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

0,3

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.

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 *)

Prog

(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

Crossrefs

Cf. A000129, A169683.

Similar to, but different from, A014418.

Similar sequences: A014417, A130310, A278038.

Sequence in context: A306960 A178361 A014418 * A089591 A064039 A255536

Adjacent sequences: A317201 A317202 A317203 * A317205 A317206 A317207

Keyword

nonn,base

Author

N. J. A. Sloane, Aug 07 2018

Extensions

More terms from Amiram Eldar, Mar 12 2022

Status

approved