Colin Linzer - cp's OEIS frontend

User: Colin Linzer

Colin Linzer has authored 11 sequences. Here are the ten most recent ones:

A377813 Decimal expansion of arctanh(phi-1).

7, 2, 1, 8, 1, 7, 7, 3, 7, 5, 8, 9, 4, 0, 5, 1, 7, 1, 2, 4, 6, 6, 3, 8, 3, 7, 0, 1, 3, 6, 5, 5, 2, 6, 3, 4, 7, 0, 2, 7, 7, 6, 5, 0, 1, 5, 7, 8, 4, 9, 0, 7, 7, 9, 4, 9, 1, 5, 2, 7, 2, 5, 3, 2, 6, 0, 2, 4, 5, 8, 0, 1, 4, 1, 2, 3, 3

Offset: 0

Colin Linzer, Nov 08 2024

arctanh(phi-1) is the solution for real valued x in tanh(x) = d/dx tanh(x).

arctanh(phi-1) is the solution for real valued x in cosh(x) * sinh(x) = 1. - Colin Linzer, Nov 22 2024

			0.721817737589405171246638370136552634702...

Index entries for transcendental numbers

Cf. A001622, A202541.

RealDigits[ArcTanh[GoldenRatio - 1], 10, 120][[1]] (* Amiram Eldar, Nov 12 2024 *)

PARI
```
atanh((1+sqrt(5))/2-1)
```

Equals (1/2)*log(2+sqrt(5)).
Equals (3/2)*log(phi).
Equals arccosh(sqrt(phi)).
Equals arcsinh(sqrt(phi-1)).
Equals f(phi-1) with f(x) = (1/2)*log((2-x+2*sqrt(1-x))/x), a branch of the converse function of the derivative of tanh(x).
Equals 3*A202541. - Hugo Pfoertner, Nov 12 2024
Equals arcsinh(2)/2. - Colin Linzer, Nov 22 2024

A371627 The x-coordinate of the point where x + y = n, x is an integer and x/y is as close as possible to 1/phi.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 13, 13, 13, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 26, 26, 26, 27, 27, 27

Offset: 1

Colin Linzer, Mar 29 2024

Each term is equal to or one greater than the previous term.
The average run length approaches 1+phi.
a(n) = x = either ceiling or floor of n/phi^2, according to which minimizes abs(x/(n-x) - phi).
The 4 following statements are equivalent for any positive integer n and any function f(x) such that for all real x, x-1  a(n) != A371626(n);
  A371625(n) != y(n);
  a(n) != n-f(n/phi) xor A371626(n) != n-f(n/phi);
  A371625(n) != f(n/phi) xor y(n) != f(n/phi).

			For n=4, the possibilities are (0,4), (1,3), (2,2), and (3,1). 1/3 is the closest to 1/phi out of them, so a(4)=1.

Cf. A094214 (1/phi), A371625 (with phi).

Elements referring to sequences that were not submitted removed by Peter Munn, Aug 04 2025

A371626 The y-coordinate of the point where x + y = n, x is an integer and x/y is as close as possible to phi (by absolute difference).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 13, 13, 13, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 26, 26, 26, 27, 27, 28

Offset: 1

Colin Linzer, Mar 29 2024

a(n) = k = n - either ceiling or floor of n/phi, according to which minimizes abs((n-k)/k - phi).
Each term is equal to or one greater than the previous term.
The average run length approaches 1+phi.
The 2 following statements are equivalent for any positive integer n and any function f(x) such that for any real x, f(x) equals a integer within the range (x-1,x+1):
  a(n) != A371627(n);
  a(n) != n-f(n/phi) xor A371627(n) != n-f(n/phi).
Let s(n) = (phi*n - 1 - sqrt(1+(n^2)*(phi^-4)))/2.
Floor(s(n)) equals the number of times that a(n) swapped from being equal to n-floor(n/phi) to being equal to n-ceiling(n/phi) when n is extended to the reals.
This is true because s(n) is the solution to the equation n = (phi/4) * (phi(2r+1) + sqrt((2r+1)^2 / phi^4 + 4/phi)) solved for w. The equation gives the n-value of w-th swap from a(n) = n-floor(n/phi) to a(n) = n-ceiling(n/phi).
s(n) is asymptotic to n/phi - 1/2.
Floor(s(n)) != floor(n/phi - 1/2) <-> a(n) != round(n).
Floor(n/phi) equals the number of that a(n) swapped from being equal to n-ceiling(n/phi) to being equal to n-floor(n/phi) when n is extended to the reals.

