A035508 -id:A035508 - cp's OEIS frontend

A027418 Duplicate of A035508.

0, 2, 7, 20, 54, 143, 376, 986, 2583, 6764, 17710, 46367, 121392, 317810

Offset: 1

dead

Bisection of A000071 and A007492. Pairwise sums of A059840.

A333599 a(n) = Fibonacci(n) * Fibonacci(n+1) mod Fibonacci(n+2).

0, 1, 2, 1, 7, 1, 20, 1, 54, 1, 143, 1, 376, 1, 986, 1, 2583, 1, 6764, 1, 17710, 1, 46367, 1, 121392, 1, 317810, 1, 832039, 1, 2178308, 1, 5702886, 1, 14930351, 1, 39088168, 1, 102334154, 1, 267914295, 1, 701408732, 1, 1836311902, 1, 4807526975, 1, 12586269024

Offset: 0

Adnan Baysal, Mar 28 2020

			a(0) = 0*1 mod 1 = 0;
a(1) = 1*1 mod 2 = 1;
a(2) = 1*2 mod 3 = 2;
a(3) = 2*3 mod 5 = 1;
a(4) = 3*5 mod 8 = 7.

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, d'Ocagne's Identity.
Index entries for linear recurrences with constant coefficients, signature (-1,3,3,-1,-1).

Equals A035508 interleaved with A000012.

Cf. A182126, A000045, A022461, A072565, A022462.

With[{f = Fibonacci}, Table[Mod[f[n] * f[n+1], f[n+2]], {n, 0, 50}]] (* Amiram Eldar, Mar 28 2020 *)

a(n) = if (n % 2, 1, fibonacci(n+2) - 1); \\ Michel Marcus, Mar 29 2020

concat(0, Vec(x*(1 + 3*x - x^3) / ((1 + x)*(1 + x - x^2)*(1 - x - x^2)) + O(x^45))) \\ Colin Barker, Mar 29 2020

def a(n):
    f1 = 0
    f2 = 1
    for i in range(n):
        f = f1 + f2
        f1 = f2
        f2 = f
    return (f1 * f2) % (f1 + f2)

a(2n+1) = 1, and a(2n) = F(2n+2) - 1, and lim(a(2n+2)/a(2n)) = phi^2 by d'Ocagne's identity.
a(n) = F(n) * F(n+1) mod (F(n) + F(n+1)) since F(n+2) := F(n+1) + F(n).
From Colin Barker, Mar 28 2020: (Start)
G.f.: x*(1 + 3*x - x^3) / ((1 + x)*(1 + x - x^2)*(1 - x - x^2)).
a(n) = -a(n-1) + 3*a(n-2) + 3*a(n-3) - a(n-4) - a(n-5) for n>4.
(End)

A152891 a(1) = b(1) = 0; for n > 1, b(n) = b(n-1) + n-1 + a(n-1) and a(n) = a(n-1) + n-1 + b(n).

Original entry on oeis.org

0, 2, 9, 29, 83, 226, 602, 1588, 4171, 10935, 28645, 75012, 196404, 514214, 1346253, 3524561, 9227447, 24157798, 63245966, 165580120, 433494415, 1134903147, 2971215049, 7778742024, 20365011048, 53316291146, 139583862417

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 14 2008

nonn

Partial sums of A035508. - R. J. Mathar, Dec 15 2008

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (5,-8,5,-1).

Cf. A000045, A001906, A035508, A054452 (b sequence).

with(combinat): seq(fibonacci(2*n+1)-n-1, n = 1 .. 27); # Emeric Deutsch, Jun 01 2009

lst={};a=b=0;Do[b+=n+a;a+=n+b;AppendTo[lst,a],{n,0,2*4!}];lst
Table[Fibonacci[2n+1]-n-1,{n,30}] (* or *) LinearRecurrence[{5,-8,5,-1},{0,2,9,29},30] (* Harvey P. Dale, Sep 24 2013 *)

From R. J. Mathar, Dec 15 2008: (Start)
G.f.: x^2*(2 - x)/((1 - 3*x + x^2)*(1 - x)^2).
a(n) = A001906(n+1) - A001906(n) - n - 1. (End)
a(n) = Fibonacci(2*n+1) - n - 1. - Emeric Deutsch, Jun 01 2009

Name corrected by Jon E. Schoenfield, Feb 19 2019

A165279 Table read by antidiagonals: T(n, k) is the k-th number with n-1 odd-indexed Fibonacci numbers in its Zeckendorf representation.

