(k*gamma) where gamma is Euler's constant (A001620).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions