A104326 -id:A104326 - cp's OEIS frontend

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

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

A350215 A048715, written in binary.

Original entry on oeis.org

0, 1, 10, 100, 1000, 1001, 10000, 10001, 10010, 100000, 100001, 100010, 100100, 1000000, 1000001, 1000010, 1000100, 1001000, 1001001, 10000000, 10000001, 10000010, 10000100, 10001000, 10001001, 10010000, 10010001, 10010010, 100000000, 100000001, 100000010

Offset: 0

A.H.M. Smeets, Dec 19 2021

Narayana weighted representation of n (the top version).

a(n) equals binary representation of m, if and only if A350311(m) = n and for all k > m A350311(k) > n.

A.H.M. Smeets, The design of greedy number representations

Cf. A000930, A048715, A350311, A350312.

Fibonacci representations: A014417 (Zeckendorf), A104326 (dual Zeckendorf).

bin[n_] := FromDigits[IntegerDigits[n, 2]]; q[n_] := BitAnd[n, 6*n] == 0; bin /@ Select[Range[0, 250], q] (* Amiram Eldar, Jan 27 2022 *)

def c(b): return not "11" in b and not "101" in b
def auptod(digits):
    return [int(b) for b in (bin(k)[2:] for k in range(2**digits)) if c(b)]
print(auptod(9)) # Michael S. Branicky, Dec 20 2021

Regular expression 0|(1000*)*10*.

A130312 Each Fibonacci number F(n) appears F(n) times.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34

Offset: 1

Edwin F. Sampang, May 21 2007

Also n-1-s(n-1), where s(n) is the length of the longest proper suffix of p, the length-n prefix of the infinite Fibonacci word (A003849), that appears twice in p. - Jeffrey Shallit, Mar 20 2017
a(n+1) = the least period of the length-n prefix of the infinite Fibonacci word (A003849).  A period of a length-n word x is an integer p, 1 <= p <= n such that x[i] = x[i+p] for 1 <= i <= n-p. - Jeffrey Shallit, May 23 2020
a(n) is the largest term in dual Zeckendorf representation of n-1 (A104326), for n >= 2. - Amiram Eldar, Aug 09 2024

			As triangle:
   1;
   1;
   2,  2;
   3,  3,  3;
   5,  5,  5,  5,  5;
   8,  8,  8,  8,  8,  8,  8,  8;
  13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13;
  ...

Cf. A000045, A003849, A072649, A104326.

T:= n-> (f-> f$f)((<<0|1>, <1|1>>^n)[1,2]):
seq(T(n), n=1..10);  # Alois P. Heinz, Nov 23 2024

Flatten[Table[#,{#}]&/@Fibonacci[Range[10]]] (* Harvey P. Dale, Apr 18 2012 *)

a(n) = my(m=0); until(fibonacci(m)>n, m++); fibonacci(m-2); \\ Michel Marcus, Nov 26 2022

from itertools import islice
def A130312_gen(): # generator of terms
    a, b = 1, 1
    while True:
        yield from (a,)*a
        a, b = b, a+b
A130312_list = list(islice(A130312_gen(),20)) # Chai Wah Wu, Oct 13 2022

def A130312(n):
    a, b = 0, 1
    while (c:=a+b) <= n: a, b = b, c
    return a # Chai Wah Wu, Nov 23 2024

a(n) = A000045(A072649(n)). - Michel Marcus, Aug 03 2022

More terms from Harvey P. Dale, Apr 18 2012

A339212 Dual-Zeckendorf self numbers: numbers not of the form k + A112310(k).

Original entry on oeis.org

1, 4, 8, 10, 14, 17, 19, 28, 31, 33, 39, 41, 50, 53, 55, 59, 63, 66, 68, 74, 76, 85, 88, 90, 97, 106, 109, 111, 115, 119, 122, 124, 130, 132, 141, 144, 146, 153, 156, 158, 164, 166, 175, 178, 180, 187, 196, 199, 201, 205, 209, 212, 214, 220, 222, 231, 234, 236

Offset: 1

Amiram Eldar, Nov 27 2020

Analogous to self numbers (A003052) using the dual Zeckendorf representation (A104326) instead of decimal expansion.

