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 A299692 #56 Mar 21 2024 08:35:31
%S A299692 3,10,20,35,51,75,97,128,159,197,231,283,323,375,429,492,544,619,677,
%T A299692 759,833,913,983,1091,1172,1266,1360,1472,1560,1692,1786,1913,2027,
%U A299692 2149,2267,2430,2542,2678,2812,2982,3106,3286,3416,3588,3756,3920,4062,4282,4437,4630,4804,5006,5166,5394,5576,5808,6002
%N A299692 a(n) is the total area that is visible in the perspective view of the stepped pyramid with n levels described in A245092.
%C A299692 a(n) is also the sum of all divisors of all positive integers <= n, plus the n-th oblong number, since A024916(n) equals the total area of the horizontal terraces of the stepped pyramid with n levels, and A002378(n) equals the total area of the vertical sides that are visible (see link).
%C A299692 a(n) is also the sum of all aliquot divisors of all positive integers <= n, plus the n-th triangular matchstick number.
%H A299692 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr05.jpg">Perspective view of the pyramid with 16 levels</a>, contains 492 cells.
%F A299692 a(n) = A024916(n) + A002378(n).
%F A299692 a(n) = A153485(n) + A045943(n).
%F A299692 a(n) = A328366(n)/2. - _Omar E. Pol_, Apr 22 2020
%F A299692 a(n) = c * n^2 + O(n*log(n)), where c = zeta(2)/2 + 1 = A072691 + 1 = 1.822467... . - _Amiram Eldar_, Mar 21 2024
%e A299692 For n = 3 the areas of the terraces of the first three levels starting from the top of the stepped pyramid are 1, 3 and 4 respectively. On the other hand the areas of the vertical sides that are visible are [1, 1], [2, 2], [2, 1, 1, 2], or in successive levels 2, 4, 6 respectively. Hence the total area that is visible is equal to 1 + 3 + 4 + 2 + 4 + 6 = 8 + 12 = 20, so a(3) = 20.
%e A299692 For n = 16 the total number of horizontal and vertical cells that are visible are 220 and 272 respectively. So a(16) = 220 + 272 = 492 (see the link).
%t A299692 Accumulate[Table[DivisorSigma[1, n] + 2*n, {n, 1, 50}]] (* _Amiram Eldar_, Mar 21 2024 *)
%o A299692 (PARI) a(n) = sum(k=1, n, n\k*k) + n*(n+1); \\ _Michel Marcus_, Jun 21 2018
%o A299692 (Python)
%o A299692 from math import isqrt
%o A299692 def A299692(n): return n*(n+1)+(-(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))>>1) # _Chai Wah Wu_, Oct 22 2023
%Y A299692 Partial sums of A224880.
%Y A299692 Cf. A002378, A024916, A045943, A072691, A153485, A196020, A236104, A237048, A237270, A237271, A237591, A237593, A244050, A245092, A262626, A328366.
%K A299692 nonn
%O A299692 1,1
%A A299692 _Omar E. Pol_, Mar 06 2018