A228285 -id:A228285 - cp's OEIS frontend

A014417 Representation of n in base of Fibonacci numbers (the Zeckendorf representation of n). Also, binary words starting with 1 not containing 11, with the word 0 added.

0, 1, 10, 100, 101, 1000, 1001, 1010, 10000, 10001, 10010, 10100, 10101, 100000, 100001, 100010, 100100, 100101, 101000, 101001, 101010, 1000000, 1000001, 1000010, 1000100, 1000101, 1001000, 1001001, 1001010, 1010000, 1010001, 1010010, 1010100, 1010101

Offset: 0

Olivier Gérard

Old name was: Representation of n in base of Fibonacci numbers (the Zeckendorf representation of n). Also, binary vectors not containing 11.
For n > 0, write n = Sum_{i >= 2} eps(i) Fib_i where eps(i) = 0 or 1 and no 2 consecutive eps(i) can be 1 (see A035517); then a(n) is obtained by writing the eps(i) in reverse order.
"One of the most important properties of the Fibonacci numbers is the special way in which they can be used to represent integers. Let's write j >> k <==> j >= k+2. Then every positive integer has a unique representation of the form n = F_k1 + F_k2 + ... + F_kr, where k1 >> k2 >> ... >> kr >> 0. (This is 'Zeckendorf's theorem.') ... We can always find such a representation by using a "greedy" approach, choosing F_k1 to be the largest Fibonacci number =< n, then choosing F_k2 to be the largest that is =< n - F_k1 and so on. Fibonacci representation needs a few more bits because adjacent 1's are not permitted; but the two representations are analogous." [Concrete Math.]
Since the binary representation of n in base of Fibonacci numbers allows only the successive bit pairs 00, 01, 10 and leaves 11 unused, we can use a ternary representation using all trits 0, 1, 2 where 00 --> 0, 01 --> 1 and 10 --> 2 (e.g. binary 1001010 as ternary 1022). - Daniel Forgues, Nov 30 2009
The same sequence also arises when considering the NegaFibonacci representations of the integers, as follows. Take the NegaFibonacci representations of n = 0, 1, 2, ... (A215022) and of n = -1, -2, -3, ... (A215023), sort the union of these two lists into increasing binary order, and we get A014417. Likewise the corresponding list of decimal representations, A003714, is the union of A215024 and A215025 sorted into increasing order.  - N. J. A. Sloane, Aug 10 2012
Also, numbers, written in binary, such that no adjacent bits are equal to 1: A one-dimensional analog of the matrices considered in A228277/A228285, A228390, A228476, A228506 etc. - M. F. Hasler, Apr 27 2014
The sequence of final bits, starting with a(1), is the complement of the Fibonacci word A005614. - N. J. A. Sloane, Oct 03 2018
This representation is named after the Belgian Army doctor and mathematician Edouard Zeckendorf (1901-1983). - Amiram Eldar, Jun 11 2021

			The Zeckendorf expansions of 1, 2, ... are 1 = 1 = Fib_2 -> 1, 2 = 2 = Fib_3 -> 10, 3 = Fib_4 -> 100, 4 = 3+1 = Fib_4 + Fib_2 -> 101, 5 = 5 = Fib_5 -> 1000, 6 = 1+5 = Fib_2 + Fib_5 -> 1001, etc.

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990.
Donald E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.3, p. 169.
Edouard Zeckendorf, 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.

T. D. Noe and Harry J. Smith, Table of n, a(n) for n = 0..10000
Paul Dalenberg and Tom Edgar, Consecutive factorial base Niven numbers, Fibonacci Quart., Vol. 56, No. 2 (2018), pp. 163-166.
Eric Duchene, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, 43 pages, no date, apparently unpublished. See Table 2.
Eric Duchene, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, unpublished, no date. [Cached copy, with permission]
Donald E. Knuth, Fibonacci multiplication, Appl. Math. Lett., Vol. 1, No. 1 (1988), pp. 57-60.
Julien Leroy, Michel Rigo and Manon Stipulanti, Counting the number of non-zero coefficients in rows of generalized Pascal triangles, Discrete Mathematics, Vol. 340, No. 5 (2017), pp. 862-881.
Casey Mongoven, Zeckendorf Representations no. 17 (a musical composition with A014417).
Wikipedia, Zeckendorf's theorem.

