Number Sequences
Trinity implements a rich collection of integer sequences in src/sacred/sequences.zig. Each sequence has deep connections to the golden ratio, ternary arithmetic, or combinatorial structures. This page provides mathematical definitions, key properties, OEIS references, and connections to Trinity's architecture.
Metallic Means Family
The golden ratio phi is the first member of an infinite family of metallic means -- algebraic numbers that generalize the self-similarity property phi^2 = phi + 1.
Definition (Metallic Means)
The k-th metallic mean delta_k is the positive root of x^2 - k*x - 1 = 0:
delta_k = (k + sqrt(k^2 + 4)) / 2
| k | Name | Value | Companion Sequence |
|---|---|---|---|
| 1 | Golden (phi) | 1.6180339887... | Fibonacci |
| 2 | Silver (delta_S) | 2.4142135623... = 1 + sqrt(2) | Pell |
| 3 | Bronze (delta_B) | 3.3027756377... | "Tribonacci-like" |
Properties Shared by All Metallic Means
delta_k^2 = k * delta_k + 1 (defining equation)
1/delta_k = delta_k - k (reciprocal relation)
delta_k + 1/delta_k = sqrt(k^2 + 4) (sum with reciprocal)
For the golden mean (k=1): delta_1 + 1/delta_1 = sqrt(5), and delta_1^2 + 1/delta_1^2 = 3 = TRINITY.
Fibonacci Numbers (OEIS A000045)
Definition
F(0) = 0, F(1) = 1, F(n) = F(n-1) + F(n-2)
Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...
Binet's Formula
F(n) = (phi^n - psi^n) / sqrt(5)
where phi = (1 + sqrt(5))/2, psi = (1 - sqrt(5))/2
Key Properties
| Property | Formula | Significance |
|---|---|---|
| Limit ratio | lim F(n+1)/F(n) = phi | Theorem 5 (see Proofs) |
| GCD | gcd(F(m), F(n)) = F(gcd(m,n)) | Fibonacci numbers form a divisibility lattice |
| Sum of squares | F(n)^2 + F(n+1)^2 = F(2n+1) | Connects to Pythagorean triples |
| Zeckendorf | Every positive integer has unique Fibonacci representation | Basis for Fibonacci coding |
| F(4) = 3 | TRINITY appears at index 4 | phi^4 = 3*phi + 2 |
Connection to Trinity
- F(4) = 3 = TRINITY: The fourth Fibonacci number is the ternary base
- F(7) = 13 = TRYTE_MAX: Maximum value of a balanced ternary tryte (3 trits: 1+3+9)
- Fibonacci encoding: Every non-negative integer can be uniquely represented as a sum of non-consecutive Fibonacci numbers (Zeckendorf's theorem, 1972). This connects to balanced ternary representation via a different basis
Reference: Koshy, T. Fibonacci and Lucas Numbers with Applications. Wiley, 2001.
Lucas Numbers (OEIS A000032)
Definition
L(0) = 2, L(1) = 1, L(n) = L(n-1) + L(n-2)
Sequence: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ...
Closed Form
L(n) = phi^n + psi^n = phi^n + (-1/phi)^n
Key Properties
| Property | Formula |
|---|---|
| Fibonacci relation | L(n) = F(n-1) + F(n+1) |
| Product | F(n) * L(n) = F(2n) |
| Identity | L(n)^2 - 5F(n)^2 = 4(-1)^n |
| L(2) = 3 | TRINITY is the second Lucas number |
Trinity Connections
- L(2) = 3 = TRINITY: The most direct appearance of 3 in the golden ratio family
- L(n) = phi^n + 1/phi^n (for even n): This generalizes the Trinity Identity. At n=2: phi^2 + 1/phi^2 = 3
- L(10) = 123: Used as a magic constant in several Trinity modules
Pell Numbers (OEIS A000129)
Definition
P(0) = 0, P(1) = 1, P(n) = 2*P(n-1) + P(n-2)
Sequence: 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, ...
Silver Ratio Connection
lim P(n+1)/P(n) = delta_S = 1 + sqrt(2) = 2.41421356...
The Pell numbers play the same role for the silver ratio (delta_S) that Fibonacci numbers play for the golden ratio (phi). They provide the best rational approximations to sqrt(2):
P(1)/P(0) → undefined
P(2)/P(1) = 2/1 = 2.000
P(3)/P(2) = 5/2 = 2.500
P(4)/P(3) = 12/5 = 2.400
P(5)/P(4) = 29/12 = 2.4166...
P(6)/P(5) = 70/29 = 2.4137... (converging to sqrt(2) + 1)
Connection to Trinity
- The silver ratio satisfies delta_S^2 = 2*delta_S + 1, analogous to phi^2 = phi + 1
- However, delta_S^2 + 1/delta_S^2 = 6 ≠ 3. The Trinity Identity is unique to the golden ratio
Reference: Sloane, N. J. A. "Sequence A000129." The On-Line Encyclopedia of Integer Sequences.
Tribonacci Numbers (OEIS A000073)
Definition
T(0) = 0, T(1) = 0, T(2) = 1, T(n) = T(n-1) + T(n-2) + T(n-3)
Sequence: 0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, ...
