How many blocks make a "truncated square-pyramid garden"?

Question

A truncated square-pyramid of height \$h\$ has \$h\$ square layers where each layer has a side \$1\$ greater than the one above it, apart from the top layer which is a square of blocks with a given side length.

Here is a truncated square-pyramid of height \$7\$ and top side-length \$4\$ viewed from the side:

             side length   blocks
   ████          4          16
  ▐████▌         5          25
  ██████         6          36
 ▐██████▌        7          49
 ████████        8          64
▐████████▌       9          81
██████████      10         100
                   Total = 371

It requires \$\sum_{i=4}^{10}i^2=371\$ blocks to construct.

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A truncated square-pyramid garden of size \$N\$ consists of truncated, square-pyramids of heights \$1\$ to \$N\$ where the \$n^{th}\$ tallest pyramid has a top side-length of \$x\$ where \$x\$ is the remainder after dividing \$N\$ by \$n\$ unless there is no remainder in which case \$x=n\$.

Here is a side-on view of a pyramid garden with \$N=6\$ pyramids, arranged from tallest to shortest:

N=6                ▐▌
                   ██     ██
                  ▐██▌   ▐██▌   ▐██▌
                  ████   ████   ████   ██
                 ▐████▌ ▐████▌ ▐████▌ ▐██▌ ▐▌
                 ██████ ██████ ██████ ████ ██  ██████
          height   6      5      4     3   2     1
               n   1      2      3     4   5     6
remainder of N/n   0      0      0     2   1     0
 top side-length   1      2      3     2   1     6  

This garden takes \$337\$ blocks to construct.

Task

Given a positive integer, \$N\$, calculate the number of blocks required to build a truncated square-pyramid garden of size \$N\$.

You are not required to handle size zero (just in case that causes an edge case for anyone).

Since this is a sequence, you may instead opt to output the block counts for all gardens up to size \$N\$ (with or without a leading zero, for size zero), or generate the block counts indefinitely without input (again a leading zero is acceptable). It's not currently in the OEIS, and neither is its two-dimensional version.

This is code-golf, so the shortest code in each language wins.

Test cases

The first \$100\$ terms of the sequence are (starting with \$N=1\$):

1, 9, 28, 80, 144, 337, 455, 920, 1251, 1941, 2581, 4268, 4494, 7065, 9049, 11440, 13299, 19005, 20655, 28544, 31140, 37673, 45305, 59360, 59126, 73289, 86256, 101124, 109647, 136805, 138364, 170780, 184520, 211485, 241157, 275528, 272869, 326729, 368320, 414692, 424823, 499261, 510708, 596140, 636361, 680537, 753508, 867036, 857345, 966889, 1027920, 1130172, 1197747, 1358369, 1393684, 1528840, 1571095, 1712605, 1860668, 2083248, 2023267, 2261821, 2445122, 2584136, 2685714, 2910217, 2980225, 3298056, 3459910, 3719313, 3824917, 4206640, 4128739, 4534965, 4846194, 5081240, 5308615, 5695545, 5827090, 6349936, 6395099, 6753185, 7173903, 7783720, 7688846, 8192521, 8679955, 9202980, 9429730, 10177969, 10090513, 10725680, 11134432, 11766133, 12407705, 13134004, 13024244, 13979357, 14523352, 15111244

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Note: there is a surprisingly terse Jelly solution to find for this one, happy hunting!

Here's a hint that might help get you thinking about a different approach:

In Jelly lists are 1-indexed and indexing is modular, so finding the top side-length could be calculated by using \$N\$ to index into a list of [1,2,...,n].

2022-03-15: If you just want to see this solution I have now posted it below.

Answer 1

Python 3, 62 bytes

-2 bytes thanks to Neil!

lambda N:sum((N%~n-~h)**2for n in range(N)for h in range(n,N))

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Answer 2

JavaScript (Node.js), 63 61 bytes

n=>eval("for(x=y=0;x++<n;)for(z=0;z++<=n-x;)y+=(~-n%x+z)**2")

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Shortened by @ophact.