Cf. A000045, A003794, A007895, A035517, A130234, A215022, A215023, A215024, A215025.
a(n) = A003714(n) converted to binary.
Cf. also A005614, A189920, A104324, A213911.
See A104326 for dual Zeckendorf representation of n.

a014417 0 = 0
a014417 n = foldl (\v z -> v * 10 + z) 0 $ a189920_row n
-- Reinhard Zumkeller, Mar 10 2013

A014417 := proc(n)
    local nshi,Z,i ;
    if n <= 1 then
        return n;
    end if;
    nshi := n ;
    Z := [] ;
    for i from A130234(n) to 2 by -1 do
        if nshi >= A000045(i) and nshi > 0 then
            Z := [1,op(Z)] ;
            nshi := nshi-A000045(i) ;
        else
            Z := [0,op(Z)] ;
        end if;
    end do:
    add( op(i,Z)*10^(i-1),i=1..nops(Z)) ;
end proc: # R. J. Mathar, Jan 31 2015

fb[n_Integer] := Block[{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-- ]; FromDigits[fr]]; Table[ fb[n], {n, 0, 30}] (* Robert G. Wilson v, May 15 2004 *)
r = Map[Fibonacci, Range[2, 12]]; Table[Total[FromDigits@ PadRight[{1}, Flatten@ #] &@ Reverse@ Position[r, #] & /@ Abs@ Differences@ NestWhileList[Function[k, k - SelectFirst[Reverse@ r, # < k &]], n + 1, # > 1 &]], {n, 0, 33}] (* Michael De Vlieger, Mar 27 2016, Version 10 *)
FromDigits/@Select[Tuples[{0,1},7],SequenceCount[#,{1,1}]==0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 14 2019 *)

Zeckendorf(n)=my(k=0,v,m); while(fibonacci(k)<=n,k=k+1); m=k-1; v=vector(m-1); v[1]=1; n=n-fibonacci(k-1); while(n>0,k=0; while(fibonacci(k)<=n,k=k+1); v[m-k+2]=1; n=n-fibonacci(k-1)); v \\ Ralf Stephan

Zeckendorf(n)= { local(k); a=0; while(n>0, k=0; while(fibonacci(k)<=n, k=k+1); a=a+10^(k-3); n=n-fibonacci(k-1); ); a }
{ for (n=0, 10000, Zeckendorf(n); print(n," ",a); write("b014417.txt", n, " ", a) ) } \\ Harry J. Smith, Jan 17 2009

from sympy import fibonacci
def a(n):
    k=0
    x=0
    while n>0:
        k=0
        while fibonacci(k)<=n: k+=1
        x+=10**(k - 3)
        n-=fibonacci(k - 1)
    return x
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 07 2017, after PARI code by Harry J. Smith

Comment layout fixed by Daniel Forgues, Dec 07 2009
Typo corrected by Daniel Forgues, Mar 25 2010
Definition expanded and Duchene et al. reference added by N. J. A. Sloane, Aug 07 2018
Name corrected by Michel Dekking, Nov 30 2020

A228277 Number of n X n binary arrays with top left value 1 and no two ones adjacent horizontally, vertically or nw-se diagonally.

Original entry on oeis.org

1, 1, 13, 133, 3631, 172082, 16566199, 3057290265, 1105411581741, 776531523355217, 1063228770141145384, 2834013489992345694498, 14712337761578682394367473, 148727865257442275211424889367

Offset: 1

R. H. Hardin, Aug 19 2013

nonn

Main diagonal of A228285.

			The thirteen solutions for n=3 correspond to the thirteen possible values of 5-bit numbers with no two adjacent bits equal to 1, namely, the matrices
( 1 0 a )
( 0 0 b )
( e d c ) ; with abcde = A014417(0,...,12) = 0, 1, 10, 100, 101, 1000, 1001, 1010, 10000, 10001, 10010, 10100, 10101 (leading zeros omitted). - _M. F. Hasler_, Apr 27 2014
Some solutions for n=4:
.1..0..0..1.  .1..0..0..0.  .1..0..0..0.  .1..0..0..0.  .1..0..0..0
.0..0..0..0.  .0..0..0..1.  .0..0..0..0.  .0..0..1..0.  .0..0..0..0
.0..0..0..0.  .0..0..0..0.  .0..0..0..0.  .0..0..0..0.  .0..0..0..1
.1..0..0..1.  .0..1..0..1.  .0..1..0..1.  .0..1..0..0.  .0..0..1..0
The last example shows that sw-ne (= anti)diagonally adjacent "1"s are allowed. See A228476, A228506 and A228390 for other variants.

