A108852 -id:A108852 - cp's OEIS frontend

A345377 Number of terms m <= n, where m is a term in A006190.

1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 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, 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

Offset: 0

Ovidiu Bagdasar, Jun 16 2021

Table 1 of Andrica 2021 paper (p. 24) refers to A006190 as the "bronze Fibonacci" numbers.

			a(0)=1, since A006190(0) = 0 and A006190(1) = 1.
a(1)=a(2)=2 since 0 and 1 are the terms in A006190 that do not exceed 1 and 2, respectively.
a(k)=3 for 3 <= k <= 9 since the first terms of A006190 are {0, 1, 3, 10}.

Michael De Vlieger, 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. See Section 5.3, pp. 33, Table 4.

Cf. A006190, A108852 (Fibonacci), A130245 (Lucas), A345378.

Block[{a = 3, b = -1, nn = 105, u, v = {}}, u = {0, 1}; Do[AppendTo[u, Total[{-b, a} u[[-2 ;; -1]]]]; AppendTo[v, Count[u, _?(# <= i &)]], {i, nn}]; {Boole[First[u] <= 0]}~Join~v] (* or *)
Accumulate@ ReplacePart[ConstantArray[0, Last[#]], Map[# -> 1 &, # + 1]] &@ Fibonacci[Range[0, 5], 3] (* Michael De Vlieger, Jun 16 2021 *)

A345378 Number of terms m <= n, where m is a term in A006497.

Original entry on oeis.org

0, 0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

Ovidiu Bagdasar, Jun 16 2021

Table 1 of Andrica 2021 paper (p. 24), refers to A006497 as "bronze Lucas" numbers.

			a(0)=a(1)=0, since the least term in A006497 is 2.
a(2)=1 since A006497(0) = 2 is followed in that sequence by 3.
a(k)=3 for 3 <= k <= 11 since the first terms of A006490 are {0, 2, 3, 11}.

Michael De Vlieger, 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. See Section 5.3, pp. 33, Table 4.

Cf. A006497, A108852 (Fibonacci), A130245 (Lucas), A345377.

Block[{a = 3, b = -1, nn = 105, u, v = {}}, u = {0, 1}; Do[AppendTo[u, Total[{-b, a} u[[-2 ;; -1]]]]; AppendTo[v, Count[u, _?(# <= i &)]], {i, nn}]; {Boole[First[u] <= 0]}~Join~v] (* or *)
{0}~Join~Accumulate@ ReplacePart[ConstantArray[0, Last[#]], Map[# -> 1 &, #]] &@ LucasL[Range[0, 4], 3] (* Michael De Vlieger, Jun 16 2021 *)

A137203 Number of Fibonacci numbers less than or equal to n^2.

Original entry on oeis.org

1, 3, 5, 7, 8, 9, 10, 10, 11, 11, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20

Offset: 0

Parthasarathy Nambi, Apr 04 2008

nonn

			When n=1, the number of Fibonacci numbers less than or equal to 1 is 3.
When n=2, the number of Fibonacci numbers less than or equal to 4 is 5.
When n=3, the number of Fibonacci numbers less than or equal to 9 is 7.
When n=4, the number of Fibonacci numbers less than or equal to 16 is 8.

David A. Corneth, Table of n, a(n) for n = 0..9999
Planet Math, list of Fibonacci numbers.

Cf. A000045, A108852.

fibPi[n_] := 1 + Floor[ Log[ GoldenRatio, 1 + n*Sqrt@ 5]]; Array[ fibPi[#^2] &, 80, 0] (* Robert G. Wilson v, Aug 03 2014 *)

first(n) = { res = vector(n+1); fibs = List([0, 1]); i = 2; n2 = n^2; f = 1; while(f <= n2, listput(fibs, f); i++; f = fibonacci(i) ); for(i = 1, #fibs, res[ceil(sqrt(fibs[i]))+1]++ ); for(i = 2, #res, res[i]+=res[i-1]); res} \\ David A. Corneth, Mar 28 2021

a(n) = A108852(n^2). - Michel Marcus, Aug 03 2014

Corrected and extended by Robert G. Wilson v, Aug 03 2014

A345347 Find the largest k with F(k) <= n, where F(k) is the k-th Fibonacci number. a(n) = F(k+2) + n.