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. L. Brown, Jr., A new characterization of the Fibonacci numbers, Fibonacci Quarterly, Vol. 3, No. 1 (1965) pp. 1-8.
Eric Weisstein's World of Mathematics, Self Number.
Wikipedia, Self number.
Index entries for Colombian or self numbers and related sequences

Cf. A003052, A010061, A010064, A010067, A010070, A104326, A112310, A328212, A339211, A339213, A339214, A339215.

fibTerms[n_] := Module[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k--]; fr]; dzs[n_] := n + Module[{v = fibTerms[n]}, nv = Length[v]; i = 1; While[i <= nv - 2, If[v[[i]] == 1 && v[[i + 1]] == 0 && v[[i + 2]] == 0, v[[i]] = 0; v[[i + 1]] = 1; v[[i + 2]] = 1; If[i > 2, i -= 3]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, 0, Total[v[[i[[1, 1]] ;; -1]]]]]; m = 240; Complement[Range[m], Array[dzs, m]]

A381579 The Chung-Graham representation of n: representation of n in base of even-indexed Fibonacci numbers.

Original entry on oeis.org

0, 1, 2, 10, 11, 12, 20, 21, 100, 101, 102, 110, 111, 112, 120, 121, 200, 201, 202, 210, 211, 1000, 1001, 1002, 1010, 1011, 1012, 1020, 1021, 1100, 1101, 1102, 1110, 1111, 1112, 1120, 1121, 1200, 1201, 1202, 1210, 1211, 2000, 2001, 2002, 2010, 2011, 2012, 2020, 2021

Offset: 0

Amiram Eldar, Feb 28 2025

Chung and Graham (1981, 1984) proved that every nonnegative integer n can be uniquely represented as a sum n = Sum_{i>=1} d_i * Fibonacci(2*i), where each d_i is in {0, 1, 2}, and if d_i = d_j = 2 with i < j, then for some k, i < k < j, we have d_k = 0.

			a(3) = 10 since 3 = 1 * 3 + 0 * 1 = 3 * Fibonacci(2*2) + 0 * Fibonacci(2*1).
a(10) = 102 since 10 = 1 * 8 + 0 * 3 + 2 * 1 = 1 * Fibonacci(2*3) + 0 * Fibonacci(2*2) + 2 * Fibonacci(2*1).

Amiram Eldar, Table of n, a(n) for n = 0..10000
Rob Burns, Chung-Graham and Zeckendorf representations, arXiv:2502.18870 [math.NT], 2025.
Hung Viet Chu, Aney Manish Kanji, and Zachary Louis Vasseur, Fixed-Term Decompositions Using Even-Indexed Fibonacci Numbers, arXiv:2501.03231 [math.GM], 2024.
F. R. K. Chung and R. L. Graham, On irregularities of distribution of real sequences, Proc. Nat. Acad. Sci. USA, Vol. 78, No. 7 (1981), p. 4001.
F. R. K. Chung and R. L. Graham, On irregularities of distribution, Finite and Infinite Sets, Colloquia Mathematica Societatis János Bolyai, Vol. 37, North-Holland, 1984, pp. 181-222; alternative link.

Cf. A000045, A001906, A291711 (sum of digits).

Similar sequences: A014417, A104326.

f[n_] := f[n] = Fibonacci[2*n]; a[n_] := Module[{s = 0, m = n, k}, While[m > 0, k = 1; While[m > f[k], k++]; If[m < f[k], k--]; If[m >= 2*f[k], s += 2*10^(k-1); m -= 2*f[k], s += 10^(k-1); m -= f[k]]]; s]; Array[a, 50, 0]

m = 20; fvec = vector(m, i, fibonacci(2*i)); f(n) = if(n <= m, fvec[n], fibonacci(2*n));
a(n) = {my(s = 0, m = n, k); while(m > 0, k = 1; while(m > f(k), k++); if(m < f(k), k--); if(m >= 2*f(k), s += 2*10^(k-1); m -= 2*f(k), s += 10^(k-1); m -= f(k))); s;}

A035508 a(n) = Fibonacci(2*n+2) - 1.