R. H. Hardin, Table of n, a(n) for n = 1..25

Cf. A228277-A228285.

See also the variants A228390, A228476, A228506, etc.

No known recurrence.

A226444 Number A(n,k) of tilings of a k X n rectangle using 1 X 1 squares and L-tiles; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 5, 6, 5, 1, 1, 1, 1, 8, 13, 13, 8, 1, 1, 1, 1, 13, 28, 42, 28, 13, 1, 1, 1, 1, 21, 60, 126, 126, 60, 21, 1, 1, 1, 1, 34, 129, 387, 524, 387, 129, 34, 1, 1, 1, 1, 55, 277, 1180, 2229, 2229, 1180, 277, 55, 1, 1

Offset: 0

Alois P. Heinz, Jun 06 2013

An L-tile is a 2 X 2 square with the upper right 1 X 1 subsquare removed and no rotations are allowed.

			A(3,3) = 6:
  ._____.  ._____.  ._____.  ._____.  ._____.  ._____.
  |_|_|_|  | |_|_|  |_|_|_|  |_| |_|  |_|_|_|  |_| |_|
  |_|_|_|  |___|_|  | |_|_|  |_|___|  |_| |_|  | |___|
  |_|_|_|  |_|_|_|  |___|_|  |_|_|_|  |_|___|  |___|_|.
Square array A(n,k) begins:
  1, 1,  1,   1,    1,     1,      1,       1,        1, ...
  1, 1,  1,   1,    1,     1,      1,       1,        1, ...
  1, 1,  2,   3,    5,     8,     13,      21,       34, ...
  1, 1,  3,   6,   13,    28,     60,     129,      277, ...
  1, 1,  5,  13,   42,   126,    387,    1180,     3606, ...
  1, 1,  8,  28,  126,   524,   2229,    9425,    39905, ...
  1, 1, 13,  60,  387,  2229,  13322,   78661,   466288, ...
  1, 1, 21, 129, 1180,  9425,  78661,  647252,  5350080, ...
  1, 1, 34, 277, 3606, 39905, 466288, 5350080, 61758332, ...

Alois P. Heinz, Antidiagonals n = 0..44, flattened

Columns (or rows) k=0+1,2-10 give: A000012, A000045(n+1), A002478, A105262, A219737(n-1) for n>2, A219738 (n-1) for n>2, A219739(n-1) for n>1, A219740(n-1) for n>2, A226543, A226544.
Main diagonal gives A066864(n-1).
See A219741 for an array with very similar entries. - N. J. A. Sloane, Aug 22 2013
Cf. A322494.

b:= proc(n, l) option remember; local k, t;
      if max(l[])>n then 0 elif n=0 or l=[] then 1
    elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
    else for k do if l[k]=0 then break fi od; b(n, subsop(k=1, l))+
        `if`(k b(max(n, k), [0$min(n, k)]):
seq(seq(A(n, d-n), n=0..d), d=0..14);
[Zeilberger gives Maple code to find generating functions for the columns - see links in A228285. - N. J. A. Sloane, Aug 22 2013]

b[n_, l_] := b[n, l] = Module[{k, t}, Which[Max[l] > n, 0, n == 0 || l == {}, 1, Min[l] > 0, t = Min[l]; b[n-t, l-t], True, k = Position[l, 0, 1][[1, 1]]; b[n, ReplacePart[l, k -> 1]] + If[k < Length[l] && l[[k+1]] == 0, b[n, ReplacePart[l, {k -> 1, k+1 -> 2}]], 0] ] ]; a[n_, k_] := b[Max[n, k], Array[0&, Min[n, k]]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 18 2013, translated from Maple *)

The k-th column satisfies a recurrence of order Fibonacci(k+1) [Zeilberger] - see links in A228285. - N. J. A. Sloane, Aug 22 2013

A228482 T(n,k)=Number of nXk binary arrays with top left value 1 and no two ones adjacent horizontally, vertically or antidiagonally.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 4, 4, 3, 5, 9, 14, 9, 5, 8, 19, 41, 41, 19, 8, 13, 41, 127, 172, 127, 41, 13, 21, 88, 386, 728, 728, 386, 88, 21, 34, 189, 1181, 3084, 4354, 3084, 1181, 189, 34, 55, 406, 3605, 13050, 25699, 25699, 13050, 3605, 406, 55, 89, 872, 11013, 55252, 152373

