A048757 -id:A048757 - cp's OEIS frontend

A094202 Integers k whose Zeckendorf representation A014417(k) is palindromic.

0, 1, 4, 6, 9, 12, 14, 22, 27, 33, 35, 51, 56, 64, 74, 80, 88, 90, 116, 127, 145, 158, 174, 184, 197, 203, 216, 232, 234, 276, 294, 326, 368, 378, 399, 425, 441, 462, 472, 493, 519, 525, 546, 572, 588, 609, 611, 679, 708, 760, 828, 847, 915, 944, 988, 1022, 1064, 1090

Offset: 1

Ron Knott, May 25 2004

			Fibonacci base columns are ...,8,5,3,2,1 with column entries 0 or 1 and no two consecutive ones (the Zeckendorf representation) so that each n has a unique representation.
12 is in the sequence because 12 = 8 + 3 + 1 = 10101 base Fib; 14 = 13 + 1 = 100001 base Fib.

C. G. Lekkerkerker, Voorstelling van natuurlijke getallen door een som van getallen van Fibonacci, Simon Stevin vol. 29, 1952, pages 190-195.
E. Zeckendorf, Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas, Bulletin de la Société Royale des Sciences de Liège vol. 41 (1972) pages 179-182.

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 129 terms from Indranil Ghosh)
Ron Knott, Fibonacci Bases.

Cf. A014417, A035517.

Gives the positions of zeros in A095734. Subsets: A095730, A048757. A006995 gives the integers whose binary expansion is palindromic.

zeck[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]]; a = {}; Do[z = zeck[n]; If[ FromDigits[ Reverse[ IntegerDigits[z]]] == z, AppendTo[a, n]], {n, 1123}]; a (* Robert G. Wilson v, May 29 2004 *)
mirror[dig_, s_] := Join[dig, s, Reverse[dig]]; select[v_, mid_] := Select[v, Length[#] == 0 || Last[#] != mid &]; fib[dig_] := Plus @@ (dig * Fibonacci[Range[2, Length[dig] + 1]]); pals = Rest[IntegerDigits /@ FromDigits /@ Select[Tuples[{0, 1}, 7], SequenceCount[#, {1, 1}] == 0 &]]; Union@Join[{0, 1}, fib /@ Join[mirror[#, {}] & /@ (select[pals, 1]), mirror[#, {1}] & /@ (select[pals, 1]), mirror[#, {0}] & /@ pals]] (* Amiram Eldar, Jan 11 2020 *)

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
def ok(n):
    x=str(a(n))
    return x==x[::-1]
print([n for n in range(1101) if ok(n)]) # Indranil Ghosh, Jun 07 2017

More terms from Robert G. Wilson v, May 28 2004

Offset changed to 1 by Alois P. Heinz, Aug 02 2017

A038183 One-dimensional cellular automaton 'sigma-minus' (Rule 90): 000,001,010,011,100,101,110,111 -> 0,1,0,1,1,0,1,0.

Original entry on oeis.org

1, 5, 17, 85, 257, 1285, 4369, 21845, 65537, 327685, 1114129, 5570645, 16843009, 84215045, 286331153, 1431655765, 4294967297, 21474836485, 73014444049, 365072220245, 1103806595329, 5519032976645, 18764712120593, 93823560602965, 281479271743489, 1407396358717445

Offset: 0

Antti Karttunen, Feb 09 1999

nonn

Generation n (starting from the generation 0: 1) interpreted as a binary number.
Observation: for n <= 15, a(n) = smallest number whose Euler totient is divisible by 4^n. This is not true for n = 16. - Arkadiusz Wesolowski, Jul 29 2012
Orbit of 1 under iteration of Rule 90 = A048725 = (n -> n XOR 4n). - M. F. Hasler, Oct 09 2017

			Successive states are:
          1
         101
        10001
       1010101
      100000001
     10100000101
    1000100010001
   101010101010101
  10000000000000001
  ...
which when converted from binary to decimal give the sequence. - _N. J. A. Sloane_, Jul 21 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200
A. J. Macfarlane, Generating functions for integer sequences defined by the evolution of cellular automata..., Fig.9
Eric Weisstein's World of Mathematics, Rule 90
Wikipedia, Rule 90
Stephen Wolfram, Geometry of Binomial Coefficients, Amer. Math. Monthly, Volume 91, Number 9, November 1984, pages 566-571.
S. Wolfram, O. Martin, and A.M. Odlyzko, Algebraic Properties of Cellular Automata (1984), Communications in Mathematical Physics, 93 (March 1984) 219-258.
Index entries for sequences related to cellular automata

