A294630 Partial sums of A294629.
4, 20, 48, 104, 172, 292, 424, 616, 844, 1140, 1448, 1888, 2340, 2876, 3488, 4224, 4972, 5892, 6824, 7936, 9140, 10460, 11792, 13416, 15092, 16900, 18816, 20960, 23116, 25612, 28120, 30880, 33764, 36812, 39968, 43568, 47180, 50972, 54904, 59240, 63588, 68372, 73168, 78288, 83676, 89276, 94888, 101112
Offset: 1
Keywords
Examples
Illustration of the top view of the pyramid with 16 levels and 4224 unit cubes: . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . | _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ | . | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | . _ _| | _ _ _ _ _ _ _ _ _ _ _ _ _ _ | |_ _ . _| _ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _ |_ . _| _| _| | _ _ _ _ _ _ _ _ _ _ _ _ | |_ |_ |_ . | _| |_ _| |_ _ _ _ _ _ _ _ _ _ _ _| |_ _| |_ | . _ _ _| | _ _| | _ _ _ _ _ _ _ _ _ _ | |_ _ | |_ _ _ . | _ _ _|_| | _| |_ _ _ _ _ _ _ _ _ _| |_ | |_|_ _ _ | . | | | _ _ _| _|_ _| _ _ _ _ _ _ _ _ |_ _|_ |_ _ _ | | | . | | | | | _ _ _| | _| |_ _ _ _ _ _ _ _| |_ | |_ _ _ | | | | | . | | | | | | | _ _|_| _| _ _ _ _ _ _ |_ |_|_ _ | | | | | | | . | | | | | | | | | _ _| |_ _ _ _ _ _| |_ _ | | | | | | | | | . | | | | | | | | | | | _ _| _ _ _ _ |_ _ | | | | | | | | | | | . | | | | | | | | | | | | | _|_ _ _ _|_ | | | | | | | | | | | | | . | | | | | | | | | | | | | | | _ _ | | | | | | | | | | | | | | | . | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | . | | | | | | | | | | | | | | | |_ _| | | | | | | | | | | | | | | | . | | | | | | | | | | | | | |_|_ _ _ _|_| | | | | | | | | | | | | | . | | | | | | | | | | | |_|_ |_ _ _ _| _|_| | | | | | | | | | | | . | | | | | | | | | |_|_ |_ _ _ _ _ _| _|_| | | | | | | | | | . | | | | | | | |_|_ _ |_ |_ _ _ _ _ _| _| _ _|_| | | | | | | | . | | | | | |_|_ _ | |_ |_ _ _ _ _ _ _ _| _| | _ _|_| | | | | | . | | | |_|_ _ |_|_ _| |_ _ _ _ _ _ _ _| |_ _|_| _ _|_| | | | . | |_|_ _ _ | |_ |_ _ _ _ _ _ _ _ _ _| _| | _ _ _|_| | . |_ _ _ | |_|_ | |_ _ _ _ _ _ _ _ _ _| | _|_| | _ _ _| . | |_ |_ _ |_ _ _ _ _ _ _ _ _ _ _ _| _ _| _| | . |_ |_ |_ | |_ _ _ _ _ _ _ _ _ _ _ _| | _| _| _| . |_ |_ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _| _| . |_ _ | |_ _ _ _ _ _ _ _ _ _ _ _ _ _| | _ _| . | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | . | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | . |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| . 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.
Links
- Muniru A Asiru, Table of n, a(n) for n = 1..2000
Crossrefs
Programs
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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
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Maple
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
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Mathematica
f[n_] := 8 (DivisorSigma[1, n] - n) + 4; Accumulate@ Accumulate@ Array[f, 48] (* Robert G. Wilson v, Dec 12 2017 *)
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Python
from math import isqrt 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
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