Offset: 1

R. H. Hardin Aug 22 2013

Table starts
..1...1.....2......3.......5.........8.........13..........21............34
..1...2.....4......9......19........41.........88.........189...........406
..2...4....14.....41.....127.......386.......1181........3605.........11013
..3...9....41....172.....728......3084......13050.......55252........233875
..5..19...127....728....4354.....25699.....152373......902042.......5342712
..8..41...386...3084...25699....211588....1748684....14433982.....119188751
.13..88..1181..13050..152373...1748684...20185842...232542935....2680777055
.21.189..3605..55252..902042..14433982..232542935..3737615288...60122232373
.34.406.11013.233875.5342712.119188751.2680777055.60122232373.1349721589622
Same recurrences as A228285 except in addition this smaller one for k=5

			Some solutions for n=4 k=4
..1..0..0..1....1..0..1..0....1..0..0..1....1..0..1..0....1..0..1..0
..0..0..0..0....0..0..0..1....0..1..0..0....0..0..0..0....0..0..0..0
..0..0..0..1....0..0..0..0....0..0..1..0....0..1..0..0....0..0..0..0
..0..0..0..0....0..1..0..0....0..0..0..1....0..0..1..0....1..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..1333

Column 1 is A000045
Column 2 is A078039(n-1).
Cf. A228476-A228481, A228277-A228285, A226444.

Recurrences for column k:
k=1: a(n) = a(n-1) +a(n-2)
k=2: a(n) = a(n-1) +2*a(n-2) +a(n-3)
k=3: a(n) = a(n-1) +5*a(n-2) +4*a(n-3) -a(n-5)
k=4: a(n) = a(n-1) +10*a(n-2) +15*a(n-3) +4*a(n-4) -6*a(n-5) -a(n-6) +3*a(n-7) -a(n-8)
k=5: a(n) = 3*a(n-1) +15*a(n-2) +16*a(n-3) -11*a(n-4) -20*a(n-5) +19*a(n-6) -8*a(n-7) +a(n-9)
k=6: a(n) = a(n-1) +42*a(n-2) +147*a(n-3) +70*a(n-4) -478*a(n-5) -449*a(n-6) +1199*a(n-7) +732*a(n-8) -2727*a(n-9) +659*a(n-10) +3827*a(n-11) -5776*a(n-12) +3926*a(n-13) -1152*a(n-14) -148*a(n-15) +154*a(n-16) +32*a(n-17) -29*a(n-18) -6*a(n-19) +3*a(n-20) +a(n-21)
k=7: a(n) = a(n-1) +85*a(n-2) +432*a(n-3) +192*a(n-4) -3711*a(n-5) -5096*a(n-6) +21164*a(n-7) +27340*a(n-8) -112654*a(n-9) -37244*a(n-10) +477721*a(n-11) -464722*a(n-12) -897815*a(n-13) +3102284*a(n-14) -4149918*a(n-15) +2761082*a(n-16) -138325*a(n-17) -1353257*a(n-18) +942033*a(n-19) +64683*a(n-20) -365483*a(n-21) +80904*a(n-22) +92350*a(n-23) -27097*a(n-24) -23292*a(n-25) +2585*a(n-26) +5635*a(n-27) +1405*a(n-28) -561*a(n-29) -545*a(n-30) -173*a(n-31) -14*a(n-32) +5*a(n-33) +a(n-34)

A219741 T(n,k) = Unmatched value maps: number of nXk binary arrays indicating the locations of corresponding elements not equal to any horizontal, vertical or antidiagonal neighbor in a random 0..1 nXk array.

