A130234 -id:A130234 - cp's OEIS frontend

A130252 Partial sums of A130250.

0, 1, 4, 7, 11, 15, 20, 25, 30, 35, 40, 45, 51, 57, 63, 69, 75, 81, 87, 93, 99, 105, 112, 119, 126, 133, 140, 147, 154, 161, 168, 175, 182, 189, 196, 203, 210, 217, 224, 231, 238, 245, 252, 259, 267, 275, 283, 291, 299, 307, 315, 323, 331, 339, 347, 355, 363, 371

Offset: 0

Hieronymus Fischer, May 20 2007

nonn

If the initial zero is omitted, partial sums of A130253.

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

Cf. A130249, A130251, A130253, A130252, A130234, A130236, A130242, A130244.

Cf. A001045, A105348.

A001045:= func< n | (2^n - (-1)^n)/3 >;
A130252:= func< n | n eq 0 select 0 else (2*n*Ceiling(Log(2, 3*n-1)) - A001045(Ceiling(Log(2,3*n-1)) +1) +1)/2 >;
[A130252(n): n in [0..70]]; // G. C. Greubel, Mar 18 2023

A001045[n_]:= (2^n - (-1)^n)/3;
A130252[n_]:= If[n==0, 0, (2*n*Ceiling[Log[2,3*n-1]] - A001045[Ceiling[Log[2,3*n-1]]+1] +1)/2];
Table[A130252[n], {n,0,70}] (* G. C. Greubel, Mar 18 2023 *)

def A130252(n): return n*(m:=(3*n-1).bit_length())-(((1<>1) # Chai Wah Wu, Apr 17 2025

def A001045(n): return (2^n - (-1)^n)/3
def A130252(n): return 0 if (n==0) else (2*n*ceil(log(3*n-1,2)) - A001045(ceil(log(3*n-1,2)) +1) +1)/2
[A130252(n) for n in range(71)] # G. C. Greubel, Mar 18 2023

a(n) = Sum_{k=0..n} A130250(k).
a(n) = n*ceiling(log_2(3n-1)) - (1/2)*( A001045(ceiling(log_2(3n-1)) +1) - 1 ).
G.f.: (1/(1-x)^2)*Sum_{k>=0} x^A001045(k).

A130256 Minimal index k of an odd Fibonacci number A001519 such that A001519(k) = Fibonacci(2*k-1) >= n (the 'upper' odd Fibonacci Inverse).

Original entry on oeis.org

0, 0, 2, 3, 3, 3, 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, 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, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 0

Hieronymus Fischer, May 24 2007, Jul 02 2007

nonn

Inverse of the odd Fibonacci sequence (A001519), nearly, since a(A001519(n))=n except for n=1 (see A130255 for another version).

a(n+1) is the number of odd Fibonacci numbers (A001519) <= n (for n >= 0).

			a(10)=4 because A001519(4) = 13 >= 10, but A001519(3) = 5 < 10.

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. partial sums A130258.
Other related sequences: A000045, A001906, A130234, A130237, A130239, A130255, A130260.
Lucas inverse: A130241 - A130248.

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

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

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

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

A130250 Minimal index k of a Jacobsthal number such that A001045(k) >= n (the 'upper' Jacobsthal inverse).

Original entry on oeis.org

0, 1, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 0

Hieronymus Fischer, May 20 2007

nonn

Inverse of the Jacobsthal sequence (A001045), nearly, since a(A001045(n))=n except for n=2 (see A130249 for another version). a(n+1) is equal to the partial sum of the Jacobsthal indicator sequence (see A105348).

			a(10)=5 because A001045(5) = 11 >= 10, but A001045(4) = 5 < 10.

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

For partial sums see A130252.

Cf. A001045, A105348, A130234, A130242, A130249, A130253.