Tribonacci Constant (Tetranacci Ratio)
lim T(n+1)/T(n) = tau_3 ≈ 1.8392867552141612
This is the real root of x^3 = x^2 + x + 1 (the tribonacci polynomial).
Properties
tau_3^3 = tau_3^2 + tau_3 + 1 (defining equation)
tau_3 is the unique real root > 1 of x^3 - x^2 - x - 1 = 0
Connection to Trinity
The tribonacci recurrence sums three previous terms, making it a natural ternary generalization of Fibonacci. Note that T(6) = 7, T(7) = 13 = TRYTE_MAX, and T(10) = 81 = 3^4.
Padovan Sequence (OEIS A000931)
Definition
P(0) = 1, P(1) = 0, P(2) = 0, P(n) = P(n-2) + P(n-3)
Sequence: 1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, ...
Plastic Number
lim P(n+1)/P(n) = rho ≈ 1.3247179572447460
The plastic number rho is the real root of x^3 = x + 1. It was named by Dom Hans van der Laan (1967), who used it in architecture as an alternative to the golden ratio.
Properties
rho^3 = rho + 1 (defining equation)
rho is the smallest Pisot-Vijayaraghavan number > 1
rho^(3n) approaches integers rapidly
Connection to Trinity
The plastic number is the smallest PV number -- an algebraic integer greater than 1 whose conjugates all lie strictly inside the unit circle. PV numbers have deep connections to:
- Quasicrystals (Penrose tilings use phi; Padovan-based tilings use rho)
- Number theory (Salem numbers, Lehmer's conjecture)
Perrin Sequence (OEIS A001608)
Definition
R(0) = 3, R(1) = 0, R(2) = 2, R(n) = R(n-2) + R(n-3)
Sequence: 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, 29, ...
Primality Connection
A remarkable property: if p is prime, then p divides R(p). The converse is almost true -- Perrin pseudoprimes are extremely rare (only 17 below 10^9).
Connection to Trinity
- R(0) = 3 = TRINITY: The Perrin sequence begins with Trinity
- Same limiting ratio as Padovan (plastic number rho)
Catalan Numbers (OEIS A000108)
Definition
C_n = (2n)! / ((n+1)! * n!) = binom(2n, n) / (n+1)
Sequence: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, ...
Combinatorial Interpretations
Catalan numbers count (among many other things):
- Binary trees with n+1 leaves
- Balanced parenthesizations of n pairs
- Triangulations of a convex (n+2)-gon
- Dyck paths of length 2n (lattice paths that never dip below the x-axis)
- Non-crossing partitions of {1, ..., n}
Generating Function
C(x) = (1 - sqrt(1 - 4x)) / (2x) = sum C_n * x^n
Asymptotic Growth
C_n ~ 4^n / (sqrt(pi) * n^(3/2))
Connection to Trinity
Catalan numbers count the number of ways to structure recursive ternary computations:
- Binary tree structures for organizing VSA bind/bundle operations
- Non-crossing partitions relate to the algebraic structure of bundle (Theorem 9)
- The generating function involves sqrt, connecting to the golden ratio's sqrt(5) definition
Reference: Stanley, R. P. Catalan Numbers. Cambridge University Press, 2015.
Bernoulli Numbers (OEIS A027642)
Definition
Bernoulli numbers B_n are defined by the generating function:
x / (e^x - 1) = sum B_n * x^n / n!
First values: B_0 = 1, B_1 = -1/2, B_2 = 1/6, B_4 = -1/30, B_6 = 1/42
(All odd Bernoulli numbers except B_1 are zero.)
Connection to Riemann Zeta
zeta(2n) = (-1)^(n+1) * B_{2n} * (2*pi)^(2n) / (2 * (2n)!)
This gives:
| n | zeta(2n) | Formula |
|---|---|---|
| 1 | pi^2/6 | B_2 = 1/6 |
| 2 | pi^4/90 | B_4 = -1/30 |
| 3 | pi^6/945 | B_6 = 1/42 |
Connection to Trinity
- B_2 = 1/6 and zeta(2) = pi^2/6: the denominator 6 = 2 * 3 = 2 * TRINITY
- Bernoulli numbers appear in the Euler-Maclaurin summation formula, which connects discrete sums to integrals -- the mathematical bridge between ternary (discrete) and continuous computation
Euler Numbers (OEIS A122045)
Definition
Euler numbers E_n are defined by:
1 / cosh(x) = sech(x) = sum E_n * x^n / n!
First values: E_0 = 1, E_2 = -1, E_4 = 5, E_6 = -61, E_8 = 1385
(All odd Euler numbers are zero.)
