Centered square numbers CS(n) = 2n^2 - 2n + 1 equal twice the quadratic f(n) = n^2 - n - 1, plus 3. The same quadratic whose roots are the golden ratio appears again, now dressed as figurate geometry. At positions 1 through 6 the values are {1, 5, 13, 25, 41, 61} -- four primes, one perfect square (25 = 5^2), and the identity.
| n | f(n) | Note | CS(n) = 2f+3 |
|---|---|---|---|
| 1 | -1 | f(1) = -1 | 1 |
| 2 | 1 | f(2) = 1 | 5 |
| 3 | 5 | f(3) = 5 (prime) | 13 |
| 4 | 11 | f(4) = 11 (prime) | 25 = 5^2 |
| 5 | 19 | f(5) = 19 (prime) | 41 |
| 6 | 29 | f(6) = 29 (prime) | 61 |
| 7 | 41 | f(7) = 41 (prime) | 85 = 5*17 |
| 11 | 109 | f(11) = 109 (prime) | 221 = 13*17 |
At chain primes, f(n) often produces the next chain prime: f(2) = 1, f(3) = 5, f(5) = 19, f(7) = 41. The centered square maps each n to 2*f(n) + 3: so 2 maps to 5, 3 maps to 13, 5 maps to 41.
Every gap is 4n. At n = 1, 2, 3, 5, 7, 11: gaps are 4, 8, 12, 20, 28, 44.
| n | CH(n) | S(n) = 2*CH-1 | Note |
|---|---|---|---|
| 1 | 1 | 1 | Identity |
| 2 | 7 | 13 | Both prime |
| 3 | 19 = f(5) | 37 | 19 = f(5); 37 prime |
| 4 | 37 | 73 | Both prime |
| 5 | 61 | 121 = 11^2 | Prime -> square |
| 7 | 127 = 2^7-1 | 253 = 11*23 | Mersenne prime -> composite |
At n = 2, the centered hexagonal is 7 and the star number is 13. At n = 3, they are 19 = f(5) and 37 (both prime). Doubling and subtracting 1 transforms centered hexagonals into star numbers.
210 sits at position 20 = 4*5 in the triangular family and position 12 in the pentagonal family. Both positions factor into small chain primes.
Enter n. See centered square CS(n), the quadratic f(n), gap to next CS, centered hexagonal CH(n), and star number S(n).
Compute figurate numbers at position:
Try: 2 (CS=5), 3 (CS=13), 5 (CS=41), 7 (CS=85=5*17), 11 (CS=221=13*17).
| Aspect | Standard view | Ring structure |
|---|---|---|
| CS values | 1, 5, 13, 25, 41, 61 -- pattern of 2n^2-2n+1 | Four of the first six are prime (5, 13, 41, 61); 25 = 5^2 |
| Identity | CS(n) = 2n^2-2n+1, a formula | = 2*f(n) + 3: twice the golden-ratio quadratic plus 3 |
| Gaps | 4n = linear in n | At chain primes: 4, 8, 12, 20, 28, 44 |
| Star-Hex | S = 2*CH - 1, algebraic | At n=2: CH=7, S=13. At n=3: CH=19=f(5), S=37 (both prime) |
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