A062878 -id:A062878 - cp's OEIS frontend

A127872 Triangle formed by reading A039599 mod 2.

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

Offset: 0

Philippe Deléham, Apr 05 2007

Also triangle formed by reading triangles A061554, A106180, A110519, A124574, A124576, A126953, A127543 modulo 2.

			Triangle begins:
1;
1, 1;
0, 1, 1;
1, 1, 1, 1;
0, 0, 0, 1, 1;
0, 0, 1, 1, 1, 1;
0, 1, 1, 0, 0, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1;
0, 0, 0, 0, 0, 0, 0, 1, 1;
0, 0, 0, 0, 0, 0, 1, 1, 1, 1;
0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1;
0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1;
0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1;
0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1;
0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1; ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A061554, A106180, A110519, A124574, A124576, A126953, A127543.

Cf. A000007, A036987, A001316, A062878, A050614, A000045, A001519

T[0, 0] := 1; T[n_, k_] := Binomial[2*n - 1, n - k] - Binomial[2*n - 1, n - k - 2]; Table[Mod[T[n, k], 2], {n,0,10}, {k,0,n}] // Flatten (* G. C. Greubel, Apr 18 2017 *)

Sum_{k=0..n} T(n,k)*x^k = A000007(n), A036987(n), A001316(n), A062878(n) for x=-1,0,1,2 respectively.

Sum_{k=0..n} T(n,k)*Fibonacci(2*k+1) = A050614(n), see A000045 and A001519. - Philippe Deléham, Aug 30 2007

A062877 0 and numbers representable as a sum of distinct odd-indexed Fibonacci numbers.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 8, 13, 14, 15, 16, 18, 19, 20, 21, 34, 35, 36, 37, 39, 40, 41, 42, 47, 48, 49, 50, 52, 53, 54, 55, 89, 90, 91, 92, 94, 95, 96, 97, 102, 103, 104, 105, 107, 108, 109, 110, 123, 124, 125, 126, 128, 129, 130, 131, 136, 137, 138, 139, 141, 142, 143, 144

Offset: 0

Antti Karttunen, Jun 26 2001

nonn

			F_1 = 1,
F_3 = 2,
F_1 + F_3 = 3,
F_5 = 5,
F_5 + F_1 = 6,
F_5 + F_3 = 7,
F_5 + F_3 + F_1 = 8,
F_7 = 13, ...

R. J. Mathar, Table of n, a(n) for n = 0..3071
A. Karttunen, On Pascal's Triangle Modulo 2 in Fibonacci Representation, The Fibonacci Quarterly, Vol. 42, #1 (2004) pp. 38-46.

A062878 gives the positions of A050614(n) in this sequence. A062879 is bisection.
A036796(n) - 1.
Cf. A022290 (even-indexed Fibonaccis), A054204.

with(combinat); [seq(A062877(j),j=0..265)]; A062877 := n -> add((floor(n/(2^i)) mod 2)*fibonacci((2*i)+1),i=0..floor_log_2(n+1));
floor_log_2 := proc(n) local nn,i; nn := n; for i from -1 to n do if(0 = nn) then RETURN(i); fi; nn := floor(nn/2); od; end;
# alternative
isA062877 := proc(n)
    local fset,fidx,ps ;
    if n = 0 then
        return true;
    end if;
    fset := {} ;
    for fidx from 1 by 2 do
        if combinat[fibonacci](fidx) >n then
            break;
        end if;
        fset := fset union {combinat[fibonacci](fidx)} ;
    end do:
    for ps in combinat[powerset](fset) do
        if n = add(fidx,fidx=ps) then
            return true;
        end if;
    end do:
    return false;
end proc: # R. J. Mathar, Aug 22 2016

Take[Union[Total/@Subsets[Fibonacci[Range[1,20,2]]]],70](* Harvey P. Dale, Dec 21 2013 *)

my(m=Mod('x,'x^2-3*'x+1)); a(n) = subst(lift(subst(Pol(binary(n)), 'x,m)), 'x,2); \\ Kevin Ryde, Nov 25 2020

A282387 Decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 462", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 3, 6, 15, 24, 60, 102, 255, 384, 960, 1632, 4080, 6168, 15420, 26214, 65535, 98304, 245760, 417792, 1044480, 1579008, 3947520, 6710784, 16776960, 25166208, 62915520, 106956384, 267390960, 404232216, 1010580540, 1717986918, 4294967295, 6442450944

Offset: 0

Robert Price, Feb 13 2017

Initialized with a single black (ON) cell at stage zero.
The first 23 terms of A062878 are the same as this sequence, but I cannot prove nor disprove that these two sequences are the same.
Agrees with A062878 for at least 1000 terms. - Sean A. Irvine, Apr 14 2023

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Robert Price, Table of n, a(n) for n = 0..126
Robert Price, Diagrams of first 20 stages
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Cf. A062878. A282385, A282386, A282388.

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 462; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[1, i]], 2], {i, 1, stages - 1}]

A128937 Triangle formed by reading A039598 mod 2.

Original entry on oeis.org

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

Offset: 0

Philippe Deléham, Apr 27 2007, May 02 2007

Also triangle formed by reading triangles A052179, A053121, A124575, A126075, A126093.

Also triangle formed by reading A065600 mod 2. - Philippe Deléham, Oct 15 2007

			Triangle begins:
  1;
  0, 1;
  1, 0, 1;
  0, 0, 0, 1;
  0, 0, 1, 0, 1;
  0, 1, 0, 0, 0, 1;
  1, 0, 1, 0, 1, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 1, 0, 1;
  0, 0, 0, 0, 0, 1, 0, 0, 0, 1;
  0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1;
  0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1;
  0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1;
  0, 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;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1; ...

Cf. A048896 (row sums).

A039598 := proc(n,k)
        binomial(2*n,n-k)-binomial(2*n,n-k-2) ;
end proc:
A128937 := proc(n,k)
        modp(A039598(n,k),2) ;
end proc: # R. J. Mathar, Apr 21 2021

Sum_{k=0..n} T(n,k) = A048896(n).
Sum_{k=0..n} T(n,k)*2^(n-k) = A101692(n). - Philippe Deléham, Oct 09 2007
Sum_{k=0..n} T(n,k)*2^k = A062878(n+1)/3. - Philippe Deléham, Aug 31 2009

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A127872 Triangle formed by reading A039599 mod 2.

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A062877 0 and numbers representable as a sum of distinct odd-indexed Fibonacci numbers.

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A282387 Decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 462", based on the 5-celled von Neumann neighborhood.

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A128937 Triangle formed by reading A039598 mod 2.

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