[0] cat [Ceiling(Log(2,3*n-1)): n in [1..120]]; // G. C. Greubel, Mar 18 2023

Table[If[n==0, 0, Ceiling[Log[2, 3*n-1]]], {n,0,120}] (* G. C. Greubel, Mar 18 2023 *)

def A130250(n): return (3*n-2).bit_length() if n else 0 # Chai Wah Wu, Apr 17 2025

def A130250(n): return 0 if (n==0) else ceil(log(3*n-1, 2))
[A130250(n) for n in range(121)] # G. C. Greubel, Mar 18 2023

a(n) = ceiling(log_2(3n-1)) = 1 + floor(log_2(3n-2)) for n >= 1.
a(n) = A130249(n-1) + 1 = A130253(n-1) for n >= 1.
G.f.: (x/(1-x))*Sum_{k>=0} x^A001045(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).

A130238 Partial sums of A130237.

Original entry on oeis.org

0, 2, 8, 20, 36, 61, 91, 126, 174, 228, 288, 354, 426, 517, 615, 720, 832, 951, 1077, 1210, 1350, 1518, 1694, 1878, 2070, 2270, 2478, 2694, 2918, 3150, 3390, 3638, 3894, 4158, 4464, 4779, 5103, 5436, 5778, 6129, 6489, 6858, 7236, 7623, 8019, 8424, 8838

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, A130239, A130240, A130243, A130246, A130248, A130239, A130251, A130253, A130257, A130261.

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

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

def A130238(n): return sum(j*int(log(3/2 +j*sqrt(5), golden_ratio)) for j in range(n+1))
[A130238(n) for n in range(71)] # G. C. Greubel, Mar 18 2023

a(n) = Sum_{k=0..n} A130237(k).
a(n) = (n*(n+1)*A130233(n) - (Fib(A130233(n)) - 1)*(Fib(A130233(n) + 1) - 1))/2.
G.f.: (1/(1-x)^3)*Sum_{k>=1} (Fib(k)*(1-x) + x)*x^Fib(k).

A130258 Partial sums of the 'upper' odd Fibonacci Inverse A130256.

Original entry on oeis.org

0, 0, 2, 5, 8, 11, 15, 19, 23, 27, 31, 35, 39, 43, 48, 53, 58, 63, 68, 73, 78, 83, 88, 93, 98, 103, 108, 113, 118, 123, 128, 133, 138, 143, 148, 154, 160, 166, 172, 178, 184, 190, 196, 202, 208, 214, 220, 226, 232, 238, 244, 250, 256, 262, 268, 274, 280, 286, 292

Offset: 0

Hieronymus Fischer, May 24 2007

nonn

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

Cf. A000045, A001519, A001906, A130234, A130235, A130236, A130255, A130257, A104162. Lucas inverse: A130241 - A130248.

[0,0] cat [(&+[Ceiling((1/2)*(1 + Log(Sqrt(5)*k-1)/Log((1+Sqrt(5))/2))): k in [2..n]]): n in [2..50]]; // G. C. Greubel, Sep 13 2018

Table[Sum[Ceiling[1/2*(1 + Log[GoldenRatio, (Sqrt[5]*k - 1)])], {k,2,n}], {n, 0, 50}] (* G. C. Greubel, Sep 13 2018 *)

for(n=0, 50, print1(if(n==0, 0, if(n==1, 0, sum(k=2, n, ceil( (1/2)*(1 + log(sqrt(5)*k - 1)/log((1+sqrt(5))/2)))))), ", ")) \\ G. C. Greubel, Sep 13 2018

a(n) = n*A130256(n) - A001906(A130256(n) -1).
a(n) = n*A130256(n) - Fib(2*A130256(n)-2) - 1.
G.f.: g(x) = x/(1-x)^2*Sum_{k>=0} x^Fib(2*k-1).

A132632 Minimal m > 0 such that Fibonacci(m) == 0 (mod n^2).

