Luc Rousseau - cp's OEIS frontend

User: Luc Rousseau

Luc Rousseau has authored 71 sequences. Here are the ten most recent ones:

A378734 Number of multisets E(n) generable by the following procedure: E(0) = { 0, 0, ... }; to get an E(n+1) from an E(n), first increment all elements then optionally choose an x and a y and replace them with 0 and x+y.

1, 2, 6, 23, 108, 579, 3447, 22190, 152407, 1103566, 8355894, 65701413

Offset: 0

Luc Rousseau, Dec 06 2024

Let there be an infinite number of producers whose initial stocks are equal to 0. At each step (n >= 1), each producer produces 1 resource and adds it to its stock; then a facetious imp may decide to move one producer's stock to another's. a(n) is the number of possible configurations after n steps.

			E(0) is:
        { 0, 0, 0, 0, 0, 0, ... }
E(1) can be:
   A := { 1, 1, 1, 1, 1, 1, ... } // A = E(0) + 1
   B := { 0, 2, 1, 1, 1, 1, ... } // B is A where a 1 moved onto a 1: B = A<1,1>
E(2) can be:
   C := { 2, 2, 2, 2, 2, 2, ... } // C = A + 1
   D := { 1, 3, 2, 2, 2, 2, ... } // D = B + 1
        { 0, 4, 2, 2, 2, 2, ... } // = C<2,2> = D<1,3> = D<3,1>
        { 0, 3, 3, 2, 2, 2, ... } // = D<1,2> = D<2,1>
        { 0, 1, 5, 2, 2, 2, ... } // = D<2,3> = D<3,2>
        { 0, 1, 3, 4, 2, 2, ... } // = D<2,2>
So, the sequence starts: 1, 2, 6, ...

A374601 Defined by: Sum_{i=1..n} i*a(i)/n^i = 1, n>=1.

Original entry on oeis.org

1, 1, 4, 28, 278, 3554, 55382, 1015750, 21401830, 508932130, 13475090126, 393026736854, 12518884854734, 432357148756210, 16092438499462630, 642170913160160710, 27351173629037613494, 1238472705706192189442, 59411223892666111129022, 3010044856761072109710262

Offset: 1

Luc Rousseau, Jul 13 2024

nonn

			1*a(1)/1^1 = 1, so a(1) = 1.
1*a(1)/2^1 + 2*a(2)/2^2 = 1, so a(2) = 1.
1*a(1)/3^1 + 2*a(2)/3^2 + 3*a(3)/3^3 = 1, so a(3) = 4.

Seiichi Manyama, Table of n, a(n) for n = 1..387

Cf. A374562.

a:= proc(n) option remember; `if`(n<1, 0,
      n^(n-1)-add(n^(n-1-i)*a(i)*i, i=1..n-1))
    end:
seq(a(n), n=1..20);  # Alois P. Heinz, Jul 13 2024

a[n_]:=a[n]=n^(n-1)-Sum[n^(n-1-i)*i*a[i],{i,1,n-1}]

a(n)=n^(n-1)-sum(i=1,n-1,n^(n-1-i)*i*a(i))

a(n) = n^(n-1) - Sum_{i=1..n-1} n^(n-1-i)*i*a(i).

a(n) = A374562(n)/n.

A374562 Defined by: Sum_{i=1..n} a(i) / n^i = 1, n >= 1.

Original entry on oeis.org

1, 2, 12, 112, 1390, 21324, 387674, 8126000, 192616470, 5089321300, 148225991386, 4716320842248, 162745503111542, 6053000082586940, 241386577491939450, 10274734610562571360, 464969951693639429398, 22292508702711459409956, 1128813253960656111451418, 60200897135221442194205240

Offset: 1

Luc Rousseau, Jul 12 2024

nonn

Constant terms of the following polynomials: P(0,x) = -1 and, for n>0, P(n,x) = x*P(n-1,x) + a(n), a(n) chosen such that P(n,n)=0.

			a(1) = 1^1 = 1.
a(2) = 2^2 - 2^1*a(1) = 2.
a(3) = 3^3 - 3^2*a(1) - 3^1*a(2) = 12.
a(1) = + 1^1                  ( 0---1 )
     = 1.
a(2) = + 2^2                  ( 0-------2 )
       - 2^1 * 1^1            ( 0---1---2 )
     = 2.
a(3) = + 3^3                  ( 0-----------3 )
       - 3^2 * 1^1            ( 0---1-------3 )
       - 3^1 * 2^2            ( 0-------2---3 )
       + 3^1 * 2^1 * 1^1      ( 0---1---2---3 )
     = 12.

