A244050 Partial sums of A243980.

4, 20, 52, 112, 196, 328, 492, 716, 992, 1340, 1736, 2244, 2808, 3468, 4224, 5104, 6056, 7164, 8352, 9708, 11192, 12820, 14544, 16508, 18596, 20852, 23268, 25908, 28668, 31716, 34892, 38320, 41940, 45776, 49804, 54196, 58740, 63524, 68532, 73900

Offset

1,1

Comments

a(n) is also the volume of a special stepped pyramid with n levels related to the symmetric representation of sigma. Note that starting at the top of the pyramid, the total area of the horizontal regions at the n-th level is equal to A239050(n), and the total area of the vertical regions at the n-th level is equal to 8*n.

From Omar E. Pol, Sep 19 2015: (Start)

Also, consider that the area of the central square in the top of the pyramid is equal to 1, so the total area of the horizontal regions at the n-th level starting from the top is equal to sigma(n) = A000203(n), and the total area of the vertical regions at the n-th level is equal to 2*n.

Also note that this stepped pyramid can be constructed with four copies of the stepped pyramid described in A245092 back-to-back (one copy in every quadrant). (End)

From Omar E. Pol, Jan 20 2021: (Start)

Convolution of A000203 and the nonzero terms of A008586.

Convolution of A074400 and the nonzero terms of A005843.

Convolution of A340793 and the nonzero terms of A046092.

Convolution of A239050 and A000027.

(End)

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 7342 terms from Robert Price)

Omar E. Pol, Illustration of a(11) = 1736, Perspective view of the stepped pyramid with 11 levels which contains 1736 unit cubes.

Formula

a(n) = 4*A175254(n).

Example

From Omar E. Pol, Aug 29 2015: (Start)
Illustration of the top view of the stepped pyramid with 16 levels. The pyramid is formed of 5104 unit cubes:
.                 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                |  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.             _ _| |  _ _ _ _ _ _ _ _ _ _ _ _ _ _  | |_ _
.           _|  _ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _  |_
.         _|  _|  _| |  _ _ _ _ _ _ _ _ _ _ _ _  | |_  |_  |_
.        |  _|   |_ _| |_ _ _ _ _ _ _ _ _ _ _ _| |_ _|   |_  |
.   _ _ _| |  _ _|     |  _ _ _ _ _ _ _ _ _ _  |     |_ _  | |_ _ _
.  |  _ _ _|_| |      _| |_ _ _ _ _ _ _ _ _ _| |_      | |_|_ _ _  |
.  | | |  _ _ _|    _|_ _|  _ _ _ _ _ _ _ _  |_ _|_    |_ _ _  | | |
.  | | | | |  _ _ _| |  _| |_ _ _ _ _ _ _ _| |_  | |_ _ _  | | | | |
.  | | | | | | |  _ _|_|  _|  _ _ _ _ _ _  |_  |_|_ _  | | | | | | |
.  | | | | | | | | |  _ _|   |_ _ _ _ _ _|   |_ _  | | | | | | | | |
.  | | | | | | | | | | |  _ _|  _ _ _ _  |_ _  | | | | | | | | | | |
.  | | | | | | | | | | | | |  _|_ _ _ _|_  | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | |  _ _  | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |   | | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |_ _| | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | |_|_ _ _ _|_| | | | | | | | | | | | | |
.  | | | | | | | | | | | |_|_  |_ _ _ _|  _|_| | | | | | | | | | | |
.  | | | | | | | | | |_|_    |_ _ _ _ _ _|    _|_| | | | | | | | | |
.  | | | | | | | |_|_ _  |_  |_ _ _ _ _ _|  _|  _ _|_| | | | | | | |
.  | | | | | |_|_ _  | |_  |_ _ _ _ _ _ _ _|  _| |  _ _|_| | | | | |
.  | | | |_|_ _    |_|_ _| |_ _ _ _ _ _ _ _| |_ _|_|    _ _|_| | | |
.  | |_|_ _ _  |     |_  |_ _ _ _ _ _ _ _ _ _|  _|     |  _ _ _|_| |
.  |_ _ _  | |_|_      | |_ _ _ _ _ _ _ _ _ _| |      _|_| |  _ _ _|
.        | |_    |_ _  |_ _ _ _ _ _ _ _ _ _ _ _|  _ _|    _| |
.        |_  |_  |_  | |_ _ _ _ _ _ _ _ _ _ _ _| |  _|  _|  _|
.          |_  |_ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _|  _|
.            |_ _  | |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |  _ _|
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
Note that the above diagram contains a hidden pattern, simpler, which emerges from the front view of every corner of the stepped pyramid.
For more information about the hidden pattern see A237593 and A245092.
(End)

Mathematica

a[n_] := 4 Sum[(n - k + 1) DivisorSigma[1, k], {k, n}]; Array[a, 40] (* Robert G. Wilson v, Aug 06 2018 *)
Nest[Accumulate, 4*DivisorSigma[1, Range[50]], 2] (* Harvey P. Dale, Sep 07 2022 *)

Prog

(PARI) a(n) = 4*sum(k=1, n, sigma(k)*(n-k+1)); \\ Michel Marcus, Aug 07 2018
(Magma) [4*(&+[(n-k+1)*DivisorSigma(1, k): k in [1..n]]): n in [1..40]]; // G. C. Greubel, Apr 07 2019
(Sage) [4*sum(sigma(k)*(n-k+1) for k in (1..n)) for n in (1..40)] # G. C. Greubel, Apr 07 2019
(Python)
from math import isqrt
def A244050(n): return (((s:=isqrt(n))**2*(s+1)*((s+1)*((s<<1)+1)-6*(n+1))>>1) + sum((q:=n//k)*(-k*(q+1)*(3*k+(q<<1)+1)+3*(n+1)*((k<<1)+q+1)) for k in range(1, s+1))<<1)//3 # Chai Wah Wu, Oct 22 2023

Crossrefs

Cf. A000203, A024916, A046092, A008586, A175254, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A239050, A239660, A239931, A239932, A239933, A239934, A243980, A245092, A262626, A340793.

Sequence in context: A367915 A368112 A108099 * A250224 A250272 A023667

Adjacent sequences: A244047 A244048 A244049 * A244051 A244052 A244053

Keyword

nonn

Author

Omar E. Pol, Jun 18 2014

Status

approved