A014437 -id:A014437 - cp's OEIS frontend

A190949 Odd Fibonacci numbers with odd index.

1, 5, 13, 89, 233, 1597, 4181, 28657, 75025, 514229, 1346269, 9227465, 24157817, 165580141, 433494437, 2971215073, 7778742049, 53316291173, 139583862445, 956722026041, 2504730781961, 17167680177565, 44945570212853, 308061521170129, 806515533049393

Offset: 1

T. D. Noe, May 24 2011

All prime Fibonacci numbers (A005478) except 2 and 3 are in this sequence. All terms equal 1 (mod 4). The indices of these Fibonacci numbers are 6k-1 or 6k+1.
This sequence can be thought of as two interlocking sequences, each of the form a(n) = 18a(n - 1) - a(n - 2).
Proof that all terms equal 1 (mod 4): From the Lucas 1876 identity Fib(2n+1) = Fib(n)^2 + Fib(n+1)^2 (see Weisstein, formula 60, or page 79 of Koshy), we see that odd-indexed Fibonacci numbers are the sum of two squares. Because a square is 0 or 1 (mod 4), the sum of two squares is 0, 1, or 2 (mod 4). All these terms are odd numbers. Hence, the only possibility is that they are 1 (mod 4). This can also be proved from the recursion formula.

Thomas Koshy, "Fibonacci and Lucas Numbers and Applications", Wiley, New York, 2001.

T. D. Noe, Table of n, a(n) for n = 1..200
Eric W. Weisstein, Fibonacci Number.
Index entries for linear recurrences with constant coefficients, signature (0,18,0,-1).

Intersection of A001519 and A014437.

Cf. A000045 (Fibonacci numbers).

LinearRecurrence[{0,18,0,-1}, {1,5,13,89}, 50]

a(n)=fibonacci(n\2*6+if(n%2,1,-1)) \\ Charles R Greathouse IV, Jun 08 2011

Vec(x*(1-x)*(1+6*x+x^2)/((1+4*x-x^2)*(1-4*x-x^2)) + O(x^30)) \\ Colin Barker, Mar 27 2016

a(n) = 18*a(n-2) - a(n-4), with a(1)=1, a(2)=5, a(3)=13, and a(4)=89.
G.f.: x*(1-x)*(1+6*x+x^2)/((1+4*x-x^2)*(1-4*x-x^2)). - Colin Barker, Jun 19 2012
a(n) = (-(2-sqrt(5))^n + (-2-sqrt(5))^n*(-2+sqrt(5)) + 2*(-2+sqrt(5))^n + sqrt(5)*(-2+sqrt(5))^n + (2+sqrt(5))^n)/(2*sqrt(5)) for n>0. - Colin Barker, Mar 27 2016
a(n) = A000045(A007310(n)). - Amiram Eldar, Jul 25 2024

A011369 a(n+1) = a(n) - F(n) if > 0, otherwise a(n) + F(n), where F() are Fibonacci numbers; a(0) = 0.

Original entry on oeis.org

0, 0, 1, 2, 4, 1, 6, 14, 1, 22, 56, 1, 90, 234, 1, 378, 988, 1, 1598, 4182, 1, 6766, 17712, 1, 28658, 75026, 1, 121394, 317812, 1, 514230, 1346270, 1, 2178310, 5702888, 1, 9227466, 24157818, 1, 39088170, 102334156, 1, 165580142, 433494438, 1, 701408734, 1836311904

Offset: 0

N. J. A. Sloane

Seiichi Manyama, Table of n, a(n) for n = 0..4000
Index entries for linear recurrences with constant coefficients, signature (1,0,4,-4,0,1,-1).

Cf. A000045, A001611, A014437.

