A130238 -id:A130238 - cp's OEIS frontend

A130235 Partial sums of the 'lower' Fibonacci Inverse A130233.

0, 2, 5, 9, 13, 18, 23, 28, 34, 40, 46, 52, 58, 65, 72, 79, 86, 93, 100, 107, 114, 122, 130, 138, 146, 154, 162, 170, 178, 186, 194, 202, 210, 218, 227, 236, 245, 254, 263, 272, 281, 290, 299, 308, 317, 326, 335, 344, 353, 362, 371, 380, 389, 398, 407, 417, 427

Offset: 0

Hieronymus Fischer, May 17 2007

nonn

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

Cf. A000045, A130233, A130234, A130236, A130238, A130240, A130243, A130246, A130244, A130246, A130248, A130251, A130257, A130261.

m:=120;
f:= func< x | (&+[x^Fibonacci(j): j in [1..Floor(3*Log(3*m+1))]])/(1-x)^2 >;
R:=PowerSeriesRing(Rationals(), m+1);
[0] cat Coefficients(R!( f(x) )); // G. C. Greubel, Mar 17 2023

nmax = 90; CoefficientList[Series[Sum[x^Fibonacci[k], {k, 1, 1 + Log[3/2 + Sqrt[5]*nmax]/Log[GoldenRatio]}]/(1-x)^2, {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 14 2020 *)

m=120
def f(x): return sum( x^fibonacci(j) for j in range(1, int(3*log(3*m+1))))/(1-x)^2
def A130235_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(x) ).list()
A130235_list(m) # G. C. Greubel, Mar 17 2023

a(n) = Sum_{k=0..n} A130233(k) = (n+1)*A130233(n) - Fib(A130233(n)+2) + 1.

G.f.: 1/(1-x)^2 * Sum_{k>=1} x^Fib(k). [corrected by Joerg Arndt, Apr 14 2020]

A130240 Partial sums of A130239.

Original entry on oeis.org

0, 2, 4, 6, 9, 12, 15, 18, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220, 225, 230, 235, 240, 245, 250, 255, 260

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, A130236, A130237, A130238, A130239, A130243, A130246, A130248, A130239, A130251, A130253, A130257, A130261.

A130233:= func< n | Floor(Log(3/2 + n*Sqrt(5))/Log((1+Sqrt(5))/2)) >;
A130240:= func< n | (&+[A130233(Floor(Sqrt(j))): j in [0..n]]) >;
[A130240(n): n in [0..70]]; // G. C. Greubel, Mar 18 2023

A130233[n_]:= Floor[Log[GoldenRatio, 3/2 + n*Sqrt[5]]];
A130240[n_]:= A130240[n]= Sum[A130233[Floor[Sqrt[j]]], {j,0,n}];
Table[A130240[n], {n,0,70}] (* G. C. Greubel, Mar 18 2023 *)

def A130233(n): return int(log(3/2 +n*sqrt(5), golden_ratio))
def A130240(n): return sum( A130233(floor(sqrt(j))) for j in range(n+1) )
[A130240(n) for n in range(71)] # G. C. Greubel, Mar 18 2023

a(n) = Sum_{k=0..n} A130239(k).
a(n) = (n+1)*A130233(sqrt(n)) - Fib(A130233(sqrt(n)) + 1) * Fib(A130232(sqrt(n))).
G.f.: (1/(1-x)^2) * Sum_{k>=1} x^(Fib(k)^2).

A130237 The 'lower' Fibonacci Inverse A130233(n) multiplied by n.

Original entry on oeis.org

0, 2, 6, 12, 16, 25, 30, 35, 48, 54, 60, 66, 72, 91, 98, 105, 112, 119, 126, 133, 140, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 256, 264, 306, 315, 324, 333, 342, 351, 360, 369, 378, 387, 396, 405, 414, 423, 432, 441, 450, 459, 468, 477, 486, 550

Offset: 0

Hieronymus Fischer, May 17 2007

nonn

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

Partial sums: A130238.

