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%I #49 Nov 01 2023 18:30:33
%S 4,20,48,104,172,292,424,616,844,1140,1448,1888,2340,2876,3488,4224,
%T 4972,5892,6824,7936,9140,10460,11792,13416,15092,16900,18816,20960,
%U 23116,25612,28120,30880,33764,36812,39968,43568,47180,50972,54904,59240,63588,68372,73168,78288,83676,89276,94888,101112
%N Partial sums of A294629.
%C 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.
%C a(n) is the number of unit cubes of the pyramid with n levels.
%H Muniru A Asiru, <a href="/A294630/b294630.txt">Table of n, a(n) for n = 1..2000</a>
%F a(n) = 4*A294017(n).
%F a(n) = A002492(n) - 8*A072481(n).
%F a(n) = A244050(n) - 4*A072481(n).
%e Illustration of the top view of the pyramid with 16 levels and 4224 unit cubes:
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%e 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.
%p 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
%t f[n_] := 8 (DivisorSigma[1, n] - n) + 4; Accumulate@ Accumulate@ Array[f, 48] (* _Robert G. Wilson v_, Dec 12 2017 *)
%o (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
%o (Python)
%o from math import isqrt
%o 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
%Y For other related diagrams see A294016 and A294629.
%Y Cf. A000203, A002492, A004125, A072481, A237593, A239050, A243980, A244050, A245092, A294015, A294017, A294628.
%K nonn
%O 1,1
%A _Omar E. Pol_, Nov 05 2017