Original entry on oeis.org

0, 2, 7, 20, 54, 143, 376, 986, 2583, 6764, 17710, 46367, 121392, 317810, 832039, 2178308, 5702886, 14930351, 39088168, 102334154, 267914295, 701408732, 1836311902, 4807526975, 12586269024, 32951280098, 86267571271, 225851433716

Offset: 0

N. J. A. Sloane

Except for 0, numbers whose dual Zeckendorf representation (A104326) has the same number of 0's as 1's. - Amiram Eldar, Mar 22 2021

Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Clark Kimberling, Interspersions and Dispersions.
Clark Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, Vol. 117, No. 2 (1993), pp. 313-321.
N. J. A. Sloane, Classic Sequences.
Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

With different offset: 2nd row of Inverse Stolarsky array A035507.

Cf. A001906, A104326, A112844, A152891 (partial sums).

[Fibonacci(2*n+2)-1: n in [0..30]]; // Vincenzo Librandi, Apr 18 2011

g:=z/(1-3*z+z^2): gser:=series(g, z=0, 43): seq(abs(coeff(gser, z, n)-1), n=1..26); # Zerinvary Lajos, Mar 22 2009
with(combinat):seq(fibonacci(4*n+2) mod fibonacci(2*n+2),n=0..25);

Fibonacci[2*Range[0, 5!]] - 1 (* Vladimir Joseph Stephan Orlovsky, May 18 2010 *)
LinearRecurrence[{4,-4,1},{0,2,7},40] (* Harvey P. Dale, Jan 15 2025 *)

makelist(fib(2*n+2)-1,n,0,30); /* Martin Ettl, Oct 21 2012 */

numlib::fibonacci(2*n)-1 $ n = 1..38; // Zerinvary Lajos, May 08 2008

[lucas_number1(n, 3, 1)-1 for n in range(1, 27)] # Zerinvary Lajos, Dec 07 2009

a(n) = A001906(n) - 1.
G.f.: x*(2 - x)/((1 - x)*(1 - 3*x + x^2)). a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3). - R. J. Mathar, Dec 15 2008; adapted to the offset by Bruno Berselli, Apr 19 2011
a(n) = Fibonacci(4*n+2) mod Fibonacci(2*n+2). - Gary Detlefs, Nov 22 2010
a(n+1) = Sum_{k=0..n} Fibonacci(2*k+3). - Gary Detlefs, Dec 24 2010
a(n) = Sum_{i=1..n} A112844(i). - R. J. Mathar, Apr 19 2011
a(n) = floor(Fibonacci(2*n+2) - Fibonacci(n+1)^2/Fibonacci(2*n+2)). - Gary Detlefs, Dec 21 2012
From Peter Bala, Nov 14 2021: (Start)
a(n) = Fibonacci(2*n+4)*(Fibonacci(2*n+1) - 1)/(Fibonacci(2*n+3) - 1).
a(n)= -2 + Sum_{k = 1..2*n+3} (-1)^(k+1)*Fibonacci(k). (End)

A352339 a(n) is the maximal (or lazy) Pell representation of n using a ternary system of vectors.

