A130310 -id:A130310 - cp's OEIS frontend

A351712 Numbers whose minimal (or greedy) Lucas representation (A130310) is palindromic.

0, 2, 6, 9, 13, 20, 24, 31, 49, 56, 64, 78, 100, 125, 136, 150, 158, 169, 201, 237, 252, 324, 342, 364, 378, 396, 404, 422, 444, 523, 581, 606, 650, 708, 845, 874, 910, 932, 961, 975, 1004, 1040, 1048, 1077, 1113, 1135, 1164, 1366, 1460, 1500, 1572, 1666, 1692, 1786

Offset: 1

Amiram Eldar, Feb 17 2022

A000211(n) = Lucas(n) + 2 is a term for all n > 2, since the representation of Lucas(n) + 2 is 10...01 with n-1 0's between the two 1's.

			The first 10 terms are:
   n  a(n) A130310(a(n))
   ---------------------
   1   0               0
   2   2               1
   3   6            1001
   4   9           10001
   5  13          100001
   6  20         1000001
   7  24         1001001
   8  31        10000001
   9  49       100000001
  10  56       100010001

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to palindromes.

Cf. A000032, A000211, A130310.
Subsequence of A054770.
Similar sequences: A002113, A006995, A014190, A094202, A331191, A351717.

lucasPalQ[n_] := Module[{s = {}, m = n, k = 1}, While[m > 0, If[m == 1, k = 1; AppendTo[s, k]; m = 0, If[m == 2, k = 0; AppendTo[s, k]; m = 0, While[LucasL[k] <= m, k++]; k--; AppendTo[s, k]; m -= LucasL[k]; k = 1]]]; PalindromeQ[IntegerDigits[Total[2^s], 2]]]; Select[Range[0, 2000], lucasPalQ]

A351714 Lucas-Niven numbers: numbers that are divisible by the number of terms in their minimal (or greedy) representation in terms of the Lucas numbers (A130310).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 14, 18, 20, 22, 24, 27, 29, 30, 32, 36, 39, 40, 42, 47, 48, 50, 54, 57, 58, 60, 64, 66, 69, 72, 76, 78, 80, 81, 84, 90, 92, 94, 96, 100, 104, 108, 120, 123, 124, 126, 129, 130, 132, 134, 135, 138, 140, 144, 152, 153, 156, 159, 160

Offset: 1

Amiram Eldar, Feb 17 2022

Numbers k such that A116543(k) | k.

			6 is a term since its minimal Lucas representation, A130310(6) = 1001, has A116543(6) = 2 1's and 6 is divisible by 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000032, A116543, A130310.
Subsequences: A351715, A351716.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351719.

lucasNivenQ[n_] := Module[{s = {}, m = n, k = 1}, While[m > 0, If[m == 1, k = 1; AppendTo[s, k]; m = 0, If[m == 2, k = 0; AppendTo[s, k]; m = 0, While[LucasL[k] <= m, k++]; k--; AppendTo[s, k]; m -= LucasL[k]; k = 1]]]; Divisible[n, Plus @@ IntegerDigits[Total[2^s], 2]]]; Select[Range[160], lucasNivenQ]

A131340 Number of 0's in minimal Lucas representation (A130310) of n.

Original entry on oeis.org

1, 0, 2, 3, 2, 2, 4, 3, 3, 3, 5, 4, 4, 4, 4, 3, 3, 6, 5, 5, 5, 5, 4, 4, 5, 4, 4, 4, 7, 6, 6, 6, 6, 5, 5, 6, 5, 5, 5, 6, 5, 5, 5, 5, 4, 4, 8, 7, 7, 7, 7, 6, 6, 7, 6, 6, 6, 7, 6, 6, 6, 6, 5, 5, 7, 6, 6, 6, 6, 5, 5, 6, 5, 5, 5, 9, 8, 8, 8, 8, 7, 7, 8, 7, 7, 7, 8

Offset: 1

Casey Mongoven, Jun 29 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ron Knott, Using the Fibonacci numbers to represent whole numbers.

Cf. A131341, A000032, A102364.

a[n_] := Module[{s = {}, m = n, k = 1}, While[m > 0, If[m == 1, k = 1; AppendTo[s, k]; m = 0, If[m == 2, k = 0; AppendTo[s, k]; m = 0, While[LucasL[k] <= m, k++]; k--; AppendTo[s, k]; m -= LucasL[k]; k = 1]]]; Total[1 - IntegerDigits[Total[2^s], 2]]]; Array[a, 100] (* Amiram Eldar, Jul 05 2025 *)

More terms from Amiram Eldar, Jul 05 2025

A133770 Number of runs (of equal bits) in the minimal Lucas binary (A130310) representation of n.

