What can you see on a hexagonal spiral?
JavaScript (ES6), 178 169 168 bytes
Returns the \$n\$ first terms as an array.
n=>(X=[1],Y=[L=1],d=5,j=x=y=0,g=a=>--n?g([...a,(X[x+=~-'210012'[d%=6]]=-~X[x])+(Y[y+=~-'1221'[j?d:d++]]=-~Y[y])+(Z[x+y]=-~Z[x+y])-2],++j%L?0:(j%=L*6)?d++:L++):a)(Z=[1])
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How?
This is a rather naive approach that actually computes the spiral with cube coordinates \$(x,y,x+y)\$ and keeps track of the number of cells on each row \$Z[x+y]\$, each diagonal \$Y[y]\$ and each anti-diagonal \$X[x]\$.
Commented
n => ( // n = input
X = [1], Y = [L = 1], // X[] = anti-diagonals, Y[] = diagonals, L = layer
d = 5, // d = direction in [0..5]
j = // j = cell index in the current layer
x = y = 0, // (x, y) = coordinates
g = a => // g = main recursive function taking the output a[]
--n ? // decrement n; if we haven't computed enough terms:
g( // recursive call:
[ ...a, // append the previous terms
( X[x += // update x and increment X[x]
~-'210012'[d %= 6] // pick dx according to d
] = -~X[x] //
) + //
( Y[y += // update y and increment Y[y]
~-'1221'[j ? d // pick dy according to d
: d++] // increment d afterwards if j = 0
] = -~Y[y] // (i.e. this is the 1st cell of the layer)
) + //
(Z[x + y] = -~Z[x + y]) // increment Z[x + y]
- 2 // subtract 2 from X[x] + Y[y] + Z[z] because
], // the current cell is counted three times
++j // increment j
% L ? // if it's not the last cell of this side:
0 // do nothing
: // else:
(j %= L * 6) ? // if it's not the last cell of the layer:
d++ // increment d
: // else:
L++ // increment L
) // end of recursive call
: // else:
a // we're done: return a[]
)(Z = [1]) // initial call to g with a[] = Z[] = [1]
APL (Dyalog Unicode), 23 bytesSBCS
⊢↑∘(∊⊢+∘(⍳¨⌊,⌈)¨÷∘3)⍳,⊢
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A tacit function that returns the first n terms of the sequence.
Observation
n m | f(n,m): m consecutive integers starting at n
---------------
1 0 |
1 1 | 1
2 0 |
2 1 | 2
3 1 | 3
3 1 | 3
4 1 | 4
4 2 | 4 5
5 1 | 5
5 2 | 5 6
6 2 | 6 7
6 2 | 6 7
7 2 | 7 8
7 3 | 7 8 9
8 2 | 8 9
8 3 | 8 9 10
9 3 | 9 10 11
9 3 | 9 10 11
---------------
Flattened: 1 2 3 3 4 4 5 5 5 6 6 7 6 7 7 8 7 8 9 8 9 8 9 10 9 10 11 9 10 11
Though I don't have a rigorous argument that the pattern is mathematically correct, the test cases show that the first 3000 terms agree with Arnauld's JS answer.
How it works: the code
Uses ⎕IO←0, so ⍳ n gives 0 1 2 .. n-1.
⊢↑∘(∊⊢+∘(⍳¨⌊,⌈)¨÷∘3)⍳,⊢ ⍝ Input: n
⍳,⊢ ⍝ Generate 0..n
( ) ⍝ For each number k...
÷∘3 ⍝ Divide k by 3
( )¨ ⍝ For each item of above,
⌊,⌈ ⍝ Collect its floor and ceil;
⍝ (0 0)(0 1)(0 1)(1 1)(1 2)(1 2)(2 2)...
⍳¨ ⍝ For each number i, generate 0..i-1
⍝ (2 3) -> ((0 1)(0 1 2))
⊢+∘ ⍝ Add k to each pair of ranges (results of f(n,m))
∊ ⍝ Enlist; collect all numbers into a vector
⊢↑∘ ⍝ Take first n numbers
MATL, 48 bytes
0i:"J_Q@q:gJJq1_J_tQ5$h@Y"&h]YsG:)&Zjyy+vt0Z)=as
Try it online! Or verify all test cases.
Explanation
This uses coordinate axes 60 degrees apart, so that cell coordinates are integer values. In addition, coordinates (r,s) are stored as a complex number r+j*s, where j is the imaginary unit.
The code first creates n layers of the spiral, which is enough (for most inputs, it is much more than enough). This is done by a loop that adds one layer in each iteration. The first cell is at the origin. For each new layer, the displacements from one cell to the next are defined (in the chosen system of complex coordinates): a displacement "outward" (to create room for the layer) and then several displacements "around" to complete a turn. Once all displacements for all layers have been specified, a cumulative sum produces the cell positions.
To see what is meant by "layer" and how the spiral is built, you can graphically depict the spiral with n layers. Note that the spiral is deformed because the plot uses orthogonal axes (real and imaginary), instead of axes with 60-degree separation.
Once the spiral has been created, the first n cells are kept. From these, the last cell is the reference. To detect which cells are in the same row/diagonal as the reference cell, the condition is that the candidate cell should have either the same value of the first coordinate as the reference cell, or the same value of the second coordinate, or the same value of the sum of coordinates.
0 % Push 0. This is the initial cell
i:" % Input n. For each k in [1 2 ... n]
J_Q % Push 1-j
@q:g % Push [1 1 1 ... 1] (k-1 entries)
J % Push j
Jq % Push -1+j
1_ % Push -1
J_ % Push -j
tQ % Duplicate, add 1: pushes 1-j
5$h % Concatenate 5 elements into a row vector: [j -1+j -1 -j 1-j]
@Y" % Repeat each entry k times
&h % Concatenate everything into a row vector
] % End
Ys % Cumulative sum
G:) % Keep first n entries. Gives a row vector of size n
&Zj % Push real and imaginary parts. Gives two row vectors
yy+ % Duplicate top two elements in the stack, add
v % Concatenate the three row vectors into a 3-row matrix
t % Duplicate
0Z) % Keep last column
= % Equal? Element-wise with broadcast. Gives a true-false matrix
a % Any: true for columns that contain at least one true entry
s % Sum. Implicit display