Original entry on oeis.org

0, 1, 10, 11, 20, 21, 22, 110, 111, 120, 121, 122, 210, 211, 220, 221, 1020, 1021, 1022, 1110, 1111, 1120, 1121, 1122, 1210, 1211, 1220, 1221, 2020, 2021, 2022, 2110, 2111, 2120, 2121, 2122, 2210, 2211, 2220, 2221, 2222, 10210, 10211, 10220, 10221, 11020, 11021

Offset: 0

Amiram Eldar, Mar 12 2022

There are 2 well-established systems of giving every nonnegative integer a unique representation as a sum of positive Pell numbers (A000129), 1, 2, 5, 12, 29, 70, ...: the minimal (or greedy) representation (A317204) in which any occurrence of the digit 2 is succeeded by a 0 (i.e., if a Pell number appears twice in the sum, its preceding term in the Pell sequence does not appear), and the maximal (or lazy) representation of n in which any occurrence of the digit 0 is succeeded by a 2 (i.e., if a Pell number does not appear in the sum, its preceding term in the Pell sequence appears twice). [Edited by Amiram Eldar and Peter Munn, Oct 04 2022]

			a(5) = 21 since 5 = 2*2 + 1.
a(6) = 22 since 6 = 2*2 + 2.
a(7) = 110 since 7 = 5 + 2.
We read the first term, 0, like the others, as a list of ternary digits. It has no 1s or 2s in it, so 0 here indicates a sum of 0 Pell numbers. This is called an "empty sum" (see Wiki link) and its total is 0. So 0 represents 0. - _Peter Munn_, Oct 04 2022

Amiram Eldar, Table of n, a(n) for n = 0..10000
A. F. Horadam, Maximal representations of positive integers by Pell numbers, The Fibonacci Quarterly, Vol. 32, No. 3 (1994), pp. 240-244.
OEIS Wiki, Empty sum

Cf. A000129, A317204.

Similar sequences: A104326 (Fibonacci), A130311 (Lucas), A352103 (tribonacci).

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]; IntegerDigits[Total[3^(s - 1)], 3]]; a[n_] := Module[{v = pellp[n]}, nv = Length[v]; i = 1; While[i <= nv - 2, If[v[[i]] > 0 && v[[i + 1]] == 0 && v[[i + 2]] < 2, v[[i ;; i + 2]] += {-1, 2, 1}; If[i > 2, i -= 3]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, 0, FromDigits[v[[i[[1, 1]] ;; -1]]]]]; Array[a, 100, 0]

A112309 Triangle read by rows: row n gives terms in lazy Fibonacci representation of n.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 2, 3, 1, 2, 3, 2, 5, 1, 2, 5, 1, 3, 5, 2, 3, 5, 1, 2, 3, 5, 1, 3, 8, 2, 3, 8, 1, 2, 3, 8, 2, 5, 8, 1, 2, 5, 8, 1, 3, 5, 8, 2, 3, 5, 8, 1, 2, 3, 5, 8, 2, 5, 13, 1, 2, 5, 13, 1, 3, 5, 13, 2, 3, 5, 13, 1, 2, 3, 5, 13, 1, 3, 8, 13, 2, 3, 8, 13, 1, 2, 3, 8, 13, 2, 5, 8, 13, 1, 2, 5, 8, 13, 1, 3

Offset: 1

N. J. A. Sloane, Dec 01 2005

Write n as a sum c_2 F_2 + c_3 F_3 + ..., where the F_i are Fibonacci numbers and the c_i are 0 or 1. The lazy expansion is the minimal one in the lexicographic order, in contrast to the Zeckendorf expansion (A035517, A007895), which is the maximal one.

In other words we give preference to the smallest Fibonacci numbers.

			Triangle begins:
1 meaning 1 = 1
2 meaning 2 = 2
1 2 meaning 3 = 1+2
1 3 meaning 4 = 1+3
2 3 meaning 5 = 2+3
1 2 3 meaning 6 = 1+2+3 (and not the Zeckendorf expansion 1+5)
2 5 meaning 7 = 2+5

Rémy Sigrist, Table of n, a(n) for n = 1..8253 (rows for n = 1..985 flattened)
Rémy Sigrist, PARI program
W. Steiner, The joint distribution of greedy and lazy Fibonacci expansions, Fib. Q., 43 (No. 1, 2005), 60-69.

Cf. A000045, A112310, A035517, A007895.