Original entry on oeis.org

1, 4, 7, 11, 12, 18, 19, 20, 29, 30, 31, 32, 33, 47, 48, 49, 50, 51, 52, 53, 54, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 199, 200, 201, 202, 203, 204

Offset: 0

Peter Munn, Jun 14 2021

The terms consist of 1 together with numbers that appear in row m of the Wythoff array (A035513) if m is in the sequence.
a(0) = 1, otherwise a(n) is the number whose Zeckendorf representation is "10" followed by the Zeckendorf representation of n.
If we define an extended Zeckendorf representation to be the Zeckendorf representation with "01" appended, then the numbers in the sequence are exactly those whose extended representation starts 101... . This extended representation is a valid Fibonacci base representation if we specify the rightmost digit to have weight F(0) = 0.
Equivalently, for positive integer m, find the largest k with F(k) <= m, where F(k) is the k-th Fibonacci number. m is in the sequence if and only if m >= F(k) + F(k-2).
Numbers given to rabbits on Rabbit 1's branch of the generation tree described in the A035513 examples.
Equivalently, take the positive integers in turn, placing runs of them alternatively into 2 sets, with run lengths from A053602/A051792 (self-interleaved Fibonacci sequence) as follows:
   set A: 1   0   1   1   2   3   5 ...
   set B:   1   1   2   3   5   8 ...
The sequence lists the numbers in set A.

			The initial Fibonacci numbers are F(0)..F(5) = 0, 1, 1, 2, 3, 5.
For n = 0, the largest k with F(k) <= 0 is k = 0, so F(k+2) = F(2) = 1, so a(0) = 1 + 0 = 1.
For n = 1, the largest k with F(k) <= 1 is k = 2, so F(k+2) = F(4) = 3, so a(1) = 3 + 1 = 4.
For n = 4, the largest k with F(k) <= 4 is k = 4, so F(k+2) = F(6) = 8, so a(4) = 8 + 4 = 12.
In the paragraph that follows we use the Wythoff array-based definition from the start of the comments.
Every positive integer appears once (only) in the Wythoff array. 0 is not positive, so does not appear in the array, so is not in the sequence. 1 is in the sequence by definition. 2 appears in Wythoff row 0, and 0 is not in the sequence, so 2 is not in the sequence. 4 appears in Wythoff row 1, and 1 is in the sequence, so 4 is in the sequence.

Encyclopedia of Mathematics, Zeckendorf representation.
OEIS Wiki, Zeckendorf representation.
N. J. A. Sloane, Classic Sequences.

Cf. A000045, A035513, A051792, A053602, A108852.

Appears to be column 1 of A194030.

kmax=12;Flatten[Table[Range[Fibonacci[k]+Fibonacci[k-2],Fibonacci[k+1]-1],{k,2,kmax}]] (* Paolo Xausa, Jan 02 2022 *)
A108852[n_]:=1+Floor[Log[GoldenRatio,1+n*Sqrt[5]]];
nterms=100;Table[n+Fibonacci[1+A108852[n]],{n,0,nterms-1}](* Paolo Xausa, Jan 02 2022 *)

a(n) = my(k=0); while(fibonacci(k)<=n, k=k+1); n+fibonacci(k+1)

a(n) = A000045(A108852(n)+1) + n.

Union_{k >= 2} {m : F(k)+F(k-2) <= m < F(k+1)}, where F(k) = A000045(k).

A345376 Number of Companion Pell numbers m <= n.

Original entry on oeis.org

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

Offset: 0

Ovidiu Bagdasar, Jun 16 2021

Table 1 of Andrica 2021 paper (p. 24) refers to A002203 as "Pell-Lucas" numbers.

			The Pell-Lucas numbers A002203 are 2, 2, 6, 14, 34, 82, ...
a(0)=a(1)=0, since there are no Pell-Lucas numbers less than or equal to 0 and 1, respectively.
a(2)=a(3)=a(4)=a(5)=2, since the first 2 Pell-Lucas numbers, 2 and 2, are less than or equal to 2, 3, 4, and 5, respectively.

Michael De Vlieger, 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. See Section 5.2, pp. 32-33, Table 3.

Cf. A002203, A108852 (Fibonacci), A130245 (Lucas), A335741 (Pell).

