A130262 -id:A130262 - cp's OEIS frontend

A130236 Partial sums of the 'upper' Fibonacci Inverse A130234.

0, 1, 4, 8, 13, 18, 24, 30, 36, 43, 50, 57, 64, 71, 79, 87, 95, 103, 111, 119, 127, 135, 144, 153, 162, 171, 180, 189, 198, 207, 216, 225, 234, 243, 252, 262, 272, 282, 292, 302, 312, 322, 332, 342, 352, 362, 372, 382, 392, 402, 412, 422, 432, 442, 452, 462, 473

Offset: 0

Hieronymus Fischer, May 17 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A000045, A130233, A130234, A130235, A130244, A130246, A130244, A130246, A130248, A130252, A130258, A130262.

m:=120;
f:= func< x | x*(&+[x^Fibonacci(j): j in [0..Floor(3*Log(3*m+1))]])/(1-x)^2 >;
R:=PowerSeriesRing(Rationals(), m+1);
[0] cat Coefficients(R!( f(x) )); // G. C. Greubel, Mar 18 2023

b[n_]:= For[i=0, True, i++, If[Fibonacci[i] >= n, Return[i]]];
b/@ Range[0, 56]//Accumulate (* Jean-François Alcover, Apr 13 2020 *)

m=120
def f(x): return x*sum( x^fibonacci(j) for j in range(1+int(3*log(3*m+1))))/(1-x)^2
def A130236_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(x) ).list()
A130236_list(m) # G. C. Greubel, Mar 18 2023

a(n) = Sum_{k=0..n} A130234(k).
a(n) = n*A130234(n) - Fibonacci(A130234(n)+1) + 1.
G.f.: (x/(1-x)^2) * Sum_{k>=0} x^Fibonacci(k).

A130244 Partial sums of the 'upper' Lucas Inverse A130242.

Original entry on oeis.org

0, 0, 0, 2, 5, 9, 13, 17, 22, 27, 32, 37, 43, 49, 55, 61, 67, 73, 79, 86, 93, 100, 107, 114, 121, 128, 135, 142, 149, 156, 164, 172, 180, 188, 196, 204, 212, 220, 228, 236, 244, 252, 260, 268, 276, 284, 292, 300, 309, 318, 327, 336, 345, 354, 363, 372, 381, 390

Offset: 0

Hieronymus Fischer, May 19 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..2500

Other related sequences: A000032, A130241, A130243, A130245, A130246, A130248, A130252, A130258, A130262. Fibonacci inverse see A130233 - A130240, A104162.

[0,0] cat [(&+[Ceiling(Log(k + 1/2)/Log((1+Sqrt(5))/2)) : k in [0..n]]): n in [1..50]]; // G. C. Greubel, Sep 12 2018

Join[{0, 0}, Table[Sum[Ceiling[Log[GoldenRatio, k + 1/2]], {k, 0, n}], {n, 1, 50}]] (* G. C. Greubel, Sep 12 2018 *)

for(n=-1,50, print1(if(n==-1, 0, if(n==0, 0, sum(k=0, n, ceil(log(k + 1/2)/log((1+sqrt(5))/2))))), ", ")) \\ G. C. Greubel, Sep 12 2018

a(n) = Sum_{k=0..n} A130242(k).
a(n) = n*A130242(n) - A000032(A130242(n) +1) for n>=3.
G.f.: x/(1-x)^2*(2*x^2 + Sum{k>=2, x^Lucas(k)}).

A130260 Minimal index k of an even Fibonacci number A001906 such that A001906(k) = Fib(2k) >= n (the 'upper' even Fibonacci Inverse).

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 0

Hieronymus Fischer, May 25 2007, May 28 2007, Jul 02 2007

nonn

Inverse of the even Fibonacci sequence (A001906), since a(A001906(n))=n (see A130259 for another version).

a(n+1) is the number of even Fibonacci numbers (A001906) <=n.

			a(10)=4 because A001906(4)=21>=10, but A001906(3)=8<10.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Dorin Andrica, Ovidiu Bagdasar, and George Cătălin Tųrcąs, On some new results for the generalised Lucas sequences, An. Şt. Univ. Ovidius Constanţa (Romania, 2021) Vol. 29, No. 1, 17-36.

Cf. partial sums A130262. Other related sequences: A000045, A001519, A130234, A130237, A130239, A130256, A130259. Lucas inverse: A130241 - A130248.

[0] cat [Ceiling(Log(Sqrt(5)*n)/(2*Log((1+ Sqrt(5))/2))): n in [1..100]]; // G. C. Greubel, Sep 12 2018

Join[{0}, Table[Ceiling[Log[GoldenRatio, Sqrt[5]*n]/2], {n, 1, 100}]] (* G. C. Greubel, Sep 12 2018 *)

for(n=0,100, print1(if(n==0, 0, ceil(log(sqrt(5)*n)/(2*log((1+ sqrt(5))/2)))), ", ")) \\ G. C. Greubel, Sep 12 2018

a(n) = ceiling(arcsinh(sqrt(5)*n/2)/(2*log(phi))) for n>=0.
a(n) = ceiling(arccosh(sqrt(5)*n/2)/(2*log(phi))) for n>=1.
a(n) = ceiling(log_phi(sqrt(5)*n)/2)=ceiling(log_phi(sqrt(5)*n-1)/2) for n>=1, where phi=(1+sqrt(5))/2.
a(n) = A130259(n-1) + 1, for n>=1.
G.f.: g(x)=x/(1-x)*Sum_{k>=0} x^Fib(2*k).

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A130236 Partial sums of the 'upper' Fibonacci Inverse A130234.

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A130244 Partial sums of the 'upper' Lucas Inverse A130242.

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A130260 Minimal index k of an even Fibonacci number A001906 such that A001906(k) = Fib(2k) >= n (the 'upper' even Fibonacci Inverse).

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