A112309 := proc(n)
    local z,d ;
    z := convert(A104326(n),base,10) ;
    for d from 1 to nops(z) do
        if op(d,z) > 0 then
            printf("%d,",combinat[fibonacci](d+1)) ;
        end if;
    end do:
end proc:
for n from 1 to 20 do
    A112309(n) ;
end do: # R. J. Mathar, Aug 28 2025

DeleteCases[IntegerDigits[Range[200], 2], {_, 0, 0, _}]
A112309 = Map[DeleteCases[Reverse[#] Fibonacci[Range[Length[#]] + 1], 0] &, DeleteCases[IntegerDigits[-1 + Range[200], 2], {_, 0, 0, _}]]
A112310 = Map[Length, A112309]
(* Peter J. C. Moses, Mar 03 2015 *)

PARI
```
See Links section.
```

Extended by Ray Chandler, Dec 01 2005

A163851 Primes of the form Fibonacci(k)-2.

Original entry on oeis.org

3, 11, 19, 53, 6763, 354224848179261915073, 6356306993006846248181, 6549923758702735390139183573072808204317151158828986996794832713116199613609647330495585293499629115294769, 131000268055609921839337880403168874871518195097038587116150308718344497444830568622793920406537331453336153249817445002421

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 05 2009

nonn

Contains the numbers A000045(k)-2 for k = 5, 7, 8, 10, 20, 100, 106, 508, 586, ...
which equal A000040(j) for j = 2, 5, 8, 16, 871,...
Repunit primes in the lazy Fibonacci representation (A104326). - A.H.M. Smeets, Jun 26 2025

Clear[lst,a,f,n,p]; a=2;lst={};Do[f=Fibonacci[n];If[PrimeQ[p=f-a]&&p>1,AppendTo[lst,p]],{n,3*6!}];lst
Select[Fibonacci[Range[4,1000]]-2,PrimeQ] (* Harvey P. Dale, Jan 12 2019 *)

Indices into Fibonacci and prime sequences added by R. J. Mathar, Aug 18 2009

A330711 Numbers that are both Zeckendorf-Niven numbers (A328208) and lazy-Fibonacci-Niven numbers (A328212).

Original entry on oeis.org

1, 2, 4, 6, 12, 16, 30, 36, 48, 55, 60, 72, 78, 84, 90, 102, 105, 126, 144, 156, 168, 180, 184, 192, 208, 238, 240, 252, 264, 304, 315, 320, 322, 344, 360, 370, 378, 396, 430, 432, 488, 528, 536, 540, 576, 590, 605, 609, 621, 639, 648, 657, 660, 672, 680, 702

Offset: 1

Amiram Eldar, Dec 27 2019

nonn

			6 is in the sequence since A007895(6) = 2 and A112310(6) = 3, and both 2 and 3 are divisors of 6.

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

Intersection of A328208 and A328212.

Cf. A007895, A014417, A104326, A112310.

zeckSum[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]];
fibTerms[n_] := Module[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k--]; fr];
dualZeckSum[n_] := Module[{v = fibTerms[n]}, nv = Length[v]; i = 1; While[i <= nv - 2, If[v[[i]] == 1 && v[[i + 1]] == 0 && v[[i + 2]] == 0, v[[i]] = 0; v[[i + 1]] = 1; v[[i + 2]] = 1; If[i > 2, i -= 3]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, 0, Total[v[[i[[1, 1]] ;; -1]]]]];
Select[Range[1000], Divisible[#, zeckSum[#]] && Divisible[#, dualZeckSum[#]] &]

cp's OEIS Frontend

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

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A350215 A048715, written in binary.

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A130312 Each Fibonacci number F(n) appears F(n) times.

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A339212 Dual-Zeckendorf self numbers: numbers not of the form k + A112310(k).

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A381579 The Chung-Graham representation of n: representation of n in base of even-indexed Fibonacci numbers.

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A035508 a(n) = Fibonacci(2*n+2) - 1.

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A352339 a(n) is the maximal (or lazy) Pell representation of n using a ternary system of vectors.

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