A080023 -id:A080023 - cp's OEIS frontend

A222334 T(n,k)=Number of binary arrays indicating the locations of trailing edge maxima of a random length-n 0..k array extended with zeros and convolved with 1,1.

2, 2, 3, 2, 3, 4, 2, 3, 4, 6, 2, 3, 4, 7, 9, 2, 3, 4, 7, 11, 13, 2, 3, 4, 7, 11, 17, 19, 2, 3, 4, 7, 11, 18, 27, 28, 2, 3, 4, 7, 11, 18, 29, 42, 41, 2, 3, 4, 7, 11, 18, 29, 46, 66, 60, 2, 3, 4, 7, 11, 18, 29, 47, 74, 104, 88, 2, 3, 4, 7, 11, 18, 29, 47, 76, 118, 163, 129, 2, 3, 4, 7, 11, 18, 29, 47

Offset: 1

R. H. Hardin Feb 15 2013

Table starts
....2....2.....2.....2.....2.....2.....2.....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
....6....7.....7.....7.....7.....7.....7.....7.....7.....7.....7
....9...11....11....11....11....11....11....11....11....11....11
...13...17....18....18....18....18....18....18....18....18....18
...19...27....29....29....29....29....29....29....29....29....29
...28...42....46....47....47....47....47....47....47....47....47
...41...66....74....76....76....76....76....76....76....76....76
...60..104...118...122...123...123...123...123...123...123...123
...88..163...189...197...199...199...199...199...199...199...199
..129..256...303...317...321...322...322...322...322...322...322
..189..402...485...511...519...521...521...521...521...521...521
..277..631...777...824...838...842...843...843...843...843...843
..406..991..1244..1328..1354..1362..1364..1364..1364..1364..1364
..595.1556..1992..2141..2188..2202..2206..2207..2207..2207..2207
..872.2443..3190..3451..3535..3561..3569..3571..3571..3571..3571
.1278.3836..5108..5563..5712..5759..5773..5777..5778..5778..5778
.1873.6023..8180..8967..9229..9313..9339..9347..9349..9349..9349
.2745.9457.13099.14454.14912.15061.15108.15122.15126.15127.15127
Empirical: for n<=2k+1, T(n,k)=A080023(n)=A169985(n), which is A000032(n) for n>=2. - Danny Rorabaugh, Mar 13 2015

			Some solutions for n=6 k=4, one extended zero followed by filtered positions
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..0....0....1....0....1....0....1....0....1....0....0....1....0....1....0....0
..0....0....0....1....0....0....0....1....0....0....0....0....0....0....1....1
..0....1....1....0....0....1....0....0....1....0....0....0....1....0....0....0
..0....0....0....0....0....0....0....0....0....1....0....0....0....1....1....0
..0....1....0....0....1....0....0....1....1....0....1....0....0....0....0....0
..1....0....0....1....0....1....1....0....0....0....0....0....0....0....0....0

R. H. Hardin, Table of n, a(n) for n = 1..9501

Column 1 is A000930(n+2).
Column 2 is A222122.
Columns 3 to 7 are A222329 to A222333.
Cf. A000032, A080023, A169985.

Empirical for column k:
k=1: a(n) = a(n-1)+a(n-3)
k=2: a(n) = a(n-1)+a(n-3)+a(n-5)
k=3: a(n) = a(n-1)+a(n-3)+a(n-5)+a(n-7)
k=4: a(n) = a(n-1)+a(n-3)+a(n-5)+a(n-7)+a(n-9)
k=5: a(n) = a(n-1)+a(n-3)+a(n-5)+a(n-7)+a(n-9)+a(n-11)
k=6: a(n) = a(n-1)+a(n-3)+a(n-5)+a(n-7)+a(n-9)+a(n-11)+a(n-13)
k=7: a(n) = a(n-1)+a(n-3)+a(n-5)+a(n-7)+a(n-9)+a(n-11)+a(n-13)+a(n-15)

A288219 a(n) = a(n-1) + a(n-2) for n >= 3, where a(0) = 2, a(1) = 4, a(2) = 7.

