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 A294629 #56 Oct 23 2023 02:00:25 %S A294629 4,16,28,56,68,120,132,192,228,296,308,440,452,536,612,736,748,920, %T A294629 932,1112,1204,1320,1332,1624,1676,1808,1916,2144,2156,2496,2508,2760, %U A294629 2884,3048,3156,3600,3612,3792,3932,4336,4348,4784,4796,5120,5388,5600,5612,6224,6292,6640,6812,7184,7196,7728,7868,8384 %N A294629 Partial sums of A294628. %C A294629 a(n) is also the volume (and the number of cubes) in the n-th level (starting from the top) of the stepped pyramid described in A294630. %C A294629 Number of terms less than 10^k, k=1,2,3,...: 1, 5, 19, 61, 195, 623, 1967, 6225, ... - _Muniru A Asiru_, Mar 04 2018 %H A294629 Iain Fox, <a href="/A294629/b294629.txt">Table of n, a(n) for n = 1..10000</a> %F A294629 a(n) = 4*A294016(n). %F A294629 a(n) = A016742(n) - 8*A004125(n). %F A294629 a(n) = A016742(n) - 4*A067436(n). %F A294629 a(n) = A243980(n) - 4*A004125(n). %F A294629 a(n) = A243980(n) - 2*A067436(n). %e A294629 Illustration of initial terms (n = 1..6): %e A294629 . _ _ _ _ _ _ %e A294629 . _ _ _ _ _| | |_ %e A294629 . _ _ _ _ _| | |_ | | | %e A294629 . _ _ | | | | _|_ | | _|_ | %e A294629 . |_|_| |_ _|_ _| |_ _| |_ _| |_ _ _| |_ _ _| %e A294629 . |_|_| | | | | |_ _| | | |_ _| | %e A294629 . |_ _|_ _| |_ | _| | | | %e A294629 . 4 |_ _|_ _| |_ | _| %e A294629 . 16 |_ _ _|_ _ _| %e A294629 . 28 %e A294629 . 56 %e A294629 . %e A294629 . _ _ _ _ _ _ _ _ %e A294629 . _ _ _ _ _ _ _| | |_ %e A294629 . | | | _| | |_ %e A294629 . _ _| | |_ _ | | | %e A294629 . | _ _|_ _ | | _|_ | %e A294629 . | | | | | _| |_ | %e A294629 . |_ _ _| |_ _ _| |_ _ _ _| |_ _ _ _| %e A294629 . | | | | | |_ _| | %e A294629 . | |_ _ _ _| | | |_ _| | %e A294629 . |_ _ | _ _| | | | %e A294629 . | | | |_ | _| %e A294629 . |_ _ _|_ _ _| |_ | _| %e A294629 . |_ _ _ _|_ _ _ _| %e A294629 . 68 %e A294629 . 120 %e A294629 . %e A294629 Note that for n >= 2 the structure has a hole (or hollow) in the center. %e A294629 a(n) is the number of ON cells in the n-th diagram. %p A294629 with(numtheory): seq(sum(8*(sigma(k)-k+(1/2)),k=1..n),n=1..1000); # _Muniru A Asiru_, Mar 04 2018 %t A294629 f[n_] := 8 (DivisorSigma[1, n] - n) + 4; Accumulate@Array[f, 56] (* _Robert G. Wilson v_, Dec 12 2017 *) %o A294629 (PARI) a(n) = 4*(sum(k=1, n, n\k*k) - sum(k=2, n, n%k)) \\ _Iain Fox_, Dec 10 2017 %o A294629 (PARI) first(n) = my(res = vector(n)); res[1] = 4; for(x=2, n, res[x] = res[x-1] + 8*(sigma(x) - x + (1/2))); res; \\ _Iain Fox_, Dec 10 2017 %o A294629 (GAP) List([1..1000],n->Sum([1..n],k->8*(Sigma(k)-k+(1/2)))); # _Muniru A Asiru_, Mar 04 2018 %o A294629 (Python) %o A294629 from math import isqrt %o A294629 def A294629(n): return -(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))-n**2<<2 # _Chai Wah Wu_, Oct 22 2023 %Y A294629 For other related diagrams see A294630 (partial sums), A294016 and A237593. %Y A294629 Cf. A000203, A004125, A016742, A024916, A067436, A239050, A243980, A294015, A294017, A294628. %K A294629 nonn %O A294629 1,1 %A A294629 _Omar E. Pol_, Nov 05 2017