Original entry on oeis.org

1, 2, 2, 4, 6, 4, 7, 13, 13, 7, 12, 28, 42, 28, 12, 21, 60, 126, 126, 60, 21, 37, 129, 387, 524, 387, 129, 37, 65, 277, 1180, 2229, 2229, 1180, 277, 65, 114, 595, 3606, 9425, 13322, 9425, 3606, 595, 114, 200, 1278, 11012, 39905, 78661, 78661, 39905, 11012, 1278, 200

Offset: 1

R. H. Hardin, Nov 26 2012

Table starts
...1.....2......4........7.........12...........21.............37
...2.....6.....13.......28.........60..........129............277
...4....13.....42......126........387.........1180...........3606
...7....28....126......524.......2229.........9425..........39905
..12....60....387.....2229......13322........78661.........466288
..21...129...1180.....9425......78661.......647252........5350080
..37...277...3606....39905.....466288......5350080.......61758332
..65...595..11012...168925....2760690.....44159095......711479843
.114..1278..33636...715072...16350693....364647622.....8201909757
.200..2745.102733..3027049...96830726...3010723330....94531063074
.351..5896.313781.12813931..573456240..24858935864..1089590912023
.616.12664.958384.54243509.3396136349.205253857220.12558669019786

			Some solutions for n=3 k=4
..0..0..0..0....1..0..0..1....0..0..1..0....0..0..1..0....0..0..0..1
..0..1..0..0....0..0..0..0....1..0..0..0....0..0..0..0....0..1..0..0
..0..0..0..0....0..1..0..1....0..0..0..1....1..0..0..1....0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..337

Column 1 is A005251(n+2).
Column 2 is A002478(n+1).
Column 3 is A105262(n+1) for n>1.
Main diagonal is A066864.
See A226444 for an array with very similar entries. - N. J. A. Sloane, Aug 22 2013

Zeilberger's Maple code (see links in A228285) would presumably give recurrences for the columns of this array. - N. J. A. Sloane, Aug 22 2013

A219737 Unmatched value maps: number of n X 4 binary arrays indicating the locations of corresponding elements not equal to any horizontal, vertical or antidiagonal neighbor in a random 0..1 n X 4 array.

Original entry on oeis.org

7, 28, 126, 524, 2229, 9425, 39905, 168925, 715072, 3027049, 12813931, 54243509, 229621433, 972024617, 4114736810, 17418344167, 73734658344, 312130693269, 1321299533915, 5593273893746, 23677229915913, 100229530526756

Offset: 1

R. H. Hardin, Nov 26 2012

nonn

Column 4 of A219741.

			Some solutions for n=3:
..0..1..0..1....0..0..1..0....0..0..0..1....1..0..1..0....1..0..0..0
..0..0..0..0....1..0..0..0....1..0..0..0....0..0..0..0....0..0..0..0
..1..0..0..0....0..1..0..1....0..1..0..0....0..1..0..1....1..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..130

Cf. A219741.

Empirical: a(n) = a(n-1) + 10*a(n-2) + 15*a(n-3) + 4*a(n-4) - 6*a(n-5) - a(n-6) + 3*a(n-7) - a(n-8) for n>9.
Zeilberger's Maple code (see links in A228285) would give a proof that this recurrence is correct. - N. J. A. Sloane, Aug 22 2013
G.f.: x*(1 + x)*(7 + 14*x + 14*x^2 - x^3 - 2*x^4 - 2*x^5 + 3*x^6 - x^7) / (1 - x - 10*x^2 - 15*x^3 - 4*x^4 + 6*x^5 + x^6 - 3*x^7 + x^8). - Colin Barker, Mar 12 2018

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A014417 Representation of n in base of Fibonacci numbers (the Zeckendorf representation of n). Also, binary words starting with 1 not containing 11, with the word 0 added.

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A228277 Number of n X n binary arrays with top left value 1 and no two ones adjacent horizontally, vertically or nw-se diagonally.

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A226444 Number A(n,k) of tilings of a k X n rectangle using 1 X 1 squares and L-tiles; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A228482 T(n,k)=Number of nXk binary arrays with top left value 1 and no two ones adjacent horizontally, vertically or antidiagonally.

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A219741 T(n,k) = Unmatched value maps: number of nXk binary arrays indicating the locations of corresponding elements not equal to any horizontal, vertical or antidiagonal neighbor in a random 0..1 nXk array.

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A219737 Unmatched value maps: number of n X 4 binary arrays indicating the locations of corresponding elements not equal to any horizontal, vertical or antidiagonal neighbor in a random 0..1 n X 4 array.

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A219738 Unmatched value maps: number of nX5 binary arrays indicating the locations of corresponding elements not equal to any horizontal, vertical or antidiagonal neighbor in a random 0..1 nX5 array.

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A219739 Unmatched value maps: number of nX6 binary arrays indicating the locations of corresponding elements not equal to any horizontal, vertical or antidiagonal neighbor in a random 0..1 nX6 array.

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