Original entry on oeis.org

2, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682, 167761, 271443, 439204, 710647, 1149851, 1860498, 3010349, 4870847, 7881196, 12752043, 20633239, 33385282, 54018521, 87403803, 141422324

Offset: 0

Clark Kimberling, Jun 19 2017

Empirically, a(n) is the number of letters (0's and 1's) in the n-th iterate of the mapping 00->1000, 10->010, starting with 00; see A288216.

Clark Kimberling, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (1, 1).

Cf. A288216, A163695, A080023, A000204.

Join[{2}, LinearRecurrence[{1, 1}, {4, 7}, 40]]

a(n) = a(n-1) + a(n-2) for n >= 3, where a(0) = 2, a(1) = 4, a(2) = 7.
a(n) = L(n+2) for n >=1, where L = A000032 (Lucas numbers).
G.f.: (-2 - 2 x - x^2)/(-1 + x + x^2).

A080021 log(n) is closer to an integer than is log(m) for any m with 2<=m

Original entry on oeis.org

2, 3, 7, 20, 148, 403, 1096, 1097, 2980, 2981, 8103, 59874, 162755, 442413, 1202604, 3269017, 8886110, 8886111, 24154952, 24154953, 65659969, 178482301, 3584912846, 9744803446, 26489122130, 72004899337, 195729609428

Offset: 1

Dean Hickerson, Jan 20 2003

nonn

Every term is floor(e^k)+r for some integers k and r with k>=1 and -1 <= r <= 1.

			log(2) = 1-0.306..., log(3) = 1+0.0986..., log(7) = 2-0.0540..., log(20) = 3-0.00426...

Suggested by Leroy Quet, Jan 19 2003

Cf. A080022, A080023.

lista(nn) = {flmin = 1; for (i=2, nn, li = log(i); fli = abs(round(li) - li); if (fli < flmin, print1(i, ", "); flmin = fli;););} \\ Michel Marcus, Aug 29 2013

More terms from Don Reble, Jan 20 2003

A359997 Irregular triangle read by rows: T(n,k) is the number of directed cycles of length k in the 2-Fibonacci digraph of order n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 4, 3, 5, 4, 7, 6, 6, 6, 4, 4, 2, 2, 1, 1, 1, 1, 2, 2, 4, 5, 5, 6, 8, 10, 15, 20, 20, 24, 23, 19, 18, 20, 30, 30, 36, 36, 16, 0, 28, 28, 28

Offset: 1

Pontus von Brömssen, Jan 21 2023

See Dalfó and Fiol (2019) or A360000 for the definition of the 2-Fibonacci digraph.
Equivalently, T(n,k) is the number of cycles of length k with no adjacent 1's that can be produced by a general n-stage feedback shift register.
Apparently, the number of terms in the n-th row (i.e., the length of the longest cycle in the 2-Fibonacci digraph of order n) is A080023(n).
Interestingly, the 2-Fibonacci digraph of order 7 has cycles of all lengths from 1 up to the maximum 29, except 26. For all other orders n <= 10, there are no such gaps, i.e., the graph is weakly pancyclic.

			Triangle begins:
  n\k| 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18
  ---+-----------------------------------------------------
  1  | 1  1
  2  | 1  1  1
  3  | 1  1  1  1
  4  | 1  1  1  1  2  1  1
  5  | 1  1  1  1  2  2  1  1  2  2  2
  6  | 1  1  1  1  2  2  4  3  5  4  7  6  6  6  4  4  2  2

Pontus von Brömssen, Table of n, a(n) for n = 1..320 (rows n = 1..10; row 10 computed by Bert Dobbelaere)
C. Dalfó and M. A. Fiol, On d-Fibonacci digraphs, arXiv:1909.06766 [math.CO], 2019.

