A317204 Expansion of n in the p-system based on convergents to sqrt(2).

0, 1, 10, 11, 20, 100, 101, 110, 111, 120, 200, 201, 1000, 1001, 1010, 1011, 1020, 1100, 1101, 1110, 1111, 1120, 1200, 1201, 2000, 2001, 2010, 2011, 2020, 10000, 10001, 10010, 10011, 10020, 10100, 10101, 10110, 10111, 10120, 10200, 10201, 11000, 11001, 11010, 11011

Offset: 0

Author

N. J. A. Sloane, Aug 07 2018

Keywords

• nonn
• base

Comments

This is the minimal (or greedy) representation of nonnegative numbers in terms of the positive Pell numbers (A000129). - Amiram Eldar, Mar 12 2022

References

• A. F. Horadam, Zeckendorf representations of positive and negative integers by Pell numbers, Applications of Fibonacci Numbers, Springer, Dordrecht, 1993, pp. 305-316.

Links

• Amiram Eldar, Table of n, a(n) for n = 0..10000
• L. Carlitz, Richard Scoville, and V. E. Hoggatt, Jr., Pellian Representations, The Fibonacci Quarterly, Vol. 10, No. 5 (1972), pp. 449-488.
• Aviezri S. Fraenkel, How to beat your Wythoff games' opponent on three fronts, Amer. Math. Monthly, Vol. 89, No. 6 (1982), pp. 353-361. See Table 2.

Crossrefs

Cf. A000129, A169683.

Similar to, but different from, A014418.

Similar sequences: A014417, A130310, A278038.

Programs

Mathematica

pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; pellp[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[pell[k] <= m, k++]; k--; AppendTo[s, k]; m -= pell[k]; k = 1]; FromDigits @ IntegerDigits[Total[3^(s - 1)], 3]]; Array[pellp, 50, 0] (* Amiram Eldar, Mar 12 2022 *)

PARI

a(n) = { my (p=[1,2]); for (k=2, oo, if (n<=p[k], my (v=0, d); while (n, v+=10^k*d=n\p[k]; n-=d*p[k]; k--); return (v/10), p = concat(p, 2*p[k]+p[k-1]))) } \\ Rémy Sigrist, Mar 12 2022

Extensions

More terms from Amiram Eldar, Mar 12 2022