This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A294630 #49 Nov 01 2023 18:30:33 %S A294630 4,20,48,104,172,292,424,616,844,1140,1448,1888,2340,2876,3488,4224, %T A294630 4972,5892,6824,7936,9140,10460,11792,13416,15092,16900,18816,20960, %U A294630 23116,25612,28120,30880,33764,36812,39968,43568,47180,50972,54904,59240,63588,68372,73168,78288,83676,89276,94888,101112 %N A294630 Partial sums of A294629. %C A294630 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 A294630 a(n) is the number of unit cubes of the pyramid with n levels. %H A294630 Muniru A Asiru, <a href="/A294630/b294630.txt">Table of n, a(n) for n = 1..2000</a> %F A294630 a(n) = 4*A294017(n). %F A294630 a(n) = A002492(n) - 8*A072481(n). %F A294630 a(n) = A244050(n) - 4*A072481(n). %e A294630 Illustration of the top view of the pyramid with 16 levels and 4224 unit cubes: %e A294630 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ %e A294630 . | _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ | %e A294630 . | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | %e A294630 . _ _| | _ _ _ _ _ _ _ _ _ _ _ _ _ _ | |_ _ %e A294630 . _| _ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _ |_ %e A294630 . _| _| _| | _ _ _ _ _ _ _ _ _ _ _ _ | |_ |_ |_ %e A294630 . | _| |_ _| |_ _ _ _ _ _ _ _ _ _ _ _| |_ _| |_ | %e A294630 . _ _ _| | _ _| | _ _ _ _ _ _ _ _ _ _ | |_ _ | |_ _ _ %e A294630 . | _ _ _|_| | _| |_ _ _ _ _ _ _ _ _ _| |_ | |_|_ _ _ | %e A294630 . | | | _ _ _| _|_ _| _ _ _ _ _ _ _ _ |_ _|_ |_ _ _ | | | %e A294630 . | | | | | _ _ _| | _| |_ _ _ _ _ _ _ _| |_ | |_ _ _ | | | | | %e A294630 . | | | | | | | _ _|_| _| _ _ _ _ _ _ |_ |_|_ _ | | | | | | | %e A294630 . | | | | | | | | | _ _| |_ _ _ _ _ _| |_ _ | | | | | | | | | %e A294630 . | | | | | | | | | | | _ _| _ _ _ _ |_ _ | | | | | | | | | | | %e A294630 . | | | | | | | | | | | | | _|_ _ _ _|_ | | | | | | | | | | | | | %e A294630 . | | | | | | | | | | | | | | | _ _ | | | | | | | | | | | | | | | %e A294630 . | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | %e A294630 . | | | | | | | | | | | | | | | |_ _| | | | | | | | | | | | | | | | %e A294630 . | | | | | | | | | | | | | |_|_ _ _ _|_| | | | | | | | | | | | | | %e A294630 . | | | | | | | | | | | |_|_ |_ _ _ _| _|_| | | | | | | | | | | | %e A294630 . | | | | | | | | | |_|_ |_ _ _ _ _ _| _|_| | | | | | | | | | %e A294630 . | | | | | | | |_|_ _ |_ |_ _ _ _ _ _| _| _ _|_| | | | | | | | %e A294630 . | | | | | |_|_ _ | |_ |_ _ _ _ _ _ _ _| _| | _ _|_| | | | | | %e A294630 . | | | |_|_ _ |_|_ _| |_ _ _ _ _ _ _ _| |_ _|_| _ _|_| | | | %e A294630 . | |_|_ _ _ | |_ |_ _ _ _ _ _ _ _ _ _| _| | _ _ _|_| | %e A294630 . |_ _ _ | |_|_ | |_ _ _ _ _ _ _ _ _ _| | _|_| | _ _ _| %e A294630 . | |_ |_ _ |_ _ _ _ _ _ _ _ _ _ _ _| _ _| _| | %e A294630 . |_ |_ |_ | |_ _ _ _ _ _ _ _ _ _ _ _| | _| _| _| %e A294630 . |_ |_ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _| _| %e A294630 . |_ _ | |_ _ _ _ _ _ _ _ _ _ _ _ _ _| | _ _| %e A294630 . | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | %e A294630 . | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | %e A294630 . |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| %e A294630 . %e A294630 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 A294630 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 A294630 f[n_] := 8 (DivisorSigma[1, n] - n) + 4; Accumulate@ Accumulate@ Array[f, 48] (* _Robert G. Wilson v_, Dec 12 2017 *) %o A294630 (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 A294630 (Python) %o A294630 from math import isqrt %o A294630 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 A294630 For other related diagrams see A294016 and A294629. %Y A294630 Cf. A000203, A002492, A004125, A072481, A237593, A239050, A243980, A244050, A245092, A294015, A294017, A294628. %K A294630 nonn %O A294630 1,1 %A A294630 _Omar E. Pol_, Nov 05 2017