Cf. A000045, A130233, A130234, A130235, A130236, A130238, A130239, A130240, A130243, A130246, A130248, A130239, A130251, A130253, A130257, A130261.

[n*Floor(Log(3/2 +n*Sqrt(5))/Log((1+Sqrt(5))/2)): n in [0..70]]; // G. C. Greubel, Mar 18 2023

Table[n*Floor[Log[GoldenRatio, 3/2 +n*Sqrt[5]]], {n,0,70}] (* G. C. Greubel, Mar 18 2023 *)

[n*int(log(3/2 +n*sqrt(5), golden_ratio)) for n in range(71)] # G. C. Greubel, Mar 18 2023

a(n) = n*A130233(n).
a(n) = n*floor(arcsinh(sqrt(5)*n/2)/log(phi)).
G.f.: (1/(1-x))*Sum_{k>=1} (Fib(k) + x/(1-x))*x^Fib(k).

A130239 Maximal index k of the square of a Fibonacci number such that Fib(k)^2 <= n (the 'lower' squared Fibonacci Inverse).

Original entry on oeis.org

0, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 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, 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

Offset: 0

Hieronymus Fischer, May 17 2007, May 28 2007

nonn

			a(10) = 4 since Fib(4)^2 = 9 <= 10 but Fib(5)^2 = 25 > 10.

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

Partial sums: A130240. Other related sequences: A000045, A130233, A130234, A130235, A130236, A130237, A130238, A130240, A130243, A130246, A130248, A130239, A130251, A130253, A130257, A130261.

A130233:= func< n | Floor(Log(3/2 + n*Sqrt(5))/Log((1+Sqrt(5))/2)) >;
[A130233(Floor(Sqrt(n))): n in [0..120]]; // G. C. Greubel, Mar 18 2023

A130233[n_]:= Floor[Log[GoldenRatio, 3/2 +n*Sqrt[5]]];
Table[A130233[Floor[Sqrt[n]]], {n, 0, 120}] (* G. C. Greubel, Mar 18 2023 *)

def A130233(n): return int(log(3/2 +n*sqrt(5), golden_ratio))
def A130239(n): return A130233(floor(sqrt(n)))
[A130239(n) for n in range(121)] # G. C. Greubel, Mar 18 2023

a(n) = max(k | Fib(k)^2 <= n) = A130233(floor(sqrt(n))).
a(n) = floor(arcsinh(sqrt(5n)/2)/log(phi)), where phi=(1+sqrt(5))/2.
G.f.: (1/(1-x))*Sum_{k>=1} x^(Fib(k)^2).

A130473 Partial sums of A087172.

Original entry on oeis.org

1, 3, 6, 9, 14, 19, 24, 32, 40, 48, 56, 64, 77, 90, 103, 116, 129, 142, 155, 168, 189, 210, 231, 252, 273, 294, 315, 336, 357, 378, 399, 420, 441, 475, 509, 543, 577, 611, 645, 679, 713, 747, 781, 815, 849, 883, 917, 951, 985, 1019, 1053, 1087, 1121, 1155, 1210

Offset: 1

Hieronymus Fischer, May 28 2007

nonn

Cf. A000045, A130233, A130234, A130235, A130236, A130238, A130240, A087172.

Accumulate[Flatten[Map[ConstantArray[Fibonacci[#],Fibonacci[#-1]]&,Range[15]]]] (* Peter J. C. Moses, May 02 2022 *)

a(n) = (1/2)*(Fib(2*b(n)+1) - 3*Fib(b(n))*Fib(b(n)+1)-1) + (n+1)*Fib(b(n)) where b(n) = A130233(n) = A130234(n+1)-1 and Fib(n)=A000045(n).

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A130235 Partial sums of the 'lower' Fibonacci Inverse A130233.

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A130240 Partial sums of A130239.

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A130237 The 'lower' Fibonacci Inverse A130233(n) multiplied by n.

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A130239 Maximal index k of the square of a Fibonacci number such that Fib(k)^2 <= n (the 'lower' squared Fibonacci Inverse).

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Magma

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A130473 Partial sums of A087172.

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Mathematica

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