Answer 3

R, 58 49 bytes

function(N){for(i in 1:N)F=F+(x=i:N+N%%-i)%*%x
F}

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Relatively straightforward implementation of the spec.

R, 58 54 bytes

function(N)sum(unlist(Map(seq,N%%-(1:N)+1:N,l=N:1))^2)

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Slightly more interesting but longer; relies on the same logic as above to generate the size of the top of each tower.

Answer 4

Jelly, 11 bytes

Rr_"N%RƊF²S

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R              Range [1..n]
 r             Range each to n: [[1..n], [2..n], …, [n]]
    N%RƊ       [(-n)%1, …, (-n)%n]
  _"           Vectorize subtract
        F²S    Flatten, square, sum

Answer 5

APL+WIN, 47 38 bytes

Prompts for N

+/∊((((n=0)×i)+n←i|N)+¯1+⌽⍳¨i←⍳N←⎕)*¨2

Try it online! Thanks to Dyalog Classic

Answer 6

05AB1E, 14 11 bytes

L(%ā.sR+˜nO

Try it online or verify the first 25 test cases.

Explanation:

    #  e.g. input=6
L   # Push a list in the range [1, (implicit) input]
    #  STACK: [1,2,3,4,5,6]
(   # Negate each value
    #  STACK: [-1,-2,-3,-4,-5,-6]
%   # Modulo the implicit input by these values
    #  STACK: [0,0,0,-2,-4,0]
ā   # Push a list in the range [1,length] (without popping)
    #  STACK: [0,0,0,-2,-4,0],
    #         [1,2,3,4,5,6]
.s  # Get its suffixes
    #  STACK: [0,-4,-2,0,0,0],
    #         [[6],[5,6],[4,5,6],[3,4,5,6],[2,3,4,5,6],[1,2,3,4,5,6]]
R   # Reverse the list of suffixes
    #  STACK: [0,-4,-2,0,0,0],
    #         [[1,2,3,4,5,6],[2,3,4,5,6],[3,4,5,6],[4,5,6],[5,6],[6]]
+   # Add the values in the two lists at the same positions together
    #  STACK: [[1,2,3,4,5,6],[2,3,4,5,6],[3,4,5,6],[2,3,4],[1,2],[6]]
˜   # Flatten to a single list
    #  STACK: [1,2,3,4,5,6,2,3,4,5,6,3,4,5,6,2,3,4,1,2,6]
n   # Square each integer
    #  STACK: [1,4,9,16,25,36,4,9,16,25,36,9,16,25,36,4,9,16,1,4,36]
O   # Sum them together
    #  STACK: 337
    # (after which the result is output implicitly)

Answer 7

JavaScript (Node.js), 61 bytes

N=>eval('for(k=n=0;n++<N;)for(c=e=N%n||n;c<=e+N-n;)k+=c*c++')

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-2 by Polichinelle.

Iteratively calculates sum, which is shorter than direct calculation.

Answer 8

Pari/GP, 37 bytes

N->sum(n=1,N,sum(i=n,N,(i-(-N)%n)^2))

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Answer 9

C (gcc), 71 bytes

s;h;n;x;f(N){for(s=n=0;N/++n;)for(h=-~N-n,x=N%n?:n;h--;++x)s+=x*x;n=s;}

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Inputs an integer \$N>0\$.
Returns the number of blocks needed to make a truncated square pyramid garden of size \$N\$.

Answer 10

JavaScript (ES6), 56 bytes

f=(N,n=N)=>n&&f(N,n-1)+(g=k=>n++<=N&&k*k+g(k+1))(N%n||n)

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---

Faster, 66 bytes

A longer version that works in \$O(N)\$:

f=(N,n=N,k=N%n||n)=>n&&f(N,n-1)-(g=x=>x*(x+++x)*x/6)(k-1)+g(k+N-n)

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Answer 11

Jelly, 8 bytes

ịṡ¥Ɱ¹²FS

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Uses the very clever tip supplied, but I'm not sure this quite inspires the description "surprisingly terse" yet... that ¹ just smells wrong!