Tangent Numbers
The closely related tangent numbers (coefficients of tan(x)) are:
T_1 = 1, T_3 = 2, T_5 = 16, T_7 = 272
Connection to Trinity
- |E_0| = 1, |E_2| = 1, |E_4| = 5: the sequence 1, 1, 5 satisfies 1 + 1 + ... = building blocks of ternary arithmetic
- Euler numbers count alternating permutations (zigzag permutations), connecting to the structure of cyclic permutations in VSA (Theorem 10)
Summary Table
| Sequence | Recurrence | Limiting Ratio | OEIS | Trinity Connection |
|---|---|---|---|---|
| Fibonacci | F(n) = F(n-1) + F(n-2) | phi = 1.618... | A000045 | F(4) = 3 = TRINITY |
| Lucas | L(n) = L(n-1) + L(n-2) | phi = 1.618... | A000032 | L(2) = 3, phi^2 + 1/phi^2 = 3 |
| Pell | P(n) = 2P(n-1) + P(n-2) | 1+sqrt(2) = 2.414... | A000129 | Silver ratio analog |
| Tribonacci | T(n) = T(n-1)+T(n-2)+T(n-3) | 1.839... | A000073 | 3-term = ternary recurrence |
| Padovan | P(n) = P(n-2) + P(n-3) | rho = 1.325... | A000931 | Smallest PV number |
| Perrin | R(n) = R(n-2) + R(n-3) | rho = 1.325... | A001608 | R(0) = 3 = TRINITY |
| Catalan | C_n = binom(2n,n)/(n+1) | 4^n growth | A000108 | Binary tree counting |
| Bernoulli | Generating function | -- | A027642 | zeta(2n), Euler-Maclaurin |
| Euler | Generating function | -- | A122045 | Alternating permutations |
Numerical Verification
const std = @import("std");
const math = std.math;
const PHI: f64 = (1.0 + math.sqrt(5.0)) / 2.0;
test "Fibonacci: F(4) = 3 = TRINITY" {
const fib = [_]u64{ 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 };
try std.testing.expectEqual(@as(u64, 3), fib[4]);
}
test "Lucas: L(2) = 3 = TRINITY" {
const lucas = [_]u64{ 2, 1, 3, 4, 7, 11, 18, 29, 47, 76 };
try std.testing.expectEqual(@as(u64, 3), lucas[2]);
}
test "Lucas closed form: L(n) = phi^n + psi^n" {
const psi = (1.0 - math.sqrt(5.0)) / 2.0;
var n: usize = 0;
const lucas = [_]u64{ 2, 1, 3, 4, 7, 11, 18, 29, 47, 76 };
while (n < 10) : (n += 1) {
const phi_n = math.pow(f64, PHI, @floatFromInt(n));
const psi_n = math.pow(f64, psi, @floatFromInt(n));
const computed = phi_n + psi_n;
try std.testing.expectApproxEqAbs(
@as(f64, @floatFromInt(lucas[n])),
computed,
0.001,
);
}
}
test "Pell: ratio converges to silver ratio" {
var a: f64 = 0;
var b: f64 = 1;
var i: usize = 0;
while (i < 30) : (i += 1) {
const temp = b;
b = 2 * b + a;
a = temp;
}
const silver = 1.0 + math.sqrt(2.0);
try std.testing.expectApproxEqAbs(b / a, silver, 1e-10);
}
test "Golden uniqueness: only phi gives identity = 3" {
// phi^2 + 1/phi^2 = 3
const golden = PHI * PHI + 1.0 / (PHI * PHI);
try std.testing.expectApproxEqAbs(golden, 3.0, 1e-10);
// Silver: delta_S^2 + 1/delta_S^2 = 6, NOT 3
const silver = 1.0 + math.sqrt(2.0);
const silver_sum = silver * silver + 1.0 / (silver * silver);
try std.testing.expectApproxEqAbs(silver_sum, 6.0, 1e-10);
// Bronze: delta_B^2 + 1/delta_B^2 = 11, NOT 3
const bronze = (3.0 + math.sqrt(13.0)) / 2.0;
const bronze_sum = bronze * bronze + 1.0 / (bronze * bronze);
try std.testing.expectApproxEqAbs(bronze_sum, 11.0, 1e-10);
}
The last test demonstrates that the Trinity Identity phi^2 + 1/phi^2 = 3 is unique to the golden ratio. No other metallic mean produces 3.
Compute with TRI CLI
tri fib 30 # F(30) = 832040
tri lucas 10 # L(10) = 123
tri math golden-function # Pellis 2025: G(x) = phi^x + phi^(-x)
tri math-compare # Side-by-side comparison table
tri math nuclear # Nuclear Fibonacci shell stability
References
- Koshy, T. Fibonacci and Lucas Numbers with Applications. Wiley, 2001.
- Graham, R. L., Knuth, D. E., and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley, 2nd edition, 1994.
- Stanley, R. P. Catalan Numbers. Cambridge University Press, 2015.
- Sloane, N. J. A. The On-Line Encyclopedia of Integer Sequences. https://oeis.org/.
- Zeckendorf, E. "Representation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas." Bulletin de la Societe Royale des Sciences de Liege 41, pp. 179--182, 1972.
- van der Laan, H. Le Nombre Plastique: Quinze Lecons sur l'Ordonnance Architectonique. Brill, 1960.
- Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers. Oxford University Press, 6th edition, 2008.
- Ireland, K. and Rosen, M. A Classical Introduction to Modern Number Theory. Springer, 2nd edition, 1990.
phi^2 + 1/phi^2 = 3 = TRINITY