Seiichi Manyama, Table of n, a(n) for n = 1..386

Cf. A374601.

a:= proc(n) option remember; `if`(n<1, 0,
       n^n-add(n^(n-i)*a(i), i=1..n-1))
    end:
seq(a(n), n=1..20);  # Alois P. Heinz, Jul 13 2024

a[n_] := a[n] = n^n - Sum[n^(n - i)*a[i], {i, 1, n - 1}]
a /@ Range[20]

PARI
```
a(n)=n^n-sum(i=1,n-1,n^(n-i)*a(i))
```

a(n) = n^n - Sum_{i=1..n-1} n^(n-i)*a(i).

a(n) = -Sum_{c composition of n} ((-1)^(#c) * Product_{k=1..#c} (n - (Sum_{i

a(n) = n * A374601(n).

A373325 Number of semi-infinite curves of the plane with n simple, transverse self-intersections and no other self-intersections, up to an orientation-preserving homeomorphism.

Original entry on oeis.org

1, 2, 10, 66, 498, 4072, 35144, 315352, 2914074, 27553880, 265387528, 2595131328

Offset: 0

Luc Rousseau, Jun 01 2024

			Curves without self-intersection are equivalent; one might for instance take the half-line y <= 0 as their representative; so a(0) = 1.
To get a curve with n+1 self-intersections, one can start from a curve with n self-intersections; identify the cycle of oriented edges that directly surrounds the finite extremity of the curve; choose an edge from that cycle and extend the curve so that it crosses that edge.
When "outside" it might help visualization to imagine that a noncrossable oriented edge "at infinity" closes the cycle.
Thus, for a transition between 0 and 1 self-intersection, the choice is between making a loop that turns left and making a loop that turns right; so a(1) = 2.
See provided illustration for n=0..3 in section 'Links'.

Luc Rousseau, Illustration for n=0..3
Luc Rousseau, A Prolog program.

Cf. A000682, A268563.

A367795 Triangle read by rows, where row n = L(n) is defined by L(1) = [1,0] and L(n+1) is obtained from L(n) by inserting their binary concatenation between elements x,y.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 6, 2, 4, 0, 1, 14, 6, 26, 2, 20, 4, 8, 0, 1, 30, 14, 118, 6, 218, 26, 106, 2, 84, 20, 164, 4, 72, 8, 16, 0, 1, 62, 30, 494, 14, 1910, 118, 950, 6, 1754, 218, 7002, 26, 3434, 106, 426, 2, 340, 84, 2708, 20, 5284, 164, 1316, 4, 584, 72, 1160, 8, 272, 16, 32, 0

Offset: 1

Luc Rousseau, Nov 30 2023

0 is considered to be a 1-bit-long number and has 0 for binary expansion.
The numbers of bits of the numbers in this triangle are provided by the A049456 triangle.
The sorted set of the numbers that occur in some row of this triangle is provided by A367745.

			Triangle begins:
  1 0
  1 2 0
  1 6 2 4 0
  1 14 6 26 2 20 4 8 0
  1 30 14 118 6 218 26 106 2 84 20 164 ...
Or the same in binary:
  1 0
  1 10 0
  1 110 10 100 0
  1 1110 110 11010 10 10100 100 1000 0
  1 11110 1110 1110110 110 11011010 11010 1101010 10 1010100 10100 10100100 ...

Luc Rousseau, SVG illustration of the nesting of the L(n) lists for n=1..9.

