A050613 -id:A050613 - cp's OEIS frontend

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).

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) );

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).

Original entry on oeis.org

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 *)

A050614 Products of distinct terms of A001566: a(n) = Product_{i=0..floor(log_2(n+1))} L(2^(i+1))^bit(n,i).

Original entry on oeis.org

1, 3, 7, 21, 47, 141, 329, 987, 2207, 6621, 15449, 46347, 103729, 311187, 726103, 2178309, 4870847, 14612541, 34095929, 102287787, 228929809, 686789427, 1602508663, 4807525989, 10749959329, 32249877987, 75249715303

Offset: 0

Antti Karttunen, Dec 02 1999

nonn

Each subset a(0..(2^k)-1) gives all the divisors of F(2^(k+1)) up to k=4 (F_32) and after that a subset of such divisors. E.g., the terms a(0)-a(7) are the divisors of F_16 = 987 (A018760).

Robert Price, Table of n, a(n) for n = 0..200
A. Karttunen, On Pascal's Triangle Modulo 2 in Fibonacci Representation, Fibonacci Quarterly, 42 (2004), 38-46.

Bisection of A075149 and A050613 (see there for the other Maple procedures), subset of A062877. Cf. also A050615.

[seq(A050614(n),n=0..30)]; A050614 := n -> product('luc(2^(i+1))^bit_i(n,i)','i'=0..floor_log_2(n+1));

Table[k = Floor[Log[2, n + 1]]; Product[j = 2^(i + 1); l = Fibonacci[j + 1] + Fibonacci[j - 1]; If[BitAnd[2^i, n] == 0, b = 0, b = 1]; l^b, {i, 0, k}], {n, 0, 200}] (* Robert Price, Feb 13 2017 *)

a(n) = Sum_{k=0..n} A127872(n,k)*Fibonacci(2*k+1), see A000045 and A001519. - Philippe Deléham, Aug 30 2007

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).

A050615 Products of distinct terms of Fibonacci(2^(i+2)): a(n) = Product_{i=0..floor(log_2(n+1))} F(2^(i+2))^bit(n,i).

Original entry on oeis.org

1, 3, 21, 63, 987, 2961, 20727, 62181, 2178309, 6534927, 45744489, 137233467, 2149990983, 6449972949, 45149810643, 135449431929, 10610209857723, 31830629573169, 222814407012183, 668443221036549, 10472277129572601

Offset: 0

Antti Karttunen, Dec 02 1999

nonn

Cf. A050613, A050614.

distinct_fib_products := n -> product('fib(2^(i+2))^bit_i(n,i)','i'=0..floor_log_2(n+1));

cp's OEIS Frontend

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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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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A050614 Products of distinct terms of A001566: a(n) = Product_{i=0..floor(log_2(n+1))} L(2^(i+1))^bit(n,i).

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Mathematica

Formula

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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Programs

Maple

Formula

A050615 Products of distinct terms of Fibonacci(2^(i+2)): a(n) = Product_{i=0..floor(log_2(n+1))} F(2^(i+2))^bit(n,i).

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Maple

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).

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).