Nicomachus's Theorem

Author

Eric W. Weisstein
January 2, 2025

This notebook downloaded from http://mathworld.wolfram.com/notebooks/NumberTheory/NicomachussTheorem.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/NicomachussTheorem.html.

©2025 Wolfram Research, Inc. except for portions noted otherwise

Partition odd numbers

Partition

l=Range[1,100,2];
split={};
Do[
AppendTo[split,Take[l,i]];
l=Drop[l,i],
{i,9}
];

split//Column

Values

(val=Table[n(n-1)-1+2i,{n,15},{i,n}])//Column

Total/@val

Table[(n^3),{n,15}]

Sum

Sum[n(n-1)-1+2i,{i,n}]

Faulhaber formula

Sum[(k^3),{k,n}]

((Sum[k,{k,n}])^2)

((Binomial[n+1,2])^2)

((Binomial[Range[0,10]+1,2])^2)