Module[{n = 0, f}, NestList[If[(f = Fibonacci[n++]) < #, # - f, # + f] &, 0, 49]] (* Paolo Xausa, Nov 08 2024 *)
Flatten[Join[{0, 0}, Table[{1, Fibonacci[{k, k+2}] + 1}, {k, 2, 49, 3}]]] (* Paolo Xausa, Nov 08 2024 *)
LinearRecurrence[{1, 0, 4, -4, 0, 1, -1}, {0, 0, 1, 2, 4, 1, 6, 14, 1}, 50] (* Paolo Xausa, Nov 08 2024 *)

a(n) = if (n==0, 0, my(d=a(n-1)-fibonacci(n-1)); if (d>0, d, d+2*fibonacci(n-1))) \\ Michel Marcus, Dec 29 2018

a(n) = if (n<=1, 0, my(m=(n % 3)); if (m==0, fibonacci(n-1)+1, if (m==1, fibonacci(n)+1, 1))); \\ Michel Marcus, Dec 29 2018

a(n) = 0, if n <= 1; F(n-1)+1, if n == 0 (mod 3); F(n)+1, if n == 1 (mod 3); 1, if n == 2 (mod 3). - David W. Wilson; corrected by Michel Marcus, Dec 29 2018
For n>=1, a(n) = F(0)<+>F(1)<+>...<+>F(n-1), where operation <+> is defined in comment in A245618. - Vladimir Shevelev, Nov 05 2014
Empirical g.f.: -x^2*(2*x^6 - x^4 + 7*x^3 - 2*x^2 - x - 1) / ((x-1)*(x^2 + x - 1)*(x^4 - x^3 + 2*x^2 + x + 1)). - Colin Barker, Nov 06 2014

Name edited by Michel Marcus, Dec 29 2018

A014728 Squares of odd Fibonacci numbers.

Original entry on oeis.org

1, 1, 9, 25, 169, 441, 3025, 7921, 54289, 142129, 974169, 2550409, 17480761, 45765225, 313679521, 821223649, 5628750625, 14736260449, 101003831721, 264431464441, 1812440220361, 4745030099481, 32522920134769

Offset: 0

Mohammad K. Azarian

nonn

Cf. A014437.

Select[Fibonacci[Range[40]],OddQ]^2 (* Harvey P. Dale, Jan 18 2012 *)

A014728(n)=A014437(n)^2 \\ M. F. Hasler, Nov 18 2018

G.f.: (-x^5-x^4+8x^3-8x^2+x+1)/[(1+x^2)(1+4x-x^2)(1-4x-x^2)].

a(n) = A014437(n)^2. - Sean A. Irvine, Nov 18 2018

More terms from James Sellers

A181156 Odd Fibonacci numbers F which have a proper Fibonacci divisor G such that F/G is a Lucas number or a product of Lucas numbers.

Original entry on oeis.org

3, 21, 55, 377, 987, 6765, 17711, 121393, 317811, 2178309, 5702887, 39088169, 102334155, 701408733, 1836311903, 12586269025, 32951280099, 225851433717, 591286729879, 4052739537881, 10610209857723, 72723460248141, 190392490709135, 1304969544928657, 3416454622906707

Offset: 1

Vladimir Shevelev, Oct 07 2010

nonn

A conjectural statement that, for odd prime p, the ratio F_{p^2}/F_{p} is never a Lucas number or a product of some Lucas numbers, yields that
a) an odd Fibonacci number F is in the sequence iff for its maximal proper Fibonacci divisor G, we have: ind G does not equal sqrt(ind F) and F/G does not have a proper Fibonacci divisor > 3;
b) an odd Fibonacci number F is in the sequence iff its index has one of the forms: 6k+2 or 6k+4 (see A047235).

			F = 3 has the proper Fibonacci divisor G=1, and F/G = 3 is a Lucas number.
F = 317811 has the proper Fibonacci divisors 3, 13, and 377, and F/377 = 843 is a Lucas number.

Cf. A000045, A000032, A014437, A014447.

A193666 F(2) odd Fib. numbers, F(3) even Fib. numbers, F(4) odd Fib. numbers..., F(n) = A000045(n), a(n) < a(n+1).

