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%I #19 Mar 15 2022 16:02:27
%S 0,1,10,11,20,100,101,110,111,120,200,201,1000,1001,1010,1011,1020,
%T 1100,1101,1110,1111,1120,1200,1201,2000,2001,2010,2011,2020,10000,
%U 10001,10010,10011,10020,10100,10101,10110,10111,10120,10200,10201,11000,11001,11010,11011
%N Expansion of n in the p-system based on convergents to sqrt(2).
%C This is the minimal (or greedy) representation of nonnegative numbers in terms of the positive Pell numbers (A000129). - _Amiram Eldar_, Mar 12 2022
%D A. F. Horadam, Zeckendorf representations of positive and negative integers by Pell numbers, Applications of Fibonacci Numbers, Springer, Dordrecht, 1993, pp. 305-316.
%H Amiram Eldar, <a href="/A317204/b317204.txt">Table of n, a(n) for n = 0..10000</a>
%H L. Carlitz, Richard Scoville, and V. E. Hoggatt, Jr., <a href="https://www.fq.math.ca/Scanned/10-5/carlitz1.pdf">Pellian Representations</a>, The Fibonacci Quarterly, Vol. 10, No. 5 (1972), pp. 449-488.
%H Aviezri S. Fraenkel, <a href="http://www.jstor.org/stable/2321643">How to beat your Wythoff games' opponent on three fronts</a>, Amer. Math. Monthly, Vol. 89, No. 6 (1982), pp. 353-361. See Table 2.
%t 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 *)
%o (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
%Y Cf. A000129, A169683.
%Y Similar to, but different from, A014418.
%Y Similar sequences: A014417, A130310, A278038.
%K nonn,base
%O 0,3
%A _N. J. A. Sloane_, Aug 07 2018
%E More terms from _Amiram Eldar_, Mar 12 2022
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