Original entry on oeis.org

1, 6, 12, 12, 25, 12, 56, 48, 108, 150, 110, 12, 91, 168, 300, 192, 153, 108, 342, 300, 168, 330, 552, 48, 625, 546, 972, 168, 406, 300, 930, 768, 660, 306, 1400, 108, 703, 342, 1092, 1200, 820, 168, 1892, 660, 2700, 552, 752, 192, 2744, 3750, 612, 1092

Offset: 1

Hieronymus Fischer, Aug 24 2007

nonn

a(n) is a divisor of the Pisano period A001175(n^2).

			a(4)=12, since Fib(12)=144==0(mod 4^2), but Fib(k) is not congruent to 0 modulo (4^2) for 1<=k<12.

Cf. A000045, A001175, A001177, A130233, A130234, A131294.

Join[{1}, Table[a = {0, 1}; k = 0; While[k++; s = Mod[Plus @@ a, n^2]; a = RotateLeft[a]; a[[2]] = s; a[[1]] != 0]; k, {n, 2, 60}]] (* T. D. Noe, Aug 08 2012 *)

a(n) = A001177(n^2)

A130262 Partial sums of the 'upper' even Fibonacci Inverse A130260.

Original entry on oeis.org

0, 1, 3, 5, 8, 11, 14, 17, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 77, 82, 87, 92, 97, 102, 107, 112, 117, 122, 127, 132, 137, 142, 147, 152, 157, 162, 167, 172, 177, 182, 187, 192, 197, 202, 207, 212, 217, 222, 227, 232, 237, 242, 248, 254, 260, 266

Offset: 0

Hieronymus Fischer, May 25 2007

nonn

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

Cf. A000045, A001519, A001906, A130234, A130235, A130236, A130256, A130259, A104161. Lucas inverse: A130241 - A130248.

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

Table[Sum[Ceiling[Log[GoldenRatio, Sqrt[5]*k]/2], {k, 1, n}], {n, 0, 60}] (* G. C. Greubel, Sep 12 2018 *)

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

a(n) = n*A130260(n) - A001519(A130260(n)) + 1.
a(n) = n*A130260(n) - Fib(2*A130260(n)-1) + 1.
G.f.: g(x)=x/(1-x)^2*Sum_{k>=0} x^Fib(2*k).

A132633 Minimal m > 0 such that Fibonacci(m) == 0 (mod n^3).

Original entry on oeis.org

1, 6, 36, 48, 125, 36, 392, 384, 972, 750, 1210, 144, 1183, 1176, 4500, 3072, 2601, 972, 6498, 6000, 3528, 3630, 12696, 1152, 15625, 7098, 26244, 2352, 11774, 4500, 28830, 24576, 21780, 5202, 49000, 3888, 26011, 6498, 42588, 48000, 33620, 3528, 81356

Offset: 1

Hieronymus Fischer, Aug 24 2007

nonn

a(n) is a divisor of the Pisano period A001175(n^3).

			a(6)=36, since Fib(36)=14930352==0(mod 6^3), but Fib(k) is not congruent to 0 modulo (6^3) for 1<=k<36.

Cf. A000045, A001175, A001177, A130233, A130234, A131294.

Join[{1}, Table[a = {0, 1}; k = 0; While[k++; s = Mod[Plus @@ a, n^3]; a = RotateLeft[a]; a[[2]] = s; a[[1]] != 0]; k, {n, 2, 50}]] (* T. D. Noe, Aug 08 2012 *)

a(n) = A001177(n^3)

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).

cp's OEIS Frontend

A130252 Partial sums of A130250.

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A130256 Minimal index k of an odd Fibonacci number A001519 such that A001519(k) = Fibonacci(2*k-1) >= n (the 'upper' odd Fibonacci Inverse).

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A130250 Minimal index k of a Jacobsthal number such that A001045(k) >= n (the 'upper' Jacobsthal inverse).

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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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A130238 Partial sums of A130237.

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A130258 Partial sums of the 'upper' odd Fibonacci Inverse A130256.

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A132632 Minimal m > 0 such that Fibonacci(m) == 0 (mod n^2).

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