Block[{a = 2, b = -1, nn = 64, u, v = {}}, u = {2, a}; Do[AppendTo[u, Total[{-b, a} u[[-2 ;; -1]]]]; AppendTo[v, Count[u, ?(# <= i &)]], {i, nn}]; {Boole[First[u] <= 0]}~Join~v]  (* _Michael De Vlieger, Jun 16 2021 *)

A345379 Number of terms m <= n, where m is a term in the bisection of Lucas numbers (A005248).

Original entry on oeis.org

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

Offset: 0

Ovidiu Bagdasar, Jun 16 2021

			a(0)=a(1)=0, since the least term in A005248 is 2.
a(2)=1 since A005248(0) = 2 is followed in that sequence by 3.
a(k)=3 for 3 <= k <= 6 since the first terms of A005248 are {0, 2, 3, 7}.

Michael De Vlieger, 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. See Section 5.4, pp. 33-34, Table 4.

Cf. A005248, A108852 (Fibonacci), A130245 (Lucas), A130260.

Block[{a = 3, b = 1, nn = 105, u, v = {}}, u = {2, a}; Do[AppendTo[u, Total[{-b, a} u[[-2 ;; -1]]]]; AppendTo[v, Count[u, _?(# <= i &)]], {i, nn}]; {Boole[First[u] <= 0]}~Join~v] (* or *)
{0}~Join~Accumulate@ ReplacePart[ConstantArray[0, Last[#]], Map[# -> 1 &, #]] &@ LucasL@ Range[0, 10, 2] (* Michael De Vlieger, Jun 16 2021 *)

A345380 Number of Jacobsthal-Lucas numbers m <= n.

Original entry on oeis.org

0, 1, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 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, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 0

Ovidiu Bagdasar, Jun 16 2021

			a(0)=0 since the least term in A014551 is 1.
a(1)=1 since A014551(0) = 2 is followed in that sequence by 1.
a(k)=2 for 2 <= k <= 4 since the first terms of A014551 are {2, 1, 5}.

Michael De Vlieger, 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. See Section 5.6, pp. 35, Table 7.

Cf. A014551, A108852 (Fibonacci), A130245 (Lucas), A130253.

Block[{a = 1, b = -2, nn = 105, u, v = {}}, u = {2, a}; Do[AppendTo[u, Total[{-b, a} u[[-2 ;; -1]]]]; AppendTo[v, Count[u, _?(# <= i &)]], {i, nn}]; {Boole[First[u] <= 0]}~Join~v]  (* or *)
{0}~Join~Accumulate@ ReplacePart[ConstantArray[0, Last[#]], Map[# -> 1 &, #]] &@ LinearRecurrence[{1, 2}, {2, 1}, 8] (* Michael De Vlieger, Jun 16 2021 *)

A128185 A097806 * A051731.

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 0, 0, 1, 1, 2, 1, 0, 1, 0, 0, 1, 1, 2, 1, 1, 1, 0, 0, 0, 1, 1, 2, 1, 1, 0, 1, 0, 0, 0, 1, 1

Offset: 1

Gary W. Adamson, Feb 17 2007

A128184 = A051731 * A097806 Row sums = 1, 3, 4, 5, 5, 6, 6, 6, 7, 7, ... (the first 10 terms of A108852).

			First few rows of the triangle:
  1;
  2, 1;
  2, 1, 1;
  2, 1, 1, 1;
  2, 1, 0, 1, 1;
  2, 1, 1, 0, 1, 1;
  2, 1, 1, 0, 0, 1, 1;
  2, 1, 0, 1, 0, 0, 1, 1;
  ...

Cf. A128184, A097806, A051731, A108852.

A097806 * A051731 as infinite lower triangular matrices.

cp's OEIS Frontend

A345377 Number of terms m <= n, where m is a term in A006190.

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A345378 Number of terms m <= n, where m is a term in A006497.

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A137203 Number of Fibonacci numbers less than or equal to n^2.

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A345347 Find the largest k with F(k) <= n, where F(k) is the k-th Fibonacci number. a(n) = F(k+2) + n.

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A345376 Number of Companion Pell numbers m <= n.

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A345379 Number of terms m <= n, where m is a term in the bisection of Lucas numbers (A005248).

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A345380 Number of Jacobsthal-Lucas numbers m <= n.

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A128185 A097806 * A051731.

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