Original entry on oeis.org

1, 2, 8, 13, 21, 55, 144, 610, 2584, 10946, 46368, 75025, 121393, 317811, 514229, 1346269, 2178309, 5702887, 9227465, 14930352, 63245986, 267914296, 1134903170, 4807526976, 20365011074, 86267571272, 365435296162, 1548008755920, 6557470319842, 27777890035288

Offset: 1

Ctibor O. Zizka, Aug 08 2011

nonn

			Taking (F(2)=) 1 odd Fib. number gives a(1)=1.
Then taking F(3)=2 even Fib. numbers starting with 2 gives a(2)=2, a(3)=8.
Then taking F(4)=3 odd Fib. numbers starting with 13 gives a(4)=13, a(5)=21, a(6)=55.
Then taking F(5)=5 even Fib. numbers starting with 144 gives a(7)=144, a(8)=610, a(9)=2584, a(10)=10946, a(11)=46368, etc...

Cf. A000045, A001614, A014437, A014445.

A384219 Areas of triangles whose three vertices are consecutive ordered pairs of consecutive odd Fibonacci numbers such that an ordered pair’s y-value is the next ordered pair’s x-value.

Original entry on oeis.org

2, 6, 24, 104, 442, 1870, 7920, 33552, 142130, 602070, 2550408, 10803704, 45765226, 193864606, 821223648, 3478759200, 14736260450, 62423800998, 264431464440, 1120149658760, 4745030099482, 20100270056686, 85146110326224, 360684711361584, 1527884955772562

Offset: 1

Angela L. Brobson, May 22 2025

As described above, the vertices of each triangle are made up exclusively of odd integers in both coordinates. The area of a parallelogram spanned by (x_1, y_1), (x_2, y_2), and (x_3, y_3) is given by abs(x_1*(y_2 - y_3) + x_2*(y_3 - y_1) + x_3*(y_1 - y_2)). This value is even since each y_i - y_j is a difference of odd integers and is thereby even. As the area of a triangle formed from three given vertices is half the area of the parallelogram spanned by those vertices, it follows that the area of each triangle formed according to the description is an integer.

			The first term is created by finding the area of the triangle formed by the ordered pairs (1,1), (1,3), and (3,5), which is 2.
The second term is created by finding the area of the triangle formed by the ordered pairs (1,3), (3,5), and (5,13), which is 6.
The third term is created by finding the area of the triangle formed by the ordered pairs (3,5), (5,13), and (13,21), which is 24.

Index entries for linear recurrences with constant coefficients, signature (4,0,4,1).

Cf. A000045, A014437.

a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <1|4|0|4>>^n.<<0, 2, 6, 24>>)[1,1]:
seq(a(n), n=1..25);  # Alois P. Heinz, May 22 2025

G.f.: 2*x*(x-1)/((x^2+1)*(x^2+4*x-1)). - Alois P. Heinz, May 22 2025

E.g.f.: (exp(2*x)*(cosh(sqrt(5)*x) + sqrt(5)*sinh(sqrt(5)*x)) - cos(x) + 3*sin(x))/5. - Stefano Spezia, May 25 2025

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A190949 Odd Fibonacci numbers with odd index.

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A011369 a(n+1) = a(n) - F(n) if > 0, otherwise a(n) + F(n), where F() are Fibonacci numbers; a(0) = 0.

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A014728 Squares of odd Fibonacci numbers.

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A181156 Odd Fibonacci numbers F which have a proper Fibonacci divisor G such that F/G is a Lucas number or a product of Lucas numbers.

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A193666 F(2) odd Fib. numbers, F(3) even Fib. numbers, F(4) odd Fib. numbers..., F(n) = A000045(n), a(n) < a(n+1).

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A384219 Areas of triangles whose three vertices are consecutive ordered pairs of consecutive odd Fibonacci numbers such that an ordered pair’s y-value is the next ordered pair’s x-value.

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