Cf. A000051, A049456, A367745.

sz(n)=if(n==0, 1, logint(n, 2)+1)
L(n)=if(n==1, List([1, 0]), my(LL=L(n-1), k=#LL); while(k>1, listinsert(LL, (LL[k-1] << sz(LL[k])) + LL[k], k); k--); LL)
for(k=1,8,my(l=L(k));for(i=1,#l,print1(l[i],", ")))

from itertools import chain, count, islice, zip_longest
def agen(): # generator of terms
    L = ["1", "0"]
    for k in count(1):
        yield from (int(t, 2) for t in L)
        Lnew = [s+t for s, t in zip(L[:-1], L[1:])]
        L = [t for t in chain(*zip_longest(L, Lnew)) if t is not None]
print(list(islice(agen(), 69))) # Michael S. Branicky, Nov 30 2023

Length of row n = #L(n) = 2^(n-1) + 1 = A000051(n-1).

A367745 Numbers which occur anywhere in A367795, i.e. in lists L(k) where L(1) = [1,0] and L(k+1) is obtained from L(k) by inserting their binary concatenation between elements x,y.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 14, 16, 20, 26, 30, 32, 62, 64, 72, 84, 106, 118, 126, 128, 164, 218, 254, 256, 272, 340, 426, 494, 510, 512, 584, 950, 1022, 1024, 1056, 1160, 1316, 1364, 1706, 1754, 1910, 2014, 2046, 2048, 2708, 3434, 4094, 4096, 4160, 4368, 4680, 5284, 5460, 6826, 7002, 7606, 7918, 8126, 8190

Offset: 1

Luc Rousseau, Nov 29 2023

0 is considered to be a 1-bit-long number and has 0 for binary expansion.

Empirically, there are A000010(n) positive terms with n binary digits. - Rémy Sigrist, Jan 01 2024

			The L(k) lists written in binary begin:
L(1) = [1, 0]
L(2) = [1, 10, 0] -- 10 inserted between 1 and 0
L(3) = [1, 110, 10, 100, 0] -- 110 inserted between 1 and 10, 100 between 10 and 0
L(4) = [1, 1110, 110, 11010, 10, 10100, 100, 1000, 0] -- etc.
0, 1, 10, 100, 110, 1000, ... are producible binary expansions, so the corresponding numbers (0, 1, 2, 4, 6, 8, ...) are in this sequence.

Rémy Sigrist, Binary plot of the terms < 2^64

Cf. A000010, A367795 (the triangle of L(k) lists).

sz(n)=if(n==0,1,logint(n,2)+1)
L(n)=if(n==1, List([1, 0]), my(LL=L(n-1), k=#LL); while(k>1, listinsert(LL, (LL[k-1] << sz(LL[k])) + LL[k], k); k--); LL)
list_a(depth)=my(aa=vecsort(L(depth)), i=1, j=2^depth); while(i<=#aa&&aa[i]

explore(w, p, s) = { my (ps=concat(p, s)); if (#ps <= w, if (nb++ > #vv, vv=concat(vv, vector(#vv))); vv[nb]=fromdigits(ps,2); explore(w, p, ps); explore(w, ps, s);); }
list_a(w) = { nb = 2; vv = [1,0]; explore(w,[1],[0]); Set(vv[1..nb]); } \\ terms < 2^w; Rémy Sigrist, Jan 01 2024

from itertools import chain, count, islice, zip_longest
def agen(): # generator of terms
    L = ["1", "0"]
    for k in count(1):
        yield from sorted(int(t, 2) for t in L if len(t) == k)
        Lnew = [s+t for s, t in zip(L[:-1], L[1:])]
        L = [t for t in chain(*zip_longest(L, Lnew)) if t is not None]
print(list(islice(agen(), 60))) # Michael S. Branicky, Nov 30 2023

A364311 Lexicographically earliest infinite sequence of nonnegative integers, {a(n)} for n>=0, such that all lines with equations y = a(n)*x + n are in general position.