Cf. A006977, A006978, A038184, A038185 (other cellular automata), A000215 (Fermat numbers).
Also alternate terms of A001317. Cf. A048710, A048720, A048757 (same 0/1-patterns interpreted in Fibonacci number system).
Equals 4*A089893(n)+1.
For right half of triangle (excluding the middle bit) see A245191.
Cf. Sierpiński's gasket, A047999.

bit_n := (x,n) -> `mod`(floor(x/(2^n)),2);
# A recursive, cellular automaton rule version:
sigmaminus := proc(n) option remember: if (0 = n) then (1)
else sum('((bit_n(sigmaminus(n-1),i)+bit_n(sigmaminus(n-1),i-2)) mod 2)*(2^i)', 'i'=0..(2*n)) fi: end:

r = 24; c = CellularAutomaton[90, {{1}, 0}, r - 1]; Table[FromDigits[c[[k, r - k + 1 ;; r + k - 1]], 2], {k, r}] (* Arkadiusz Wesolowski, Jun 09 2013 *)
a[ n_] := Sum[ 4^(n - k) Mod[Binomial[2 n, 2 k], 2], {k, 0, n}]; (* Michael Somos, Jun 30 2018 *)
a[ n_] := If[ n < 0, 0, Product[ BitGet[n, k] (2^(2^(k + 1))) + 1, {k, 0, n}]]; (* Michael Somos, Jun 30 2018 *)

vector(100,i,a=if(i>1,bitxor(a<<2,a),1)) \\ M. F. Hasler, Oct 09 2017

{a(n) = sum(k=0, n, binomial(2*n, 2*k)%2 * 4^(n-k))}; /* Michael Somos, Jun 30 2018 */

a=1
for n in range(55):
    print(a, end=",")
    a ^= a*4
# Alex Ratushnyak, May 04 2012

def A038183(n): return sum((bool(~(m:=n<<1)&m-k)^1)<Chai Wah Wu, May 02 2023

a(n) = Product_{i>=0} bit_n(n, i)*(2^(2^(i+1)))+1: A direct algebraic formula!
a(n) = Sum_{k=0..n} (C(2*n, 2*k) mod 2)*4^(n-k). - Paul Barry, Jan 03 2005
a(2*n+1) = 5*a(2n); a(n+1) = a(n) XOR 4*a(n) where XOR is binary exclusive OR operator. - Philippe Deléham, Jun 18 2005
a(n) = A001317(2n). - Alex Ratushnyak, May 04 2012

A070886 Triangle read by rows giving successive states of cellular automaton generated by "Rule 90".

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0

Offset: 0

N. J. A. Sloane, May 19 2002

If either neighbor is 1 then new state is 1, otherwise new state is 0.
Row n has length 2n+1.
Rules #18, #26, #82, #90, #146, #154, #210, #218 all give rise to this sequence. - Hans Havermann

			1; 1,0,1; 1,0,0,0,1; 1,0,1,0,1,0,1; ...

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 25.

Robert Price, Table of n, a(n) for n = 0..9999
Eric Weisstein's World of Mathematics, Rule 90
S. Wolfram, A New Kind of Science
Index to Elementary Cellular Automata
Index entries for sequences related to cellular automata

Cf. A070950, A070887. Alternate rows of A047999. Interpreted as binary numbers: A038183. Interpreted as Zeckendorf-expansions: A048757. Drawn as binary trees: A080263.

rows = 10; ca = CellularAutomaton[90, {{1}, 0}, rows-1]; Flatten[ Table[ca[[k, rows-k+1 ;; rows+k-1]], {k, 1, rows}]] (* Jean-François Alcover, May 24 2012 *)

More terms from Hans Havermann, May 26 2002

A050609 Table T(n,k) = Sum_{i=0..2n} (C(2n,i) mod 2)F(i+k) = Sum_{i=0..n} (C(n,i) mod 2)F(2i+k).

Original entry on oeis.org

0, 1, 1, 3, 3, 1, 12, 6, 4, 2, 21, 21, 9, 7, 3, 77, 35, 33, 15, 11, 5, 168, 126, 56, 54, 24, 18, 8, 609, 273, 203, 91, 87, 39, 29, 13, 987, 987, 441, 329, 147, 141, 63, 47, 21, 3572, 1598, 1596, 714, 532, 238, 228, 102, 76, 34, 7755, 5781, 2585, 2583, 1155, 861, 385

Offset: 0

Antti Karttunen, Dec 02 1999

Listed antidiagonalwise as T(0,0), T(1,0), T(0,1), T(2,0), T(1,1), T(0,2), ...

