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
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...
A225799
a(n) = Sum_{k=0..n} binomial(n,k) * 10^(n-k) * Fibonacci(n+k).
Original entry on oeis.org
0, 11, 143, 3058, 55341, 1052755, 19717984, 371084087, 6973353387, 131101759514, 2464418392865, 46327530894271, 870879506447808, 16371134451297043, 307750614069672631, 5785211638097121890, 108752568228856901349, 2044371455527726003547, 38430858858805840293152
Offset: 0
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Table[Sum[Binomial[n, k]*10^(n - k)*Fibonacci[n + k], {k, 0, n}], {n, 0, 25}]
FullSimplify[Table[((13 + 11 Sqrt[5])^n - (13 - 11 Sqrt[5])^n)/(2^n Sqrt[5]), {n, 0, 25}]]
LinearRecurrence[{13,109},{0,11},30] (* Harvey P. Dale, Jul 31 2018 *)
A272123
a(n) = Fibonacci(3n) - Fibonacci(2n).
Original entry on oeis.org
0, 1, 5, 26, 123, 555, 2440, 10569, 45381, 193834, 825275, 3506867, 14883984, 63124593, 267596485, 1134071130, 4805348667, 20359308187, 86252640920, 365396207993, 1547906421765, 6557202405546, 27777188626555, 117667194149091, 498449204352288
Offset: 0
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a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>,
<1|1|-12|7>>^n. <<0, 1, 5, 26>>)[1, 1]:
seq(a(n), n=0..30); # Alois P. Heinz, Apr 21 2016
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Table[Fibonacci[3n] - Fibonacci[2n], {n, 0, 25}] (* Robert Price, Apr 21 2016 *)
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a(n) = fibonacci(3*n) - fibonacci(2*n); \\ Michel Marcus, Apr 21 2016
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