			For n=5, the possibilities are (0,5), (1,4), (2,3), (3,2) and (4,1). Of those, 3/2 is the closest to phi, so a(5)=3.

Cf. A001622 (phi), A371625 (x_coordinate).

a(n) = n - A371625(n).
Let s(n) = (phi*n - 1 - sqrt(1+(n^2) / phi^4))/2.
Floor(s(n))+floor(n/phi) is even -> a(n) = n-ceiling(n/phi) = (n mod 1) + floor(n/phi^2).
Floor(s(n))+floor(n/phi) is odd -> a(n) = n-floor(n/phi) = (n mod 1) + ceiling(n/phi^2).
a(n) = -a(-n).

Elements referring to sequences that were not submitted removed by Peter Munn, Aug 04 2025

A371625 The x-coordinate of the point (x,y) where x + y = n, x is an integer, and x/y is as close as possible to phi (by absolute difference).

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 9, 9, 10, 10, 11, 12, 12, 13, 14, 14, 15, 15, 16, 17, 17, 18, 19, 19, 20, 20, 21, 22, 22, 23, 23, 24, 25, 25, 26, 27, 27, 28, 28, 29, 30, 30, 31, 32, 32, 33, 33, 34, 35, 35, 36, 36, 37, 38, 38, 39, 40, 40, 41, 41, 42, 43

Offset: 1

Colin Linzer, Mar 29 2024

nonn

a(n) = x = either ceiling or floor of n/phi, according to which minimizes abs(x/(n-x) - phi).
Each term is equal to or one greater than the previous term.
The average run length approaches phi.
The 2 following statements are equivalent for any real n and any function f(x) such that for any real x, f(x) equals an integer within the range (x-1,x+1) (e.g., round(x), ceiling(x), floor(x)):
  A371626(n) != A371627(n);
  A371626(n) != n-f(n/phi) xor A371627(n) != n-f(n/phi).
Let s(n) = (phi*n - 1 - sqrt(1+(n^2)*(phi^-4)))/2.
Floor(s(n)) equals the number of times that a(n) swapped from being equal to floor(n/phi) to being equal to ceiling(n/phi) when n is extended to the reals.
This is true because s(n) is the solution to the equation n = (phi/4)(phi(2w+1)+sqrt((2w+1)^2 * phi^-4 + 4/phi)) solved for w. The equation gives the n-value of w-th swap from a(n) = floor(n/phi) to a(n) = ceiling(n/phi).
s(n) is asymptotic to n/phi - 1/2.
floor(s(n)) != floor(n/phi - 1/2) <-> a(n) != round(n).
Floor(n/phi) equals the number of times that a(n) swapped from being equal to ceiling(n/phi) to being equal to floor(n/phi) when n is extended to the reals.

			For n=5, the possibilities are (0,5), (1,4), (2,3), (3,2) and (4,1). Of those, 3/2 is the closest to phi, so a(5)=3.

Cf. A001622 (phi), A371626 (y_coordinate), A371627 (with 1/phi), A002163 (sqrt(5)).

a(n) = n - A371626(n).
a(n) = ceiling(n/phi) if floor(s(n)) + floor(n/phi) is even.
a(n) = floor(n/phi) if floor(s(n)) + floor(n/phi) is odd.
a(n) = ceiling(n/phi) - (floor(s(n))+floor(n/phi) mod 2).
a(n) = round(n/phi) + floor(s(n)) - floor(n/phi+1/2).

A370892 Least positive integer k such that 4 numbers starting with k in an arithmetic progression with difference n have a solution in the game 24.

