A178521 -id:A178521 - cp's OEIS frontend

A354044 a(n) = 2(-i)^n(nsin(c(n+1)) - isin(-cn))/sqrt(5) where c = arccos(i/2).

0, 2, 5, 11, 23, 45, 86, 160, 293, 529, 945, 1673, 2940, 5134, 8917, 15415, 26539, 45525, 77842, 132716, 225685, 382877, 648165, 1095121, 1846968, 3109850, 5228261, 8777315, 14716223, 24643389, 41220110, 68873848, 114964805, 191719849, 319436697, 531789785

Offset: 0

Peter Luschny, May 16 2022

Jianing Song, Table of n, a(n) for n = 0..1000
Peter Luschny, Illustration of A354044.
Peter Luschny, The Fibonacci Function.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A000045 (the Fibonacci numbers), A007502, A088209, A094588, A136391, A178521, A264147, A353595.

function fibrec(n::Int)
    n == 0 && return (BigInt(0), BigInt(1))
    a, b = fibrec(div(n, 2))
    c = a * (b * 2 - a)
    d = a * a + b * b
    iseven(n) ? (c, d) : (d, c + d)
end
function A354044(n)
    n == 0 && return BigInt(0)
    a, b = fibrec(n + 1)
    a*(n - 1) + b
end
println([A354044(n) for n in 0:35])

c := arccos(I/2): a := n -> 2*(-I)^n*(n*sin(c*(n+1)) - I*sin(-c*n))/sqrt(5):
seq(simplify(a(n)), n = 0..35);

a(n) = fibonacci(n) + n*fibonacci(n+1) \\ Jianing Song, May 16 2022

a(n) = [x^n] ((2 - x)*x*(x + 1))/(x^2 + x - 1)^2.
a(n) = (((-1 - sqrt(5))^(-n)*(sqrt(5)*n - n - 2) + (-1 + sqrt(5))^(-n)*(sqrt(5)*n + n + 2)))/(2^(1 - n)*sqrt(5)).
a(n) = (-1)^(n - 1)*(Fibonacci(-n) - n*Fibonacci(-n - 1)).
a(n) = (-1)^(n - 1)*A353595(-n, -n). (A353595 is defined for all n in Z.)
a(n) = ((-42*n^2 + 259*n - 350)*a(n - 3) + (123*n^2 - 76*n - 446)*a(n - 2) + (207*n^2 - 885*n + 412)*a(n - 1)) / ((165*n - 542)*(n - 1)) for n >= 4.
a(n) = Fibonacci(n) + n*Fibonacci(n+1). - Jianing Song, May 16 2022

A178525 The sum of the costs of all nodes in the Fibonacci tree of order n.

Original entry on oeis.org

0, 0, 3, 8, 22, 49, 104, 208, 403, 760, 1406, 2561, 4608, 8208, 14499, 25432, 44342, 76913, 132808, 228416, 391475, 668840, 1139518, 1936513, 3283392, 5555424, 9381699, 15815528, 26618518, 44733745, 75073256, 125827696, 210642643

Offset: 0

Emeric Deutsch, Jun 15 2010

nonn

A Fibonacci tree of order n (n>=2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node. In a Fibonacci tree the cost of a left (right) edge is defined to be 1 (2). The cost of a node in a Fibonacci tree is defined to be the sum of the costs of the edges that form the path from the root to this node.

A178525 is the 1-sequence of reduction of the odd number sequence (2n-1) by x^2 -> x+1; as such it is related to 0-sequence of this reduction, A192304. See A192232 for definition of "k-sequence of reduction of [sequence] by [substitution]". - Clark Kimberling, Jun 27 2011

D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Y. Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly, 20, No. 2, 1982, 168-178.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,1,1).

Cf. A000045, A094584, A178521.

List([0..40], n -> 3 +(2*n-3)*Fibonacci(n-1) +(2*n-5)*Fibonacci(n)); # G. C. Greubel, Jan 30 2019

[3 +(2*n-3)*Fibonacci(n-1) +(2*n-5)*Fibonacci(n): n in [0..40]]; // G. C. Greubel, Jan 30 2019

with(combinat): seq(3+(2*n-3)*fibonacci(n-1)+(2*n-5)*fibonacci(n), n = 0 .. 32);

Table[3 +(2*n-3)*Fibonacci[n-1] +(2*n-5)*Fibonacci[n], {n,0,40}] (* G. C. Greubel, Jan 30 2019 *)

a(n) = 3+(2*n-3)*fibonacci(n-1) + (2*n-5)*fibonacci(n); \\ Michel Marcus, Jan 21 2019