A. Karttunen, On Pascal's Triangle Modulo 2 in Fibonacci Representation, Fibonacci Quarterly, 42 (2004), 38-46.

Transpose of A050610. First row: A051656, second row: A050611, third row: A048757, fourth row: A050612. A050613 gives other Maple procedures. Cf. A025581, A002262.

A050609_as_sum := proc(n,k) local i; RETURN(add(((binomial(n,i) mod 2)*fibonacci(k+2*i)),i=0..n)); end;
A050609_as_product := (n,k) -> (`if`(1 = (n mod 2),luc(n+k),fibonacci(n+k)))*product('luc(2^i)^bit_i(n,i)','i'=1..floor_log_2(n+1)); # Produces same answers.
[seq(A050609_as_sum(A025581(n), A002262(n)),n=0..119)];

Also a(n) = A075148(n, k)*A050613(n).

A054433 Numbers formed by interpreting the reduced residue set of every even number as a Zeckendorf Expansion.

Original entry on oeis.org

1, 4, 9, 33, 80, 174, 588, 1596, 3135, 9950, 28512, 56268, 196040, 496496, 888300, 3524577, 9224880, 18118362, 63239220, 150527400, 310190454, 1129200138, 2971168704, 5834056536, 18513646430, 53213956640, 104687896833

Offset: 1

Antti Karttunen

nonn

Amiram Eldar, Table of n, a(n) for n = 1..2392

Cf. A054432, A048757, A051258, A063683.

with(combinat,fibonacci); # one_or_zero given at A054431.
A054433_as_sum := proc(n) local i; RETURN(add((one_or_zero(igcd(n,i))*fibonacci(i+1)),i=1..(n-1))); end;

r[n_] := Sum[If[GCD[n, k] == 1, Fibonacci[n + 1 - k], 0], {k, 1, n}]; r /@ (2*Range[27]) (* Amiram Eldar, Oct 19 2019 *)

a(n) = A054433_as_sum(2*n).

A051656 Sum_{i=0..n} (C(n,i) mod 2)Fibonacci(2i).

Original entry on oeis.org

0, 1, 3, 12, 21, 77, 168, 609, 987, 3572, 7755, 28059, 47376, 171409, 372099, 1346268, 2178309, 7881197, 17108664, 61899729, 104512485, 378129724, 820851717, 2969869413, 4809706272, 17401680769, 37775923491, 136674575148

Offset: 0

Antti Karttunen, Nov 30 1999

Positions in the first column (A003622) of Wythoff array of the terms which have their Zeckendorf Expansion patterned as row[2n+1] in Pascal's Triangle computed modulo 2 (A047999)

Proof in preparation, to be published (see A048757).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A048757, A047999, A035513, A038183, A051256. First row of A050609, First column of A050610.
a(n) = A019586[A048757[n]]. A048757[n] = SS(Athis_sequence[n])+1, where SSx means the second Fibonacci Successor of x (= x's Z.E. shifted left twice).
Cf. A001906.

a051656 = sum . zipWith (*) a001906_list . a047999_row
-- Reinhard Zumkeller, Feb 27 2011

Table[Sum[Mod[Binomial[n,i],2]Fibonacci[2i],{i,0,n}],{n,0,30}] (* Harvey P. Dale, Oct 30 2011 *)

a(n)=sum(i=0,n,if(!bitand(i,n-i),fibonacci(2*i))) \\ Charles R Greathouse IV, Jan 04 2013

a(n) = sum_{i=0..n} (C(2n, 2i) mod 2)*F(2*i) = FL(n)product_{i=0..inf} L(2^i)^bit(n, i) where L is n-th Lucas number (A000032) and FL is defined as in A048757: FL(n) = n-th fibonacci number if n even, n-th Lucas number if n odd.

A051256 Numbers formed from binomial coefficients (mod 2) interpreted as digits in factorial base.

Original entry on oeis.org

1, 3, 7, 33, 121, 843, 5167, 46233, 362881, 3991683, 40279687, 522910113, 6227383801, 93409304523, 1313941673647, 22324392524313, 355687428096001, 6758061133824003, 122000787836928007, 2561305169719296033

Offset: 0

Antti Karttunen, Oct 24 1999

			a(5) = 1! + 2! + 5! + 6! = 843 (only the first, second, fifth and sixth terms are odd in row 5 of Pascal's Triangle).