Original entry on oeis.org

1, 3, 2, 4, 5, 7, 8, 6, 15, 20, 9, 10, 18, 41, 54, 11, 13, 19, 49, 109, 143, 12, 14, 28, 52, 130, 287, 376, 21, 16, 36, 53, 138, 342, 753, 986, 22, 17, 39, 75, 141, 363, 897, 1973, 2583, 24, 23, 40, 96, 142, 371, 952, 2350, 5167, 6764, 25, 26, 44, 104, 198, 374

Offset: 1

Clark Kimberling, Sep 13 2009

For n>=0, row n is the monotonic sequence of positive integers m such that the number of odd-indexed Fibonacci numbers in the Zeckendorf representation of m is n.
We begin the indexing at 2; that is, 1=F(2), 2=F(3), 3=F(4), 5=F(5),...
Every positive integer occurs exactly once in the array, so that as a sequence it is a permutation of the positive integers.
For counts of even-indexed Fibonacci numbers, see A165278.
Essentially, (row 0)=A054204, (column 1)=A035508.

			Northwest corner:
1....3....4....8....9...11...12...21...22...
2....5....6...10...13...14...16...17...23...
7...15...18...19...28...36...39...40...44...
20..41...49...52...53...75...96..104..107...
Examples:
12=8+3+1=F(6)+F(4)+F(2), zero odds, so 12 is in row 0.
28=21+5+2=F(8)+F(5)+F(3), two odds, so 28 is in row 2.

Cf. A165276, A165277, A165278.

f[n_] := Module[{i = Ceiling[Log[GoldenRatio, Sqrt[5]*n]], v = {}, m = n}, While[i > 1, If[Fibonacci[i] <= m, AppendTo[v, 1]; m -= Fibonacci[i], If[v != {}, AppendTo[v, 0]]]; i--]; Total[Reverse[v][[1 ;; -1 ;; 2]]]]; T = GatherBy[SortBy[ Range[10^4], f], f]; Table[Table[T[[n - k + 1, k]], {k, n, 1, -1}], {n, 1, Length[T]}] // Flatten (* Amiram Eldar, Feb 04 2020 *)

More terms from Amiram Eldar, Feb 04 2020

A210619 Triangle of numbers with n 1's and n 0's in their representation in base of Fibonacci numbers (A014417).

Original entry on oeis.org

2, 6, 7, 17, 19, 20, 46, 51, 53, 54, 122, 135, 140, 142, 143, 321, 355, 368, 373, 375, 376, 842, 931, 965, 978, 983, 985, 986, 2206, 2439, 2528, 2562, 2575, 2580, 2582, 2583, 5777, 6387, 6620, 6709, 6743, 6756, 6761, 6763, 6764, 15126, 16723, 17333, 17566, 17655, 17689, 17702, 17707, 17709, 17710

Offset: 1

Alex Ratushnyak, May 07 2012

There are n such 2n-bit numbers. For example, 17, 19, and 20 all require six bits: 100101, 101001, 101010. The least number in each group is Fib(2n+1) + Fib(2n-1) - 1, which is A005592(n). The greatest number in each group is Fib(2n+2) - 1, which is A035508(n). - T. D. Noe, May 08 2012

			Representation of 20 is 101010, three 1's and three 0's, so 20 is in the sequence.
Representation of 22 is 1000001, two 1's and five 0's, so 22 is not in the sequence.

T. D. Noe, Table of rows 1 to 49, flattened

Cf. A014417, A003714, A000045 (Fibonacci numbers).

