A294629 Partial sums of A294628.

4, 16, 28, 56, 68, 120, 132, 192, 228, 296, 308, 440, 452, 536, 612, 736, 748, 920, 932, 1112, 1204, 1320, 1332, 1624, 1676, 1808, 1916, 2144, 2156, 2496, 2508, 2760, 2884, 3048, 3156, 3600, 3612, 3792, 3932, 4336, 4348, 4784, 4796, 5120, 5388, 5600, 5612, 6224, 6292, 6640, 6812, 7184, 7196, 7728, 7868, 8384

Offset: 1

Author

Omar E. Pol, Nov 05 2017

Keywords

• nonn

Comments

a(n) is also the volume (and the number of cubes) in the n-th level (starting from the top) of the stepped pyramid described in A294630.

Number of terms less than 10^k, k=1,2,3,...: 1, 5, 19, 61, 195, 623, 1967, 6225, ... - Muniru A Asiru, Mar 04 2018

Examples

			Illustration of initial terms (n = 1..6):
.                                                  _ _ _ _ _ _
.                                _ _ _ _         _|     |     |_
.                _ _ _ _       _|   |   |_      |       |       |
.      _ _      |   |   |     |    _|_    |     |      _|_      |
.     |_|_|     |_ _|_ _|     |_ _|   |_ _|     |_ _ _|   |_ _ _|
.     |_|_|     |   |   |     |   |_ _|   |     |     |_ _|     |
.               |_ _|_ _|     |_    |    _|     |       |       |
.       4                       |_ _|_ _|       |_      |      _|
.                  16                             |_ _ _|_ _ _|
.                                  28
.                                                      56
.
.                                        _ _ _ _ _ _ _ _
.             _ _ _ _ _ _              _|       |       |_
.            |     |     |           _|         |         |_
.         _ _|     |     |_ _       |           |           |
.        |      _ _|_ _      |      |          _|_          |
.        |     |       |     |      |        _|   |_        |
.        |_ _ _|       |_ _ _|      |_ _ _ _|       |_ _ _ _|
.        |     |       |     |      |       |_     _|       |
.        |     |_ _ _ _|     |      |         |_ _|         |
.        |_ _      |      _ _|      |           |           |
.            |     |     |          |_          |          _|
.            |_ _ _|_ _ _|            |_        |        _|
.                                       |_ _ _ _|_ _ _ _|
.                 68
.                                              120
.
Note that for n >= 2 the structure has a hole (or hollow) in the center.
a(n) is the number of ON cells in the n-th diagram.
		

Links

• Iain Fox, Table of n, a(n) for n = 1..10000

Crossrefs

For other related diagrams see A294630 (partial sums), A294016 and A237593.

Cf. A000203, A004125, A016742, A024916, A067436, A239050, A243980, A294015, A294017, A294628.

Programs

GAP

List([1..1000],n->Sum([1..n],k->8*(Sigma(k)-k+(1/2)))); # Muniru A Asiru, Mar 04 2018

Maple

with(numtheory): seq(sum(8*(sigma(k)-k+(1/2)),k=1..n),n=1..1000); # Muniru A Asiru, Mar 04 2018

Mathematica

f[n_] := 8 (DivisorSigma[1, n] - n) + 4; Accumulate@Array[f, 56] (* Robert G. Wilson v, Dec 12 2017 *)

PARI

a(n) = 4*(sum(k=1, n, n\k*k) - sum(k=2, n, n%k)) \\ Iain Fox, Dec 10 2017

PARI

first(n) = my(res = vector(n)); res[1] = 4; for(x=2, n, res[x] = res[x-1] + 8*(sigma(x) - x + (1/2))); res; \\ Iain Fox, Dec 10 2017

Python

from math import isqrt
def A294629(n): return -(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))-n**2<<2 # Chai Wah Wu, Oct 22 2023

Formula

a(n) = 4*A294016(n).

a(n) = A016742(n) - 8*A004125(n).

a(n) = A016742(n) - 4*A067436(n).

a(n) = A243980(n) - 4*A004125(n).

a(n) = A243980(n) - 2*A067436(n).