Pell Number

The numbers obtained by the $U_n$s in the Lucas Sequence with $P=2$ and $Q=-1$. They and the Pell-Lucas numbers (the $V_n$s in the Lucas Sequence) satisfy the recurrence relation

$$P_n=2P_{n-1}+P_{n-2}.$$ (1)

Using $P_i$ to denote a Pell number and $Q_i$ to denote a Pell-Lucas number,

$$P_{m+n}=P_mP_{n+1}+P_{m-1}P_n$$ (2)

$$P_{m+n}=2P_mQ_n-(-1)^nP_{m-n},$$ (3)

$$P_{2^tm}=P_m(2Q_m)(2Q_{2m})(2Q_{4m})\cdots(2Q_{2^{t-1}m})$$ (4)

$${Q_m}^2=2{P_m}^2+(-1)^m$$ (5)

$$Q_{2m}=2{Q_m}^2-(-1)^m.$$ (6)

The Pell numbers have $P_0=0$ and $P_1=1$ and are 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, ... (Sloane's A000129). The Pell-Lucas numbers have $Q_0=2$ and $Q_1=2$ and are 2, 2, 6, 14, 34, 82, 198, 478, 1154, 2786, 6726, ... (Sloane's A002203).

The only Triangular Pell number is 1 (McDaniel 1996).

See also Brahmagupta Polynomial, Pell Polynomial

References

McDaniel, W. L. ``Triangular Numbers in the Pell Sequence.'' Fib. Quart. 34, 105-107, 1996.

Sloane, N. J. A. Sequences A000129/M1413 and A002203/M0360 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.