Cf. A005592 (column k=1), A035508 (main diagonal), A249450 (second diagonal), A346434 (in Fibonacci base).

nn = 10; f = Join[{0}, Accumulate[Fibonacci[Range[2, 2*nn, 2] - 1]]]; t = Table[hi = f[[n+1]] - 1; Reverse[Table[hi - f[[i]], {i, n - 1}]], {n, 2, nn}]; t = Flatten[t] (* T. D. Noe, May 08 2012 *)

Numbers with equal counts of 1's and 0's in their Zeckendorf representation.
From Kevin Ryde, Jul 24 2021: (Start)
T(n,k) = Fibonacci(2*n+2) - Fibonacci(2*(n-k)) - 1.
G.f.: x*y*(2 - 2*x + x^2 - (1 + x + x^2)*x*y + x^3*y^2) / ( (1-x) * (1 - 3*x + x^2) * (1 - x*y) * (1 - 3*x*y + (x*y)^2) ).
(End)

A364007 Numbers k such that k and k+1 are both Wythoff-Niven numbers (A364006).

Original entry on oeis.org

3, 6, 7, 20, 39, 51, 54, 55, 90, 135, 143, 294, 305, 321, 356, 365, 369, 374, 375, 376, 784, 800, 924, 978, 979, 980, 986, 1904, 1945, 1970, 2043, 2199, 2232, 2289, 2394, 2424, 2439, 2499, 2525, 2562, 2580, 2583, 4185, 4598, 4707, 4774, 4790, 4796, 4879, 5004

Offset: 1

Amiram Eldar, Jul 01 2023

A035508(n) = Fibonacci(2*n+2) - 1 is a term for n >= 2 since A135818(Fibonacci(2*n+2) - 1) = A135818(Fibonacci(2*n+2)) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A035508, A135818.
Subsequence of A364006.
Subsequences: A364008, A364009.
Similar sequences: A330927, A328205, A328209, A328213, A330931, A331086, A333427, A334309, A331820, A342427, A344342, A351715, A351720, A352090, A352108, A352321, A352509.

seq[count_, nConsec_] := Module[{cn = wnQ /@ Range[nConsec], s = {}, c = 0, k = nConsec + 1}, While[c < count, If[And @@ cn, c++; AppendTo[s, k - nConsec]]; cn = Join[Rest[cn], {wnQ[k]}]; k++]; s]; seq[50, 2] (* using the function wnQ[n] from A364006 *)

A371592 Irregular table T(n, k), n >= 0, k = 1..2^n, read by rows; the n-th row lists the nonnegative integers whose dual Zeckendorf-binary representation has n ones.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 12, 13, 15, 20, 11, 14, 16, 17, 18, 21, 22, 23, 25, 26, 28, 33, 34, 36, 41, 54, 19, 24, 27, 29, 30, 31, 35, 37, 38, 39, 42, 43, 44, 46, 47, 49, 55, 56, 57, 59, 60, 62, 67, 68, 70, 75, 88, 89, 91, 96, 109, 143

Offset: 0

Rémy Sigrist, Mar 28 2024

As a flat sequence, this is a permutation of the nonnegative integers with inverse A371593.

			Table T(n, k) begins:
    0
    1, 2
    3, 4, 5, 7
    6, 8, 9, 10, 12, 13, 15, 20
    11, 14, 16, 17, 18, 21, 22, 23, 25, 26, 28, 33, 34, 36, 41, 54
    ...

Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

See A371590 for a similar sequence.

Cf. A001911, A035508, A112310, A371593 (inverse).

PARI
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\\ See Links section.
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A112310(T(n, k)) = n.
T(n, 1) = A001911(n).
T(n, 2^n) = A035508(n).

cp's OEIS Frontend

A027418 Duplicate of A035508.

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A333599 a(n) = Fibonacci(n) * Fibonacci(n+1) mod Fibonacci(n+2).

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A152891 a(1) = b(1) = 0; for n > 1, b(n) = b(n-1) + n-1 + a(n-1) and a(n) = a(n-1) + n-1 + b(n).

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A165279 Table read by antidiagonals: T(n, k) is the k-th number with n-1 odd-indexed Fibonacci numbers in its Zeckendorf representation.

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A210619 Triangle of numbers with n 1's and n 0's in their representation in base of Fibonacci numbers (A014417).

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A364007 Numbers k such that k and k+1 are both Wythoff-Niven numbers (A364006).

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Mathematica

A371592 Irregular table T(n, k), n >= 0, k = 1..2^n, read by rows; the n-th row lists the nonnegative integers whose dual Zeckendorf-binary representation has n ones.

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