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Pattern

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Recursive Sequences

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Geometric Progression

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G(n) = G(n-1) * r with base case G(1) = a

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Fibonacci Sequence

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F(n) = F(n - 1) + F(n - 2) with base cases F(0) = 0 and F(1) = 1

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Bell Numbers

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B(n) = \sum_{k=0}^{n-1} \binom{n-1}{k} B(k) with base case B(0) = 1

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Padovan Sequence

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P(n) = P(n-2) + P(n-3) with base cases P(0) = P(1) = P(2) = 1

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Pell Numbers

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P(n) = 2 * P(n-1) + P(n-2) with base cases P(0) = 0 and P(1) = 1

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Catalan Numbers

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C(n) = \frac{(2n)!}{(n+1)!n!} for n≥0 (recursive formula can be defined in terms of earlier values as well)

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Harmonic Sequence

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H(n) = 1/n with base case H(1) = 1

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Arithmetic Progression

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A(n) = A(n-1) + d with base case A(1) = a

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Lucas Numbers

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L(n) = L(n-1) + L(n-2) with base cases L(0) = 2 and L(1) = 1

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Factorial

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n! = n * (n-1)! with base case 0! = 1

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