cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: José María Grau Ribas

José María Grau Ribas's wiki page.

José María Grau Ribas has authored 387 sequences. Here are the ten most recent ones:

A373724 Number of totally positive 3 X 3 matrices having all terms in {1,...,n}.

Original entry on oeis.org

1, 61, 797, 6490, 32744, 146441, 492277, 1521123, 4105795, 10194558, 22922408, 49594408, 98935110, 190221734, 350417949, 621178227, 1058404994, 1764873413, 2845696865, 4506618651, 6966717779, 10552756376, 15670141644, 22984055065, 33094853060, 47016605050, 65934960254, 91414399149
Offset: 1

Views

Author

José María Grau Ribas, Jun 15 2024

Keywords

Comments

A matrix is totally positive if all its minors are nonnegative.

Links

Crossrefs

Cf. A211059, A373723.

Programs

  • Mathematica
    ispositive2[M_]:=ispositive1[M]=Union@Table[Select[Union@Flatten@Minors[M,r],(#<= 0)&]=={},{r,1,Length[M]}]=={True};
W[n_]:=W[n]=Flatten[Table[{{a11,a12,a13},{a21,a22,a23},{a31,a32,a33}},{a11,1,n},{a12,1,n},{a13,1,n},{a21,1,n},{a22,1,n},{a23,1,n},{a31,1,n},{a32,1,n},{a33,1,n}],8];
Table[Length@Select[W[n],ispositive2[#]&],{n,1,6}]
  • Sage
    import itertools
def a(n):
    ans, W = 0, itertools.product(range(1,n+1), repeat=9)
    for w in W:
        M = Matrix(ZZ, 3, 3, w)
        if (min(M.minors(2)) >= 0) and (M.det() >= 0): ans += 1
    return ans  # Robin Visser, Apr 18 2025

Extensions

More terms from Robin Visser, Apr 18 2025

A373723 Number of strictly totally positive 3 X 3 matrices having all terms in {1,...,n}.

Original entry on oeis.org

0, 0, 22, 597, 7178, 43090, 207494, 748801, 2321973, 6267631, 15596170, 34784307, 74017706, 147072570, 277965322, 503711791, 884612799, 1491687919, 2458600175, 3925566799, 6133712065, 9388594434, 14121653942, 20783339478, 30178942357, 43156537147, 60868287839, 84699183224, 116688767652
Offset: 1

Views

Author

José María Grau Ribas, Jun 15 2024

Keywords

Comments

A matrix is strictly totally positive if all its minors are greater than zero.

Links

Crossrefs

Cf. A211059, A373724.

Programs

  • Mathematica
    ispositive1[M_]:=ispositive1[M]=Union@Table[Select[Union@Flatten@Minors[M,r],(#<= 0)&]=={},{r,1,Length[M]}]=={True}; W[n_]:=W[n]=Flatten[Table[{{a11,a12,a13},{a21,a22,a23},{a31,a32,a33}},{a11,1,n},{a12,1,n},{a13,1,n},{a21,1,n},{a22,1,n},{a23,1,n},{a31,1,n},{a32,1,n},{a33,1,n}],8];  Table[Length@Select[W[n],ispositive1[#]&],{n,1,7}]
  • Sage
    import itertools
def a(n):
    ans, W = 0, itertools.product(range(1,n+1), repeat=9)
    for w in W:
        M = Matrix(ZZ, 3, 3, w)
        if (min(M.minors(2)) > 0) and (M.det() > 0): ans += 1
    return ans  # Robin Visser, Apr 18 2025

Extensions

More terms from Robin Visser, Apr 18 2025

A370498 Number of paths from (0, 0) to (2n, 2n) in an n X n grid using only steps north, northeast and east (i.e., steps (1, 0), (1, 1), and (0, 1)) and that do not pass through diagonal points with odd coordinates.

