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A244050 Partial sums of A243980. 72

%I

%S 4,20,52,112,196,328,492,716,992,1340,1736,2244,2808,3468,4224,5104,

%T 6056,7164,8352,9708,11192,12820,14544,16508,18596,20852,23268,25908,

%U 28668,31716,34892,38320,41940,45776,49804,54196,58740,63524,68532,73900

%N Partial sums of A243980.

%C a(n) is also the volume of a special step 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 From _Omar E. Pol_, Sep 19 2015: (Start)

%C 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 Also note that this step pyramid can be constructed with four copies of the step pyramid described in A245092 (one copy in every quadrant). (End)

%H 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)

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

%e From _Omar E. Pol_, Aug 29 2015: (Start)

%e Illustration of the top view of the pyramid with 16 levels. The pyramid is formed of 5104 unit cubes:

%e . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

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%e . | | | | | | | |_|_ _ |_ |_ _ _ _ _ _| _| _ _|_| | | | | | | |

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%e . | |_|_ _ _ | |_ |_ _ _ _ _ _ _ _ _ _| _| | _ _ _|_| |

%e . |_ _ _ | |_|_ | |_ _ _ _ _ _ _ _ _ _| | _|_| | _ _ _|

%e . | |_ |_ _ |_ _ _ _ _ _ _ _ _ _ _ _| _ _| _| |

%e . |_ |_ |_ | |_ _ _ _ _ _ _ _ _ _ _ _| | _| _| _|

%e . |_ |_ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _| _|

%e . |_ _ | |_ _ _ _ _ _ _ _ _ _ _ _ _ _| | _ _|

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%e . |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|

%e .

%e Note that the above diagram contains a hidden pattern, simpler, which emerges from the front view of every corner of the step pyramid.

%e For more information about the hidden pattern see A237593 and A245092.

%e (End)

%t a[n_] := 4 Sum[(n - k + 1) DivisorSigma[1, k], {k, n}]; Array[a, 40] (* _Robert G. Wilson v_, Aug 06 2018 *)

%o (PARI) a(n) = 4*sum(k=1, n, sigma(k)*(n-k+1)); \\ _Michel Marcus_, Aug 07 2018

%o (MAGMA) [4*(&+[(n-k+1)*DivisorSigma(1,k): k in [1..n]]): n in [1..40]]; // _G. C. Greubel_, Apr 07 2019

%o (Sage) [4*sum(sigma(k)*(n-k+1) for k in (1..n)) for n in (1..40)] # _G. C. Greubel_, Apr 07 2019

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

%K nonn

%O 1,1

%A _Omar E. Pol_, Jun 18 2014

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Last modified September 18 20:36 EDT 2019. Contains 327181 sequences. (Running on oeis4.)