Original entry on oeis.org

2, 1, 2, 2, 4, 3, 2, 4, 3, 4, 2, 4, 3, 4, 4, 6, 5, 2, 4, 3, 4, 4, 6, 5, 4, 6, 5, 6, 2, 4, 3, 4, 4, 6, 5, 4, 6, 5, 6, 4, 6, 5, 6, 6, 8, 7, 2, 4, 3, 4, 4, 6, 5, 4, 6, 5, 6, 4, 6, 5, 6, 6, 8, 7, 4, 6, 5, 6, 6, 8, 7, 6, 8, 7, 8, 2, 4, 3, 4, 4, 6, 5, 4, 6, 5, 6, 4, 6, 5, 6, 6, 8, 7, 4, 6, 5, 6, 6, 8, 7, 6, 8, 7, 8, 4

Offset: 1

Casey Mongoven, Sep 23 2007

nonn

			A130310(17)=101001 because 11 + 4 + 2 = 17 (a sum of Lucas numbers); this representation has five runs: 1,0,1,00,1. So a(17)=5.

Zeckendorf, E., Représentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liège 41, 179-182, 1972.

Casey Mongoven, Table of n, a(n) for n = 1..199
Ron Knott, Using Powers of Phi to represent Integers.

Cf. A133771, A130310.

A178245 If one lines up the columns of the binary representations in A130310, the minimal (or "greedy") Lucas representation of n, this sequence gives the column number (starting with 0) of the starting point of the blocks of 1s which occur.

Original entry on oeis.org

1, 0, 2, 3, 1, 0, 4, 1, 0, 2, 5, 1, 0, 2, 3, 1, 0, 6, 1, 0, 2, 3, 1, 0, 4, 1, 0, 2, 7, 1, 0, 2, 3, 1, 0, 4, 1, 0, 2, 5, 1, 0, 2, 3, 1, 0, 8, 1, 0, 2, 3, 1, 0, 4, 1, 0, 2, 5, 1, 0, 2, 3, 1, 0, 6, 1, 0, 2, 3, 1, 0, 4, 1, 0, 2, 9

Offset: 1

Casey Mongoven, May 23 2010

nonn

C. Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly 33 (1995) 3-8.

Cf. A130310, A035614.

A278038 Binary vectors not containing three consecutive 1's; or, representation of n in the tribonacci base.

Original entry on oeis.org

0, 1, 10, 11, 100, 101, 110, 1000, 1001, 1010, 1011, 1100, 1101, 10000, 10001, 10010, 10011, 10100, 10101, 10110, 11000, 11001, 11010, 11011, 100000, 100001, 100010, 100011, 100100, 100101, 100110, 101000, 101001, 101010, 101011, 101100, 101101, 110000, 110001, 110010, 110011, 110100, 110101, 110110, 1000000

Offset: 0

N. J. A. Sloane, Nov 16 2016

These are the nonnegative numbers written in the tribonacci numbering system.

			The tribonacci numbers (as in A000073(n), for n >= 3) are 1, 2, 4, 7, 13, 24, 44, 81, ... In terms of this base, 7 is written 1000, 8 is 1001, 11 is 1100, 12 is 1101, 13 is 10000, etc. Zero is 0.

N. J. A. Sloane, Table of n, a(n) for n = 0..25280
L. Carlitz, Richard Scoville, and V. E. Hoggatt, Jr., Fibonacci Representations of Higher Order, Part 2, The Fibonacci Quarterly, Vol. 10, No. 1 (1972), pp. 43-69, 94.
F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.
Eric Duchêne and Michel Rigo, A morphic approach to combinatorial games: the Tribonacci case. RAIRO - Theoretical Informatics and Applications, 42, 2008, pp 375-393. See Table 2. [Also available from Numdam]
V. E. Hoggatt, Jr. and Marjorie Bicknell-Johnson, Lexicographic Ordering and Fibonacci Representations, The Fibonacci Quarterly, Vol. 20, No. 3 (1982), pp. 193-218.
Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.

Cf. A000073, A080843 (tribonacci word, tribonacci tree).
See A003726 for the decimal representations of these binary strings.
Similar sequences: A014417 (Fibonacci), A130310 (Lucas).

