A050609 -id:A050609 - cp's OEIS frontend

A048757 Sum_{i=0..2n} (C(2n,i) mod 2)Fibonacci(i+2) = Sum_{i=0..n} (C(n,i) mod 2)Fibonacci(2i+2).

1, 4, 9, 33, 56, 203, 441, 1596, 2585, 9353, 20304, 73461, 124033, 448756, 974169, 3524577, 5702888, 20633243, 44791065, 162055596, 273617239, 989956471, 2149017696, 7775219067, 12591974497, 45558191716, 98898651657

Offset: 0

Antti Karttunen, Jul 13 1999

The history of 1-D CA Rule 90 starting from the seed pattern 1 interpreted as Zeckendorffian expansion.

Also, product of distinct terms of A001566 and appropriate Fibonacci or Lucas numbers: a(n) = FL(n+2)Product(L(2^i)^bit(n,i),i=0..) Here L(2^i) = A001566 and FL(n) = n-th Fibonacci number if n even, n-th Lucas number if n odd. bit(n,i) is the i-th digit (0 or 1) in the binary expansion of n, with the least significant digit being bit(n,0).

			1 = Fib(2) = 1;
101 = Fib(4) + Fib(2) = 3 + 1 = 4;
10001 = Fib(6) + Fib(2) = 8 + 1 = 9;
1010101 = Fib(8) + Fib(6) + Fib(4) + Fib(2) = 21 + 8 + 3 + 1 = 33; etc.

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

a(n) = A022290(A038183(n)) = A022290(A048723(5, n)) = A003622(A051656(n)) = A075148(n, 2)*A050613(n). Third row of A050609, third column of A050610.

Cf. A054433.

Table[Sum[Mod[Binomial[2n, i], 2] Fibonacci[i + 2], {i, 0, 2n}], {n, 0, 19}] (* Alonso del Arte, Apr 27 2014 *)

A050613 Products of distinct terms of 1 and rest from A001566: a(n) = Product_{i=0..floor(log_2(n+1))} L(2^i)^bit(n,i).

Original entry on oeis.org

1, 1, 3, 3, 7, 7, 21, 21, 47, 47, 141, 141, 329, 329, 987, 987, 2207, 2207, 6621, 6621, 15449, 15449, 46347, 46347, 103729, 103729, 311187, 311187, 726103, 726103, 2178309, 2178309, 4870847, 4870847, 14612541, 14612541, 34095929, 34095929

Offset: 0

Antti Karttunen, Dec 02 1999

nonn

Used to produce the rows of A050609.

Also Sum(((C(2((n+((n+1) mod 2)) mod (2^floor(log_2(n)))),i) mod 2)*F(n+((n+1) mod 2)-i)),i=0..2((n+((n+1) mod 2)) mod (2^floor(log_2(n))))) or Sum(((C(2((n-(n mod 2)) mod (2^floor(log_2(n)))),i) mod 2)*L(n-(n mod 2)-i)),i=0..2((n-(n mod 2)) mod (2^floor(log_2(n))))) for all n > 1. Here F(n) and L(n) are n-th Fibonacci (A000045) and Lucas (A000032) numbers respectively.

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

Bisection: A050614.

with(combinat); A050613 := n -> product('luc(2^i)^bit_i(n,i)','i'=0..floor_log_2(n+1));
luc := n -> (fibonacci(n-1)+fibonacci(n+1));
bit_i := (n,i) -> `mod`(floor(n/(2^i)),2);
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;

A050610 Sum_{i=0..y} (C(y,i) mod 2)F(2i+x) = FL(y+x)A050613(y), where A050613(y) = Product_{i=0..floor(log_2(y+1))} L(2^i)^bit(y,i).

Original entry on oeis.org

0, 1, 1, 1, 3, 3, 2, 4, 6, 12, 3, 7, 9, 21, 21, 5, 11, 15, 33, 35, 77, 8, 18, 24, 54, 56, 126, 168, 13, 29, 39, 87, 91, 203, 273, 609, 21, 47, 63, 141, 147, 329, 441, 987, 987, 34, 76, 102, 228, 238, 532, 714, 1596, 1598, 3572, 55, 123, 165, 369, 385, 861, 1155, 2583

Offset: 0

Antti Karttunen, Oct 24 1999

Rows cut from column 2 onward form a subset of Wythoff array (A035513), where the terms of column 0 (A051656) give the positions of those rows in that array.

Transpose of A050609. First row: A000045, second row: A000032, third row: A022086.

a(n) = generic_bincoeff_fibsum_as_sum( (n-((trinv(n)*(trinv(n)-1))/2)),(((trinv(n)-1)*(((1/2)*trinv(n))+1))-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.

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))

A050611 Sum_{i=0..n} (C(n,i) mod 2)*Fibonacci(2i+1) = FL(n+1)Product(L(2^i)^bit(n,i),i=0..).

Original entry on oeis.org

1, 3, 6, 21, 35, 126, 273, 987, 1598, 5781, 12549, 45402, 76657, 277347, 602070, 2178309, 3524579, 12752046, 27682401, 100155867, 169104754, 611826747, 1328165979, 4805349654, 7782268225, 28156510947, 61122728166, 221144107989

Offset: 0

Antti Karttunen, Oct 24 1999

nonn

Second row of A050609, second column of A050610.

A050612 Sum_{i=0..n} (C(n,i) mod 2)*Fibonacci(2i+3) = FL(n+3)Product(L(2^i)^bit(n,i),i=0..).

Original entry on oeis.org

2, 7, 15, 54, 91, 329, 714, 2583, 4183, 15134, 32853, 118863, 200690, 726103, 1576239, 5702886, 9227467, 33385289, 72473466, 262211463, 442721993, 1601783218, 3477183675, 12580568721, 20374242722, 73714702663

Offset: 0

Antti Karttunen, Oct 24 1999

nonn

Fourth row of A050609, fourth column of A050610.

cp's OEIS Frontend

A048757 Sum_{i=0..2n} (C(2n,i) mod 2)*Fibonacci(i+2) = Sum_{i=0..n} (C(n,i) mod 2)*Fibonacci(2i+2).

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A050613 Products of distinct terms of 1 and rest from A001566: a(n) = Product_{i=0..floor(log_2(n+1))} L(2^i)^bit(n,i).

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A050610 Sum_{i=0..y} (C(y,i) mod 2)*F(2i+x) = FL(y+x)*A050613(y), where A050613(y) = Product_{i=0..floor(log_2(y+1))} L(2^i)^bit(y,i).

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

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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.

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A050611 Sum_{i=0..n} (C(n,i) mod 2)*Fibonacci(2i+1) = FL(n+1)Product(L(2^i)^bit(n,i),i=0..).

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A050612 Sum_{i=0..n} (C(n,i) mod 2)*Fibonacci(2i+3) = FL(n+3)Product(L(2^i)^bit(n,i),i=0..).

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A048757 Sum_{i=0..2n} (C(2n,i) mod 2)Fibonacci(i+2) = Sum_{i=0..n} (C(n,i) mod 2)Fibonacci(2i+2).

A050610 Sum_{i=0..y} (C(y,i) mod 2)F(2i+x) = FL(y+x)A050613(y), where A050613(y) = Product_{i=0..floor(log_2(y+1))} L(2^i)^bit(y,i).

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