Original entry on oeis.org

0, 1, 3, 2, 5, 4, 8, 11, 6, 13, 12, 7, 22, 16, 17, 21, 9, 14, 10, 27, 18, 15, 19, 28, 31, 43, 34, 38, 23, 39, 25, 36, 41, 20, 55, 63, 42, 30, 24, 33, 26, 32, 65, 66, 51, 59, 29, 56, 35, 62, 85, 81, 37, 49, 46, 68, 74, 78, 88, 48, 44, 75, 40, 47, 97, 76, 93, 79, 92, 54, 58, 100, 61, 101, 107, 52

Offset: 0

Luc Rousseau, Sep 22 2023

nonn

By "in general position" we mean that no two lines are parallel or coincident, and no three are concurrent.

			Line #0 is y = 0*x + 0, as no constraint other than the default one (lexicographically earliest) applies yet, so a(0)=0;
Line #1 cannot be y = 0*x + 1, as the reuse of the slope of a previously defined line implies parallelism; but it can be y = 1*x + 1, so a(1)=1;
Line #2 cannot be y = 2*x + 2, as lines #0, #1 and #2 would be concurrent at (-1,0); but Line #2 can be y = 3*x + 2, so a(2)=3; ...

Luc Rousseau, Illustration, the first five lines

Cf. A000217 (number of intersection points to avoid after line #n is created).

g(n,an,p)=my(b=1);for(i=1,#p,if(p[i][2]==an*p[i][1]+n,b=0;break));b
c(n,s,p)=my(an=0);while((setsearch(s,an,0)!=0)||!g(n,an,p),an++);an
w(n,a,an,p)=for(m=0,#a-1,my(am=a[1+m]);listput(p,[(m-n)/(an-am),(am*n-an*m)/(am-an)]));p
list_a(N)=my(a=List(),s=List(),p=List(),n=0,an);while(n<=N,an=c(n,s,p);p=w(n,a,an,p);listput(a,an);listinsert(s,an,setsearch(s,an,1));n++);Vec(a)
list_a(75)

A362978 The y-coordinates of the even-ranked elements of the lexicographically earliest sequence of points satisfying staircase and no-three-in-a-line conditions. Companion sequence of A362977.

Original entry on oeis.org

0, 1, 3, 2, 4, 7, 5, 8, 12, 11, 10, 6, 13, 23, 20, 14, 28, 21, 15, 24, 9, 35, 43, 38, 37, 52, 33, 45, 36, 32, 22, 44, 46, 59, 53, 54, 64, 19, 34, 27, 72, 70, 84, 74, 55, 49, 26, 50, 17, 31, 77, 56, 107, 85, 47, 60, 16, 62, 29, 25, 69, 41, 42, 106, 82, 61, 65

Offset: 0

Luc Rousseau, May 11 2023

nonn

This sequence is tightly coupled to A362977, which is the main entry. See A362977.

Cf. A362977 (the corresponding x-coordinates).

A362977 The x-coordinates of the even-ranked elements of the lexicographically earliest sequence of points satisfying staircase and no-three-in-a-line conditions (see comments).

Original entry on oeis.org

0, 1, 2, 5, 6, 3, 12, 13, 4, 15, 8, 27, 19, 11, 9, 32, 25, 16, 33, 35, 59, 30, 24, 7, 51, 45, 10, 18, 29, 26, 54, 58, 14, 53, 23, 34, 37, 49, 44, 60, 43, 31, 41, 66, 17, 72, 80, 81, 78, 88, 87, 46, 83, 22, 76, 79, 130, 97, 108, 111, 94, 119, 153, 89, 39, 112

Offset: 0

Luc Rousseau, May 11 2023

nonn

We construct the lexicographically earliest sequence of points M(i), for i >= 0, with the following rules:
- their (x, y) coordinates are taken among the nonnegative integers;
- if i > 0 and i is odd, then "move horizontally to a free column"; i.e., M(i) must have the same y as M(i-1), and M(i) is not allowed to have the same x as any M(k) for k < i;
- if i > 0 and i is even, then "move vertically to a free row"; i.e., M(i) must have the same x as M(i-1), and M(i) is not allowed to have the same y as any M(k) for k < i;
- three points are not allowed to be aligned.
See SVG illustration, Links section.
Then a(n) (resp. A362978(n)) is defined as the x-coordinate (resp. y-coordinate) of M(i), where i := 2n (to eliminate duplicates).