Cf. A006206 (main diagonal), A080023, A344018, A359998 (last element in each row), A359999, A360000 (row sums).

import networkx as nx
from collections import Counter
def F(n): return nx.DiGraph(((0,0),(0,1),(1,0))) if n == 1 else nx.line_graph(F(n-1))
def A359997_row(n):
    a = Counter(len(c) for c in nx.simple_cycles(F(n)))
    return [a[k] for k in range(1,max(a)+1)]

T(n,k) = A006206(k) for n >= k-1.

A169614 Triangular array: T(n,k)=integer nearest F(n)/F(n-k), k=1,2,...,n-2; n>=3, where F=A000045 (Fibonacci numbers).

Original entry on oeis.org

2, 2, 3, 2, 3, 5, 2, 3, 4, 8, 2, 3, 4, 7, 13, 2, 3, 4, 7, 11, 21, 2, 3, 4, 7, 11, 17, 34, 2, 3, 4, 7, 11, 18, 28, 55, 2, 3, 4, 7, 11, 18, 30, 45, 89, 2, 3, 4, 7, 11, 18, 29, 48, 72, 144, 2, 3, 4, 7, 11, 18, 29, 47, 78, 117, 233, 2, 3, 4, 7, 11, 18, 29, 47, 75, 126, 189, 377, 2, 3, 4, 7, 11

Offset: 1

Clark Kimberling, Dec 03 2009

Combinatorial limit of row n is A080023, which is essentially A000032.

			The first 6 rows:
2
2 3
2 3 5
2 3 4 8
2 3 4 7 13
2 3 4 7 11 21

Cf. A000032, A000045, A080023, A169613, A169615, A169616.

A080022 Numbers n such that log_pi(n) is closer to an integer than is log_pi(m) for any m with 2<=m

Original entry on oeis.org

2, 3, 10, 31, 306, 9488, 9489, 29808, 29809, 93648, 294204, 9122171, 28658146, 888582403, 8769956796, 27551631843, 86556004192, 854273519914, 2683779414318, 8431341691876, 26487841119103, 26487841119104

Offset: 1

Dean Hickerson, Jan 20 2003

nonn

Every term is floor(pi^k)+r for some integers k and r with k>=1 and -1 <= r <= 1.

			log_pi(2) = 1-0.394..., log_pi(3) = 1-0.0402..., log_pi(10) = 2+0.0114..., log_pi(31) = 3-0.000176...

Suggested by Neil Fernandez, Jan 19 2003

Cf. A080021, A080023.

lista(nn) = {flmin = 1; for (i = 2, nn, li = log(i)/log(Pi); fli = abs(round(li) - li); if (fli < flmin, print1(i, ", "); flmin = fli;););} \\ Michel Marcus, Aug 29 2013

A359998 Number of longest directed cycles in the 2-Fibonacci digraph of order n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 28, 216, 65200, 167084480

Offset: 1

Pontus von Brömssen, Jan 21 2023

See Dalfó and Fiol (2019) or A360000 for the definition of the 2-Fibonacci digraph.

The longest cycles appear to have length A080023(n).

C. Dalfó and M. A. Fiol, On d-Fibonacci digraphs, arXiv:1909.06766 [math.CO], 2019.

Cf. A016031, A080023, A359997, A360000.

a(10) from Bert Dobbelaere, Jan 24 2023

A254729 Number of numbers j + ksqrt(2) of length n, where the length is the least number of steps to reach 0, the allowable steps being x -> x + 1 and x -> xsqrt(2).

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682, 167761, 271443, 439204, 710647, 1149851, 1860498, 3010349, 4870847, 7881196, 12752043, 20633239, 33385282, 54018521, 87403803

Offset: 0

Clark Kimberling, Feb 06 2015

See the MathOverflow link for a proof that the sequence coincides with the Lucas sequence, A000032, beginning at 4.

Therefore also the same as A080023 (beginning at 2). - Georg Fischer, Oct 09 2018

			One can view the minimal paths in a tree having generation g(0) = {0} followed by generations g(1) = {1}, g(2) = {2, sqrt(2)}, g(3) = {3, 2*sqrt(2), 1+sqrt(2)}, and so on. Duplicates are removed as they occur. Also, a(n) = |g(n)| for n >= 0.

