A051656 -id:A051656 - 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 *)

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).

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) );

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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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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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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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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Mathematica

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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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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Maple

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).

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).

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).