Original entry on oeis.org

3, 1, 1, 2, 1, 1, 1, 1, 2, 3, 2, 1, 1, 2, 2, 3, 2, 12, 9, 12, 2, 6, 2, 2, 12, 2, 2, 12, 4, 12, 5, 12, 4, 12, 12, 10, 12, 12, 12, 12, 8, 12, 12, 12, 2, 12, 2, 1, 8, 12, 2, 12, 12, 12, 12, 12, 12, 12, 12, 12, 8, 12, 12, 12, 12, 12, 12, 12, 12, 6, 5, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12

Offset: 0

Colin Linzer, Mar 04 2024

nonn

24 is a game in which, given 4 four numbers, you must find an expression that evaluates to 24 and contains each of those four numbers exactly once, no other numbers and only the operators of addition, subtraction, multiplication and division.
No term exceeds 12 as for any arithmetic progression 12, 12+n, 12+2n, 12+3n, there is the solution (12)+(12+n)+(12+2n)-(12+3n) = 24.
a(n) = 12 for all n > 144. - Robert Israel, Mar 06 2024

			For n=4, the arithmetic sequence 1, 5, 9, 13 has the solution (1+5)*(13-9) = 24 so a(4)=1.

Colin Linzer, Java program for the sequence.

nv:= proc(V1,V2)
  local a1,a2;
  {seq(seq(op([a1+a2,a1*a2,a1-a2,a2-a1]),a1 = V1),a2 = V2),
       seq(seq(a1/a2,a1 = V1),a2=V2 minus {0}),
       seq(seq(a2/a1,a1 = V1 minus {0}),a2 = V2)}
end proc:
s24:= proc(a,b,c,d) local t;
  for t in combinat:-permute([a,b,c,d]) do
    if member(24, nv(nv(nv({t[1]},{t[2]}),{t[3]}),{t[4]})) then return true fi;
    if member(24, nv(nv({t[1]},{t[2]}),nv({t[3]},{t[4]}))) then return true fi;
  od;
  false
end proc:
f:= proc(n) local k;
  for k from 1 do
     if s24(k, k+n, k+2*n, k+3*n) then return k fi;
  od;
end proc:
map(f, [$0..100]); # Robert Israel, Mar 05 2024

A367329 The y-coordinate of the point where x + y = n, x & y are integers and x/y is as close as possible to e.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 19, 19, 19, 19, 20, 20

Offset: 1

Colin Linzer, Nov 14 2023

a(n) is nondecreasing; lim_{n->oo} a(n) = oo.

Swapping the x and y coordinate of the sequence does not yield the sequence defined as the point where x + y = n, x and y are integers and x/y is as close as possible to 1/e even when excluding terms that would lead to a division by 0.

			For n = 3, the possible points are (0,3), (1,2), (2,1), as any negative fraction would would be further from e than 0/3. The closest fraction to e out of these is 2/1 so a(3) = 1.

Cf. A001113 (e), A367328 (x-coordinate), A007677.

A367328 The x-coordinate of the point where x + y = n, x and y are integers and x/y is as close as possible to e.

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 18, 18, 19, 20, 20, 21, 22, 23, 23, 24, 25, 26, 26, 27, 28, 28, 29, 30, 31, 31, 32, 33, 34, 34, 35, 36, 37, 37, 38, 39, 39, 40, 41, 42, 42, 43, 44, 45, 45, 46, 47, 48, 48, 49, 50

Offset: 1

Colin Linzer, Nov 14 2023

a(n) is nondecreasing; lim_{n->oo} a(n) = oo.

			For n = 3, the possible points are (0,3), (1,2), (2,1) as any negative value would would be further from e than 0/3. The closest value to e out of these is 2/1 so a(3) = 2.

Cf. A001113 (e), A367329 (y-coordinate), A007676.

a(n) is always either ceiling(n*e/(1 + e)) or floor(n*e/(1 + e)) = A076538(n).

A367194 The y-coordinate of the point where x + y = n, x and y are integers and x/y is as close as possible to Pi.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18

Offset: 1

Colin Linzer, Nov 13 2023

a(n) is nondecreasing; lim_{n->oo} a(n) = oo.

Swapping the x and y coordinate of the sequence does not yield the sequence defined as the point where x + y = n, x and y are integers and x/y is as close as possible to 1/Pi even when excluding terms that would lead to a division by 0.