Clark Kimberling, Table of n, a(n) for n = 0..1000
MathOverflow, A possibly surprising appearance of Lucas numbers
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000032, A080023, A169985, A252864.

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x^4)/(1-x-x^2))); // G. C. Greubel, Sep 30 2018

t = NestList[DeleteDuplicates[Flatten[Map[{# + {0, 1}, {Last[#], 2*First[#]}} &, #], 1]] &, {{0, 0}}, 25] ; s[0] = t[[1]]; s[n_] := s[n] = Union[t[[n + 1]], s[n - 1]]; g[n_] := Complement[s[n], s[n - 1]]; g[0] = {{0, 0}}; Table[Length[g[z]], {z, 0, 25}]
CoefficientList[Series[(-1 + x^4)/(-1 + x + x^2), {x, 0, 39}], x] (* Robert G. Wilson v, Feb 28 2015 *)

x='x+O('x^40); Vec((1-x^4)/(1-x-x^2)) \\ G. C. Greubel, Sep 30 2018

a(n) = a(n-1) + a(n-2) for n >= 6.

G.f.: (-1 + x^4)/(-1 + x + x^2).

A090244 a(0) = 1; a(1) = 2; a(n) = { a(n-1) + a(n-2) for n even, a(n-1) - a(n-2) for n odd }.

Original entry on oeis.org

1, 2, 3, 1, 4, 3, 7, 4, 11, 7, 18, 11, 29, 18, 47, 29, 76, 47, 123, 76, 199, 123, 322, 199, 521, 322, 843, 521, 1364, 843, 2207, 1364, 3571, 2207, 5778, 3571, 9349, 5778, 15127, 9349, 24476, 15127, 39603, 24476, 64079

Offset: 0

Felix Tubiana, Jan 23 2004

Variant of Fibonacci sequence.

With the exception of the number 2, all numbers which occur in this sequence occur twice. The second occurrence is always 3 places after the first, e.g., a(0) = a(3) = 1; a(7) = a(10) = 7. In addition, if we take only one occurrence of each number and sort them, we get the ascending list: 1,2,3,4,7,11, ... [see A000032 or A080023].

G := (1+2*z+2*z^2-z^3)/(1-z^2-z^4): Gser := series(G, z = 0, 53): seq(coeff(Gser, z, n), n = 0 .. 50); # Emeric Deutsch, Jul 25 2009

G.f.: (1 + 2z + 2z^2 - z^3)/(1 - z^2 - z^4). [Emeric Deutsch, Jul 25 2009]

a(2n) = A000032(n+1) = A000204(n+1); a(2n+1) = A000032(n). [R. J. Mathar, Mar 22 2010]

Previous a(32)-a(34) removed by Georg Fischer, Apr 16 2020

cp's OEIS Frontend

A222334 T(n,k)=Number of binary arrays indicating the locations of trailing edge maxima of a random length-n 0..k array extended with zeros and convolved with 1,1.

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A288219 a(n) = a(n-1) + a(n-2) for n >= 3, where a(0) = 2, a(1) = 4, a(2) = 7.

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A080021 log(n) is closer to an integer than is log(m) for any m with 2<=m

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A359997 Irregular triangle read by rows: T(n,k) is the number of directed cycles of length k in the 2-Fibonacci digraph of order n.

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A169614 Triangular array: T(n,k)=integer nearest F(n)/F(n-k), k=1,2,...,n-2; n>=3, where F=A000045 (Fibonacci numbers).

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A080022 Numbers n such that log_pi(n) is closer to an integer than is log_pi(m) for any m with 2<=m

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A359998 Number of longest directed cycles in the 2-Fibonacci digraph of order n.

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A254729 Number of numbers j + k*sqrt(2) of length n, where the length is the least number of steps to reach 0, the allowable steps being x -> x + 1 and x -> x*sqrt(2).

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A254729 Number of numbers j + ksqrt(2) of length n, where the length is the least number of steps to reach 0, the allowable steps being x -> x + 1 and x -> xsqrt(2).