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A244050 Partial sums of A243980.

%I A244050 #89 Oct 23 2023 01:59:46
%S A244050 4,20,52,112,196,328,492,716,992,1340,1736,2244,2808,3468,4224,5104,
%T A244050 6056,7164,8352,9708,11192,12820,14544,16508,18596,20852,23268,25908,
%U A244050 28668,31716,34892,38320,41940,45776,49804,54196,58740,63524,68532,73900
%N A244050 Partial sums of A243980.
%C A244050 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.
%C A244050 From _Omar E. Pol_, Sep 19 2015: (Start)
%C A244050 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.
%C A244050 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)
%C A244050 From _Omar E. Pol_, Jan 20 2021: (Start)
%C A244050 Convolution of A000203 and the nonzero terms of A008586.
%C A244050 Convolution of A074400 and the nonzero terms of A005843.
%C A244050 Convolution of A340793 and the nonzero terms of A046092.
%C A244050 Convolution of A239050 and A000027.
%C A244050 (End)
%H A244050 Robert G. Wilson v, <a href="/A244050/b244050.txt">Table of n, a(n) for n = 1..10000</a> (first 7342 terms from Robert Price)
%H A244050 Omar E. Pol, <a href="/A237270/a237270.jpg">Illustration of a(11) = 1736</a>, Perspective view of the stepped pyramid with 11 levels which contains 1736 unit cubes.
%F A244050 a(n) = 4*A175254(n).
%e A244050 From _Omar E. Pol_, Aug 29 2015: (Start)
%e A244050 Illustration of the top view of the stepped pyramid with 16 levels. The pyramid is formed of 5104 unit cubes:
%e A244050 .                 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
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%e A244050 .
%e A244050 Note that the above diagram contains a hidden pattern, simpler, which emerges from the front view of every corner of the stepped pyramid.
%e A244050 For more information about the hidden pattern see A237593 and A245092.
%e A244050 (End)
%t A244050 a[n_] := 4 Sum[(n - k + 1) DivisorSigma[1, k], {k, n}]; Array[a, 40] (* _Robert G. Wilson v_, Aug 06 2018 *)
%t A244050 Nest[Accumulate,4*DivisorSigma[1,Range[50]],2] (* _Harvey P. Dale_, Sep 07 2022 *)
%o A244050 (PARI) a(n) = 4*sum(k=1, n, sigma(k)*(n-k+1)); \\ _Michel Marcus_, Aug 07 2018
%o A244050 (Magma) [4*(&+[(n-k+1)*DivisorSigma(1,k): k in [1..n]]): n in [1..40]]; // _G. C. Greubel_, Apr 07 2019
%o A244050 (Sage) [4*sum(sigma(k)*(n-k+1) for k in (1..n)) for n in (1..40)] # _G. C. Greubel_, Apr 07 2019
%o A244050 (Python)
%o A244050 from math import isqrt
%o A244050 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
%Y A244050 Cf. A000203, A024916, A046092, A008586, A175254, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A239050, A239660, A239931, A239932, A239933, A239934, A243980, A245092, A262626, A340793.
%K A244050 nonn
%O A244050 1,1
%A A244050 _Omar E. Pol_, Jun 18 2014