A294630 Partial sums of A294629.

4, 20, 48, 104, 172, 292, 424, 616, 844, 1140, 1448, 1888, 2340, 2876, 3488, 4224, 4972, 5892, 6824, 7936, 9140, 10460, 11792, 13416, 15092, 16900, 18816, 20960, 23116, 25612, 28120, 30880, 33764, 36812, 39968, 43568, 47180, 50972, 54904, 59240, 63588, 68372, 73168, 78288, 83676, 89276, 94888, 101112

Offset

1,1

Comments

a(n) is also the volume of a stepped pyramid with n levels which is another version of the stepped pyramid described in A244050. Both pyramids have the same top view and the same front view, that is to say externally both pyramids are equal, but this pyramid with n levels contains a central chamber whose volume is 4*A072481(n). For more information about the central chamber see the diagrams in A294629.

a(n) is the number of unit cubes of the pyramid with n levels.

Links

Muniru A Asiru, Table of n, a(n) for n = 1..2000

Formula

a(n) = 4*A294017(n).

a(n) = A002492(n) - 8*A072481(n).

a(n) = A244050(n) - 4*A072481(n).

Example

Illustration of the top view of the pyramid with 16 levels and 4224 unit cubes:
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.
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.

Maple

with(numtheory): seq(sum(sum(8*(sigma(j)-j+(1/2)), j=1..k), k=1..n), n=1..50); # Muniru A Asiru, Mar 04 2018

Mathematica

f[n_] := 8 (DivisorSigma[1, n] - n) + 4; Accumulate@ Accumulate@ Array[f, 48] (* Robert G. Wilson v, Dec 12 2017 *)

Prog

(GAP) List([1..50], n->Sum([1..n], m->Sum([1..m], k->8*(Sigma(k)-k+(1/2))))); # Muniru A Asiru, Mar 04 2018
(Python)
from math import isqrt
def A294630(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))<<2)-(n*(n+1)*((n<<1)+1)<<1))//3 # Chai Wah Wu, Nov 01 2023

Crossrefs

For other related diagrams see A294016 and A294629.

Cf. A000203, A002492, A004125, A072481, A237593, A239050, A243980, A244050, A245092, A294015, A294017, A294628.

Sequence in context: A145194 A164924 A033579 * A160799 A187274 A367915

Adjacent sequences: A294627 A294628 A294629 * A294631 A294632 A294633

Keyword

nonn

Author

Omar E. Pol, Nov 05 2017

Status

approved