[3 +(2*n-3)*fibonacci(n-1) +(2*n-5)*fibonacci(n) for n in range(40)] # G. C. Greubel, Jan 30 2019

a(n) = 3 + (2*n-3)*F(n-1) + (2*n-5)*F(n), where F(k)=A000045(k) are the Fibonacci numbers.
a(n) = a(n-1) + a(n-2) + 2*F(n+1) + 2*F(n-1) - 3 (n>=2), F(0)=0, F(1)=0.
G.f.: z^2*(3-z+z^2)/((1-z)*(1-z-z^2)^2).

A264147 a(n) = nF(n+1) - (n+1)F(n), where F = A000045.

Original entry on oeis.org

0, -1, 1, 1, 5, 10, 22, 43, 83, 155, 285, 516, 924, 1639, 2885, 5045, 8773, 15182, 26162, 44915, 76855, 131119, 223101, 378696, 641400, 1084175, 1829257, 3081193, 5181893, 8702290, 14594830, 24446971, 40902299, 68359619, 114132765, 190373580, 317258388, 528265207

Offset: 0

Bruno Berselli, Nov 04 2015

a(n) is prime for n = 4, 7, 8, 26, 28, 52, 86, 87, 93, 97, 158, 196, 303, 2908, 3412, 4111, 4208, 6183, 6337, 9878, ...

Bruno Berselli, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A000045, A001629, A007502, A354044.
Cf. A178521: n*F(n+1) + (n+1)*F(n).
Cf. A094588: n*F(n-1) + F(n).
Cf. A099920: Sum_{i=0..n} F(i)*L(n-i).
Cf. A023607: Sum_{i=0..n} F(i)*L(n+1-i).

# The function 'fibrec' is defined in A354044.
function A264147(n)
    n == 0 && return BigInt(0)
    a, b = fibrec(n)
    n*b - a*(n + 1)
end # Peter Luschny, May 16 2022

[n*Fibonacci(n+1)-(n+1)*Fibonacci(n): n in [0..40]];

A264147 := proc(n)
    n*combinat[fibonacci](n+1)-(n+1)*combinat[fibonacci](n) ;
end proc:
seq(A264147(n),n=0..10) ; # R. J. Mathar, Jun 02 2022

Table[n Fibonacci[n + 1] - (n + 1) Fibonacci[n], {n, 0, 40}]

makelist(n*fib(n+1)-(n+1)*fib(n), n, 0, 40);

for(n=0, 40, print1(n*fibonacci(n+1)-(n+1)*fibonacci(n)", "));

concat(0, Vec(-x*(1 - 3*x) / (1 - x - x^2)^2 + O(x^50))) \\ Colin Barker, Jul 27 2017

[n*fibonacci(n+1)-(n+1)*fibonacci(n) for n in (0..40)]

G.f.:  x*(-1 + 3*x)/(1 - x - x^2)^2.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4).
a(n) = n*F(n-1) - F(n).
a(n) = Sum_{i=0..n} F(i)*L(n-1-i), where L() is a Lucas number (A000032).
a(n) = 3*A001629(n) - A001629(n+1).
a(n) = -(-1)^n*A178521(-n).
a(n+2) - a(n) = A007502(n+1).
Sum_{i>0} 1/a(i) = 1.39516607051636028893879220294180374...
a(n) = (-((1+sqrt(5))/2)^n*(2*sqrt(5) + (-5+sqrt(5))*n) + ((1-sqrt(5))/2)^n*(2*sqrt(5) + (5+sqrt(5))*n)) / 10. - Colin Barker, Jul 27 2017
a(n) = (-i)^n*(n*sin(c*(n+1)) - (n+1)*sin(c*n)*i)/sqrt(5/4) where c = arccos(i/2). - Peter Luschny, May 16 2022

cp's OEIS Frontend

A354044 a(n) = 2*(-i)^n*(n*sin(c*(n+1)) - i*sin(-c*n))/sqrt(5) where c = arccos(i/2).

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A178525 The sum of the costs of all nodes in the Fibonacci tree of order n.

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A264147 a(n) = n*F(n+1) - (n+1)*F(n), where F = A000045.

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A354044 a(n) = 2(-i)^n(nsin(c(n+1)) - isin(-cn))/sqrt(5) where c = arccos(i/2).

A264147 a(n) = nF(n+1) - (n+1)F(n), where F = A000045.