House number

This article covers the number. For historical forest dimensions see: Jachenau # Jachenau forest conditions in the course of history .

A house number is a number that can be written as the sum of a cube number and the square pyramidal number with the next smaller edge length.

The general formula for the -th house number is: ${\displaystyle n}$ ${\displaystyle a(n)}$

${\displaystyle a(n)=(n+1)^{3}+\sum _{i=1}^{n}i^{2}={\frac {1}{6}}(n+1)(8n^{2}+13n+6).}$

The first house numbers are (for ): 1, 9, 32, 78, 155, 271, 434, 652, 933, 1285, 1716, 2234, 2847, 3563, 4390, 5336, 6409, 7617, 8968, 10470, 12131, 13959, 15962, 18148, 20525, 23101, 25884, 28882, 32103, 35555, 39246, 43184, 47377, 51833, 56560, 61566, 66859, 72447, 78338, 84540. ${\displaystyle n=0,1,2,\dotsc }$

Its name comes from the fact that in three dimensions the cube forms a “house” with the square pyramid with the next smaller edge length (case ). The pyramid is the roof. ${\displaystyle n=2}$

Individual evidence

1. ↑ Follow A051662 in OEIS
2. ↑ Numbers Aplenty: House Numbers