A050611 -id:A050611 - cp's OEIS frontend

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).

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).

A063683 Integers formed from the reduced residue sets of even numbers and Fibonacci numbers.

Original entry on oeis.org

1, 3, 6, 21, 50, 108, 364, 987, 1938, 6150, 17622, 34776, 121160, 306852, 549000, 2178309, 5701290, 11197764, 39083988, 93031050, 191708244, 697884066, 1836283246, 3605645232, 11442062750, 32888033880, 64700678454

Offset: 1

Antti Karttunen, Jul 31 2001

nonn

a(2n) = L(2n)*a(n), where L(2n) is the 2n-th Lucas number = A000032(2n).

			The reduced residue set of 2*6 = 12 is {1,5,7,11}, thus a(6) = F_1 + F_5 + F_7 + F_11 = 108.

Cf. A054432, A054433, A050611.

A063683 := [seq(A063683_as_sum(2*n), n=1..101)]; A063683_as_sum := proc(n) local i; RETURN(add((one_or_zero(igcd(n,i))*fibonacci(i)),i=1..(n-1))); end; Yours, Antti Karttunen

a(n) = Sum_{i | gcd(i, 2n)=1} Fib(i) (where Fib(i) = A000045[i])

cp's OEIS Frontend

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).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Formula

A063683 Integers formed from the reduced residue sets of even numbers and Fibonacci numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

cp's OEIS Frontend

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).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Formula

A063683 Integers formed from the reduced residue sets of even numbers and Fibonacci numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

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).