House Numbers

When turning 25884 days old today, I discovered that 25884 is a house number. This was new to me and not surprising because the previous such number was 23101. I turned 23101 days old over 7.6 years ago. So I needed to discover what house numbers are. They are named not after some mathematician named House but because they can be represented as houses. A diagram is helpful at this point. See Figure 1.

Figure 1

Clearly such numbers are figurate but consist of two conjoined shapes: a cube on top of which sits a square-based pyramid. Here a definition from Numbers Aplenty:

The \(n\)-th house number \(h_n\) is a figurate number made by a cube of side \(n+1\), surmounted by a square pyramidal number with side \(n\), thus:$$h_n = (n+1)^3 + \sum_{k=1}^nk^2$$An alternative way of representing this is as: $$h_n=\frac{8n^3+21n^2+19n+6}{6}$$The generating function for this sequence is:$$\frac{1+5x+2x^2}{(x-1)^4}$$In Figure 2, the SageMath code is shown that makes use of this function to generate the coefficients of the polynomial that are the terms of the sequence (the output is truncated for display purposes):

Figure 2: permalink

Figure 3 shows the SageMath code to generate the coefficients by themselves, removed from the polynomial. The output has been truncated to fit the display. These are the house numbers. The sequence of house numbers forms  OEIS A051662. Below is the list of terms, up to and including 25884:

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, ...

Figure 3: permalink

The next house number is 28882, so it's quite some way off. There's nothing very deep about house numbers but they form an interesting variation on the figurate numbers that I'm familiar with. They are the sum of two different and independent figurate numbers that have been combined to form a composite shape, in this case a house. There's no reason that one couldn't combine other shapes together from these or different components.

For example, adding a square-based triangular pyramid to each of the six faces of the cube would produce a sort of stellated cube. The formula would be:$$h_n = (n+1)^3 + 6 \sum_{k=1}^nk^2$$ADDENDUM: May 7th 2020

Today I turned 25967 days old and this number turns out to be a 24-hedral number with formula:$$(2n + 1)(8n^2 + 14n + 7) \text{ where }n=11$$It can be considered as being made up out of a central cube and six square pyramids. The central cube number is given by OEIS A016755:$$CCub(n) = (2n + 1)^3$$The figurate number represented by the square pyramids is given by OEIS A000447:$$ P^{(4)}(n) = \frac{(1 + n)(1 + 2n)(3 + 2n)}{3}$$Therefore, for our 24-hedral number, we have:$$24H(n) = CCub(n) + 6P^{(4)}(n)$$ $$24H(n)=(2n+1)^3 + \frac{6(1+n)(1+2n)(3+2n)}{3}$$ $$24H(n)= (2n+1)(8n^2 +14n+7)$$Figure 4 shows the example for when \(n=11\):

Figure 4: link