			  y
  ^
  | . . . 9 . . 8
  | . . 4 . . 5 .
  | . . . . . 6 7
  | . 2 3 . . . .
  | 0 1 . . . . .
  +-------------------> x
    0 1 2 3 4 5 6
Abscissas of the points 0, 2, 4, 6, 8, ...: 0, 1, 2, 5, 6, ...

Luc Rousseau, SVG illustration, n = 0..126
Luc Rousseau, Java program for the combined computation of A362977 and A362978.

Cf. A362978 (the corresponding y-coordinates).

Cf. A161680 (number of nonalignment checks to pass).

Java
```
// See Rousseau link.
```

A360303 a(n) = Sum_{k=1..floor(sqrt(n))} 2^floor(n/k-k).

Original entry on oeis.org

0, 1, 2, 4, 9, 17, 34, 66, 132, 261, 521, 1033, 2066, 4114, 8226, 16420, 32837, 65605, 131209, 262281, 524554, 1048850, 2097682, 4194834, 8389668, 16778277, 33556517, 67110981, 134221897, 268439625, 536879242, 1073750154, 2147500178, 4294983954, 8589967634, 17179902228, 34359804453

Offset: 0

Luc Rousseau, Feb 02 2023

This sequence corresponds to the left half of a drawing, the whole drawing being reconstituted by symmetry (see the Illustration link). The divisors of n are closely related to the occurrences of the bit pattern "01 over 10" in the 2 X 2 squares along the (n-1)th and n-th lines (see the pattern link). In particular, n is a prime number if and only if a(n) - a(n-1) = 2^(n-2).

			For n = 5, floor(sqrt(n)) = 2. So, two bits are set in a(n); they are the bits number floor(5/1-1)=4 and floor(5/2-2)=0, so a(n) = 10001_2 = 17.

Luc Rousseau, Illustration in black and white, n = 0..100.
Luc Rousseau, Illustrated bit pattern for the detection of divisors, n = 1..9.

Cf. A034729.

a(n)=sum(k=1,floor(sqrt(n)),2^floor(n/k-k))

a(n) = Sum_{k=1..floor(sqrt(n))} 2^floor(n/k - k).

cp's OEIS Frontend

User: Luc Rousseau

A378734 Number of multisets E(n) generable by the following procedure: E(0) = { 0, 0, ... }; to get an E(n+1) from an E(n), first increment all elements then optionally choose an x and a y and replace them with 0 and x+y.

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A374601 Defined by: Sum_{i=1..n} i*a(i)/n^i = 1, n>=1.

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Maple

Mathematica

PARI

Formula

A374562 Defined by: Sum_{i=1..n} a(i) / n^i = 1, n >= 1.

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Maple

Mathematica

PARI

Formula

A373325 Number of semi-infinite curves of the plane with n simple, transverse self-intersections and no other self-intersections, up to an orientation-preserving homeomorphism.

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Author

Keywords

Examples

Links

Crossrefs

A367795 Triangle read by rows, where row n = L(n) is defined by L(1) = [1,0] and L(n+1) is obtained from L(n) by inserting their binary concatenation between elements x,y.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

Formula

A367745 Numbers which occur anywhere in A367795, i.e. in lists L(k) where L(1) = [1,0] and L(k+1) is obtained from L(k) by inserting their binary concatenation between elements x,y.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Python

A364311 Lexicographically earliest infinite sequence of nonnegative integers, {a(n)} for n>=0, such that all lines with equations y = a(n)*x + n are in general position.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A362978 The y-coordinates of the even-ranked elements of the lexicographically earliest sequence of points satisfying staircase and no-three-in-a-line conditions. Companion sequence of A362977.

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