# maximum index in A73 such that A73 <= n.
A73floorIdx := proc(n)
    local k ;
    for k from 3 do
        if A000073(k) = n then
            return k ;
        elif A000073(k) > n then
            return k -1 ;
        end if ;
    end do:
end proc:
A278038 := proc(n)
    local k,L,nres ;
    if n = 0 then
        0;
    else
        k := A73floorIdx(n) ;
        L := [1] ;
        nres := n-A000073(k) ;
        while k >= 4 do
            k := k-1 ;
            if nres >= A000073(k) then
                L := [1,op(L)] ;
                nres := nres-A000073(k) ;
            else
                L := [0,op(L)] ;
            end if ;
        end do:
        add( op(i,L)*10^(i-1),i=1..nops(L)) ;
    end if;
end proc:
seq(A278038(n),n=0..40) ; # R. J. Mathar, Jun 08 2022

t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; a[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[t[k] <= m, k++]; k--; AppendTo[s, k]; m -= t[k]; k = 1]; FromDigits @ IntegerDigits[Total[2^(s - 1)], 2]]; Array[a, 100, 0] (* Amiram Eldar, Mar 04 2022 *)

A116543 Number of terms in greedy representation of n in terms of the Lucas numbers.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 3, 1, 2, 2, 2, 2, 3, 3, 2, 3, 3, 3, 1, 2, 2, 2, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 4, 4, 1, 2, 2, 2, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 4, 4, 2, 3, 3, 3, 3, 4, 4, 3, 4, 4, 4, 1, 2, 2, 2, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 4, 4, 2, 3, 3, 3, 3, 4, 4, 3, 4, 4, 4, 2

Offset: 0

James E Davis, Mar 28 2006, Jun 07 2006

I have been studying A007895 and similar sequences and created this sequence as an analog of A007895 for the Lucas sequence (A000032).

			a(12)=2 because 12=11+1.

Clark Kimberling, Table of n, a(n) for n = 0..10000
Ron Knott, Using the Fibonacci numbers to represent whole numbers.

Cf. A000032, A007895, A007953, A130310, A131343.

s = Reverse[Sort[Table[LucasL[n - 1], {n, 1, 22}]]];
t = Map[Length[Select[Reap[FoldList[(Sow[Quotient[#1, #2]]; Mod[#1, #2]) &, #, s]][[2,1]], # > 0 &]] &, Range[1000]] (* Peter J. C. Moses, Oct 18 2012 *)

Let L(n) = max(Lucas numbers < n). Then a(0) = 0, a(n) = 1 + a(n-L(n)).

a(n) = A007953(A130310(n)). - Amiram Eldar, Feb 17 2022

Edited by N. J. A. Sloane, Aug 10 2007

a(0) added by Amiram Eldar, Feb 17 2022

A054770 Numbers that are not the sum of distinct Lucas numbers 1,3,4,7,11, ... (A000204).

Original entry on oeis.org

2, 6, 9, 13, 17, 20, 24, 27, 31, 35, 38, 42, 46, 49, 53, 56, 60, 64, 67, 71, 74, 78, 82, 85, 89, 93, 96, 100, 103, 107, 111, 114, 118, 122, 125, 129, 132, 136, 140, 143, 147, 150, 154, 158, 161, 165, 169, 172, 176, 179, 183, 187, 190, 194, 197, 201, 205, 208, 212

Offset: 1

Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), May 28 2000

Alternatively, Lucas representation of n includes L_0 = 2. - Fred Lunnon, Aug 25 2001
Conjecture: this is the sequence of numbers for which the base phi representation includes phi itself, where phi = (1 + sqrt(5))/2 = the golden ratio. Example: let r = phi; then 6 = r^3 + r + r^(-4). - Clark Kimberling, Oct 17 2012
This conjecture is proved in my paper 'Base phi representations and golden mean beta-expansions', using the formula by Wilson/Agol/Carlitz et al. - Michel Dekking, Jun 25 2019
Numbers whose minimal Lucas representation (A130310) ends with 1. - Amiram Eldar, Jan 21 2023

G. C. Greubel, Table of n, a(n) for n = 1..5000
L. Carlitz, R. Scoville, and V. E. Hoggatt, Jr., Lucas representations, Fibonacci Quart. 10 (1972), 29-42, 70, 112.
Weiru Chen and Jared Krandel, Interpolating Classical Partitions of the Set of Positive Integers, arXiv:1810.11938 [math.NT], 2018. See sequence D2 p. 4.
Michel Dekking, Base phi representations and golden mean beta-expansions, arXiv:1906.08437 [math.NT], 2019.
Jared Krandel and Weiru Chen, Interpolating classical partitions of the set of positive integers, The Ramanujan Journal (2020).