			For n = 3, the possible fractions are (0,3), (1,2), (2,1) as any negative values would would be further from Pi than 0/3. The closest fraction to Pi out of these is 2/1 so a(3) = 1.

Cf. A367193 (x-coordinate), A000796, A002486.

f:= proc(n) local x;
   x:= floor(n/(1+Pi));
   if x = 0 then return 1 fi;
   if is((n-x)/x + (n-x-1)/(x+1) < 2*Pi) then x else x+1 fi
end proc:
map(f, [$1..100]); # Robert Israel, Nov 13 2023

a(n) is always either ceiling(n/(1+Pi)) or floor(n/(1+Pi)).

A367193 The x-coordinate of the point where x + y = n, x and y are integers and x/y is as close as possible to Pi.

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 5, 6, 7, 7, 8, 9, 10, 11, 11, 12, 13, 14, 14, 15, 16, 17, 17, 18, 19, 20, 20, 21, 22, 23, 23, 24, 25, 26, 27, 27, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 36, 36, 37, 38, 39, 39, 40, 41, 42, 42, 43, 44, 45, 45, 46, 47, 48, 49, 49, 50, 51, 52

Offset: 1

Colin Linzer, Nov 09 2023

a(n) is nondecreasing; lim_{n->oo} a(n) = oo.

			For n = 3, the possible fractions are (0,3), (1,2), (2,1) as any negative values would would be further from Pi than 0/3. The closest fraction to Pi out of these is 2/1 so a(3) = 2.

Cf. A367194 (y-coordinate), A000796, A002485.

a(n) is either ceiling(n*Pi/(1+Pi)) or floor(n*Pi/(1+Pi)).

a(n) = round((2*n*Pi + n - sqrt(Pi^2 + 2*Pi + n^2 + 1))/(2*Pi + 2)). - Jon E. Schoenfield, Nov 17 2023

Corrected information and made it in line with A367194.

A365553 Starting with a plane on which two parallel lines and two additional lines have been drawn such that the four lines form two noncongruent isosceles triangles, a(n) is the total number of intersections on the plane after the n-th step, where each step consists of drawing lines that connect every intersection of two lines. If more than 2 lines intersect at the same point it is only counted once.

Original entry on oeis.org

5, 6, 8, 20, 861

Offset: 1

Colin Linzer, Sep 08 2023

Definition is better understood when viewing the linked Desmos graph.

The parity of a term is the parity of the number of intersections along the median from the base of the isosceles triangles. This is because the median is an axis of symmetry of the plane, the number of intersections not on it must be even.

Colin Linzer, Doodle Sequence: Intersections (a Desmos graph).

Cf. A050534, A364725.

A recursive formula for an upper bound:
a(n+1) <= (a(n)^4 - a(n)^2)/8 + (a(n) - a(n)^3)/4 which is equivalent to
a(n+1) <= binomial(binomial(a(n),2),2) (proven).
The proof of the above formula comes from the fact that if there are o points on a graph, then at most (o^2-o)/2 lines that can be drawn between them. If there are m lines on a graph, then there are at most (m^2-m)/2 intersections between them; substituting and simplifying leads to the former upper limit.

cp's OEIS Frontend

User: Colin Linzer

A377813 Decimal expansion of arctanh(phi-1).

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A371627 The x-coordinate of the point where x + y = n, x is an integer and x/y is as close as possible to 1/phi.

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A371626 The y-coordinate of the point where x + y = n, x is an integer and x/y is as close as possible to phi (by absolute difference).

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A371625 The x-coordinate of the point (x,y) where x + y = n, x is an integer, and x/y is as close as possible to phi (by absolute difference).

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A370892 Least positive integer k such that 4 numbers starting with k in an arithmetic progression with difference n have a solution in the game 24.

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A367329 The y-coordinate of the point where x + y = n, x & y are integers and x/y is as close as possible to e.

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A367328 The x-coordinate of the point where x + y = n, x and y are integers and x/y is as close as possible to e.

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A367194 The y-coordinate of the point where x + y = n, x and y are integers and x/y is as close as possible to Pi.

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Maple

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A367193 The x-coordinate of the point where x + y = n, x and y are integers and x/y is as close as possible to Pi.

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