Chai Wah Wu, Table of n, a(n) for n = 0..448

Cf. A001317, A001339, A048757, A047999.

A051256(n) := proc(n) local i; RETURN(add(((binomial(n,i) mod 2)*((i+1)!)),i=0..n)); end;

Table[Sum[(k+1)!Mod[Binomial[n,k],2],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Feb 14 2013 *)

from math import factorial
def A051256(n):
    return sum(0 if ~n & k else factorial(k+1) for k in range(n+1)) # Chai Wah Wu, Feb 08 2016

a(n) = Sum_{k=0..n} (k+1)!(C(n, k) mod 2).

A075148 Table E(n,k) (listed antidiagonalwise as E(0,0), E(1,0), E(0,1), E(2,0), E(1,1), E(0,2), ...) where E(n,k) is F(n+k) for all even n and L(n+k) for all odd n. F(n) and L(n) are the n-th Fibonacci (A000045) and Lucas (A000032) numbers respectively.

Original entry on oeis.org

0, 1, 1, 1, 3, 1, 4, 2, 4, 2, 3, 7, 3, 7, 3, 11, 5, 11, 5, 11, 5, 8, 18, 8, 18, 8, 18, 8, 29, 13, 29, 13, 29, 13, 29, 13, 21, 47, 21, 47, 21, 47, 21, 47, 21, 76, 34, 76, 34, 76, 34, 76, 34, 76, 34, 55, 123, 55, 123, 55, 123, 55, 123, 55, 123, 55, 199, 89, 199, 89, 199, 89, 199

Offset: 0

Antti Karttunen, Sep 05 2002

Used to construct A050609 & A048757. Cf. A025581, A002262.

with(combinat); [seq(A075148(A025581(n), A002262(n)),n=0..119)]; A075148 := (n,k) -> (fibonacci(n+k-(n mod 2))+fibonacci(n+k+(n mod 2)))/(2-(n mod 2));

E(n, k) = (fibonacci(n+k-(n mod 2))+fibonacci(n+k+(n mod 2)))/(2-(n mod 2))

A051257 Numbers formed from binomial coefficients (mod 2+k) interpreted as digits in factorial base.

Original entry on oeis.org

1, 3, 11, 43, 231, 1337, 9739, 76209, 706109, 6914977, 78150249, 920172983, 12216376453, 168531536319, 2571960399839, 40581616143967, 701349512411763, 12460393480873445, 240094506439569631, 4749510978132662277

Offset: 0

Antti Karttunen, Oct 24 1999

			a(5) = (1 mod 2)1!+(5 mod 3)2!+(10 mod 4)3!+(10 mod 5)4!+(5 mod 6)5!+(1 mod 7)6! = 1*1+2*2+2*6+0*24+5*120+1*720 = 1337

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A001317, A001339, A048757, A047999, A051256.

a(n) := proc(n) local i; RETURN(add(((binomial(n,i)mod(i+2))*((i+1)!)),i=0..n)); end;

a[n_] := Sum[(k+1)!*Mod[Binomial[n, k], 2+k], {k, 0, n}]; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Sep 09 2013 *)

a(n) = Sum (k+1)!(C(n, k) mod (2+k)), k=0..n

cp's OEIS Frontend

A094202 Integers k whose Zeckendorf representation A014417(k) is palindromic.

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A038183 One-dimensional cellular automaton 'sigma-minus' (Rule 90): 000,001,010,011,100,101,110,111 -> 0,1,0,1,1,0,1,0.

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A070886 Triangle read by rows giving successive states of cellular automaton generated by "Rule 90".

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A050609 Table T(n,k) = Sum_{i=0..2n} (C(2n,i) mod 2)*F(i+k) = Sum_{i=0..n} (C(n,i) mod 2)*F(2i+k).

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A054433 Numbers formed by interpreting the reduced residue set of every even number as a Zeckendorf Expansion.

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A051656 Sum_{i=0..n} (C(n,i) mod 2)*Fibonacci(2*i).

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A051256 Numbers formed from binomial coefficients (mod 2) interpreted as digits in factorial base.

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A050609 Table T(n,k) = Sum_{i=0..2n} (C(2n,i) mod 2)F(i+k) = Sum_{i=0..n} (C(n,i) mod 2)F(2i+k).

A051656 Sum_{i=0..n} (C(n,i) mod 2)Fibonacci(2i).