Complement of A063732.

Cf. A003263, A003622, A022342, A130310.

[Floor(n*(Sqrt(5)+5)/2)-1: n in [1..60]]; // Vincenzo Librandi, Oct 30 2018

A054770 := n -> floor(n*(sqrt(5)+5)/2)-1;

Complement[Range[220],Total/@Subsets[LucasL[Range[25]],5]] (* Harvey P. Dale, Feb 27 2012 *)
Table[Floor[n (Sqrt[5] + 5) / 2] - 1, {n, 60}] (* Vincenzo Librandi, Oct 30 2018 *)

PARI
```
a(n)=floor(n*(sqrt(5)+5)/2)-1
```

from math import isqrt
def A054770(n): return (n+isqrt(5*n**2)>>1)+(n<<1)-1 # Chai Wah Wu, Aug 17 2022

a(n) = floor(((5+sqrt(5))/2)*n)-1 (conjectured by David W. Wilson; proved by Ian Agol (iagol(AT)math.ucdavis.edu), Jun 08 2000)
a(n) = A000201(n) + 2*n - 1. - Michel Dekking, Sep 07 2017
G.f.: x*(x+1)/(1-x)^2 + Sum_{i>=1} (floor(i*phi)*x^i), where phi = (1 + sqrt(5))/2. - Iain Fox, Dec 19 2017
Ian Agol tells me that David W. Wilson's formula is proved in the Carlitz, Scoville, Hoggatt paper 'Lucas representations'. See Equation (1.12), and use A(A(n))+n = B(n)+n-1 = A(n)+2*n-1, the well known formulas for the lower Wythoff sequence A = A000201, and the upper Wythoff sequence B = A001950. - Michel Dekking, Jan 04 2018

More terms from James Sellers, May 28 2000

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

Original entry on oeis.org

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

N. J. A. Sloane, Aug 07 2018

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

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

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.

Cf. A000129, A169683.
Similar to, but different from, A014418.
Similar sequences: A014417, A130310, A278038.

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 *)

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

More terms from Amiram Eldar, Mar 12 2022

A130311 Maximal (or "lazy") Lucas representation of n. Binary system for representing integers using Lucas numbers (A000032) as a base.

Original entry on oeis.org

0, 10, 1, 11, 110, 101, 111, 1011, 1110, 1101, 1111, 10110, 10101, 10111, 11011, 11110, 11101, 11111, 101011, 101110, 101101, 101111, 110110, 110101, 110111, 111011, 111110, 111101, 111111, 1010110, 1010101, 1010111, 1011011, 1011110, 1011101, 1011111, 1101011, 1101110

Offset: 0

Casey Mongoven, May 21 2007; corrected Mar 23 2008

			a(7) = 1110 because 4 + 3 + 1 = 8.
a(8) = 1101 because 4 + 3 + 2 = 9.

Edouard Zeckendorf, Représentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liège, Vol. 41 (1972), pp. 179-182.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Ron Knott, Using Fibonacci Numbers to Represent Whole Numbers.

Cf. A000032, A104326, A130310, A131343.

lazy = Select[IntegerDigits[Range[10^2], 2], SequenceCount[#, {0, 0}] == 0 &]; t = Total[# * Reverse@LucasL[Range[0, Length[#] - 1]]] & /@ lazy; Join[{0}, FromDigits /@ lazy[[TakeWhile[Flatten[FirstPosition[t, #] & /@ Range[Max[t]]], NumberQ]]]] (* Amiram Eldar, Feb 17 2022 *)

a(0) and more terms from Amiram Eldar, Feb 17 2022

cp's OEIS Frontend

A351712 Numbers whose minimal (or greedy) Lucas representation (A130310) is palindromic.

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A351714 Lucas-Niven numbers: numbers that are divisible by the number of terms in their minimal (or greedy) representation in terms of the Lucas numbers (A130310).

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A131340 Number of 0's in minimal Lucas representation (A130310) of n.

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A133770 Number of runs (of equal bits) in the minimal Lucas binary (A130310) representation of n.

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A178245 If one lines up the columns of the binary representations in A130310, the minimal (or "greedy") Lucas representation of n, this sequence gives the column number (starting with 0) of the starting point of the blocks of 1s which occur.

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A278038 Binary vectors not containing three consecutive 1's; or, representation of n in the tribonacci base.

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A116543 Number of terms in greedy representation of n in terms of the Lucas numbers.

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A054770 Numbers that are not the sum of distinct Lucas numbers 1,3,4,7,11, ... (A000204).

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