More equivalent 8-byters:

• ṡR⁸ịⱮ²FS
• ịⱮṡR$²FS
• ịⱮṡⱮ`²FS
• RɓịⱮṡ²FS

   Ɱ¹       For each n in [1 .. N],
 ṡ          get a list of all substrings of [1 .. N] of length n,
ị ¥         and take the one at modular 1-index N.
     ²      Square each,
      F     flatten,
       S    and sum.

Answer 12

Charcoal, 19 bytes

ＮθＩΣＥθΣＸ⁺⊕…ιθ﹪θ±⊕ι²

Try it online! Link is to verbose version of code. Explanation: Inspired by @KevinCruijssen's and @ovs's answers. Explanation:

Ｎθ                  Input `N` as a number
     θ              Input `N`
    Ｅ               Map over implicit range
          …         Range from
           ι        Current value
            θ       To input `N`
         ⊕          Vectorised Increment
        ⁺           Vectorised Plus
              θ     Input `N`
             ﹪      Modulo
                 ι  Current value
                ⊕   Incremented
               ±    Negated
       Ｘ          ² Vectorised Squared
      Σ             Take the sum
   Σ                Take the sum
  Ｉ                 Cast to string
                    Implicitly print

Answer 13

Ruby, 54 46 bytes

->n{(1..n).sum{|x|(x..n).sum{|y|(n%-x+y)**2}}}

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Answer 14

Burlesque, 27 bytes

roJngsaj.%jiS~]q?+Z]FLqS[ms

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Port of Kevin Cruijssen's answer

ro    # Range [1..N]
J     # Dup
ng    # Negate
sa    # Non-destructive length
j.%   # Modulo (implicit zip)
j     # Reorder stack
iS    # Generate tails
~]    # Drop final
q?+Z] # Zip and sum
FL    # Flatten
qS[   # Square
ms    # Map sum

Answer 15

Jelly, 5 bytes

Much of this post is in spoiler blocks, just in case anyone wants to try to find it for themselves since I mentioned it in the question.

ịⱮẆ²S

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How?

Unrelated String got to the method in their answer, just not to the golfed code!

ịⱮẆ²S - Link: integer, N
  Ẇ   - all sublists of [1..N]
          e.g. N=6 -> [[1],[2],[3],[4],[5],[6]
                      ,[1,2],[2,3],[3,4],[4,5],[5,6]
                      ,[1,2,3],[2,3,4],[3,4,5],[4,5,6]
                      ,[1,2,3,4],[2,3,4,5],[3,4,5,6]
                      ,[1,2,3,4,5],[2,3,4,5,6]
                      ,[1,2,3,4,5,6]
                      ]
          i.e. for size in [1..N]:
                 for start in [1..N+1-len]
                   [start..start+size]
 Ɱ    - for each sublist:
ị     -   N index into sublist
          - indexing in Jelly is 1-indexed and modular so...
          e.g 6 ị [3,4,5,6] -> 4
          i.e N ị [start..start+size]
              -> (N mod size)+start-1 if (N mod size) != 0 else start+size
        e.g. 6 ịⱮ (6Ẇ) ->  [1,2,3,4,5,6,2,3,4,5,6,3,4,5,6,2,3,4,1,2,6]
              ...if these were grouped in lengths of [N..1] they
                 are our tower level side lengths:
                   [[1,2,3,4,5,6]  (height 6: [top..bottom])
                   ,[2,3,4,5,6]    (height 5: [top..bottom])
                   ,[3,4,5,6]      (height 4: [top..bottom])
                   ,[2,3,4]        (height 3: [top..bottom])
                   ,[1,2]          (height 2: [top..bottom])
                   ,[6]            (height 1: [top..bottom])
                   ]
   ²  - square (vectorises)
    S - sum