Original entry on oeis.org

1, 4, 60, 1204, 27724, 691812, 18198492, 496924692, 13951437804, 400212569284, 11679079547260, 345621250279284, 10347645099250060, 312857431914558244, 9538937406065229084, 292961855077076241108, 9054857076142602126636, 281439018537947499788676, 8791174819979940884130492
Offset: 0

Views

Author

José María Grau Ribas, Feb 20 2024

Keywords

Crossrefs

Cf. A001850, A008288, A138462.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, 4^n, ((8*n-6)*(34*n^2-51*n+11)
      *a(n-1)-(n-2)*(4*n-1)*(2*n-3)*a(n-2))/(n*(4*n-5)*(2*n+1)))
    end:
seq(a(n), n=0..18);  # Alois P. Heinz, Feb 20 2024
  • Mathematica
    A[n_, m_] :=  A[n, m] = Which[m*n == 0, 1, m == n && Mod[m, 2] == 1, 0, True, A[n - 1, m] + A[n, m - 1] + A[n - 1, m - 1]]; Table[A[2 n, 2 n], {n, 0, 45}]

Formula

a(n) = b(2*n,2*n) where b(n,0) = b(0,m) = 1, b(2n+1,2n+1) = 0 and b(n,m) = b(n-1,m-1) + b(n-1,m) + b(n,m-1) otherwise.
a(n) = ceiling((2/3)*A138462(n)).
a(n) ~ (1 + sqrt(2))^(4*n+1) / (3 * 2^(5/4) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 14 2024
D-finite with recurrence +n*(2*n+1)*a(n) +(-70*n^2+73*n-15)*a(n-1) +(70*n^2-277*n+270)*a(n-2) -(2*n-5)*(n-3)*a(n-3)=0. - R. J. Mathar, Mar 25 2024

A367430 a(1)=a(2)=a(3)=1; for n > 3, a(n) = a(n-1)*a(n-2)*a(n-3) + a(n-1) + a(n-2) + a(n-3).

Original entry on oeis.org

1, 1, 1, 4, 10, 55, 2269, 1250284, 156030444388, 442641893445738546589, 86351628807475712878787449679327520349, 5963928033713995675288097975918984841058670353057557384156171322944994
Offset: 1

Views

Author

José María Grau Ribas, Nov 18 2023

Keywords

Crossrefs

Cf. A063896 (a(n) = a(n-1)*a(n-2) + a(n-1) + a(n-2)), A000073.

Programs

  • Mathematica
    a[1] = 1; a[2] = 1; a[3] = 1;
a[n_] := a[n] = a[n - 1]*a[n - 2]*a[n - 3] + a[n - 1] + a[n - 2] + a[n - 3];
Table[a[n], {n, 1, 12}]

A360067 a(n) = det(M) where M is an n X n matrix with M[i,j] = i^j*(i-j).

Original entry on oeis.org

1, 0, 2, 12, 2304, 898560, 4827340800, 143219736576000, 49230909076930560000, 149334225705682285363200000, 5482643392499167214520238080000000, 2322479608280149573505226859610112000000000, 13283541711093841017468807905468592685056000000000000
Offset: 0

Views

Author

José María Grau Ribas, Jan 24 2023

Keywords

Crossrefs

Cf. A060238, A174890, A176005, A176001, A000178, A089064, A152653.

Programs

  • Maple
    a:= n-> LinearAlgebra[Determinant](Matrix(n, (i,j) -> i^j*(i-j))):
seq(a(n), n=0..12);  # Alois P. Heinz, Jan 25 2023
  • Mathematica
    a[n_] := Det@Table[i^j (i - j), {i, n}, {j, n}]; Table[a[n], {n, 1, 15}]
  • PARI
    a(n) = matdet(matrix(n, n, i, j, i^j*(i-j))); \\ Michel Marcus, Jan 24 2023
  • Python
    from sympy import Matrix
def A360067(n): return Matrix(n,n,[i**j*(i-j) for i in range(1,n+1) for j in range(1,n+1)]).det() # Chai Wah Wu, Jan 27 2023

Formula

For n>=1, a(n) = A000178(n-1) * A089064(n). - Vaclav Kotesovec, Apr 19 2024

A354557 Denominators of a sequence related to the Secretary Problem with Multiple Stoppings.

Original entry on oeis.org

1, 2, 24, 1152, 1474560, 117413668454400, 193003573558876719588311040000, 74500758812993473612938854416966977838930799571763200000000, 11100726127423649454784549321327362347631758176882955145554591521918123315624957195621435513013513748480000000000000000
Offset: 1

Views

Author

José María Grau Ribas, May 28 2022

Keywords

Links

Crossrefs

Cf. A354556 (numerators).

Programs

  • Mathematica
    la[1]=1;alfa[k_,1]=1/k!;alfa[k_,k_]:=1;la[k_]:=la[k]=1-alfa[k,k-1];
alfa[k_,kp_]:=alfa[k,kp]= la[kp]^(1+k-kp)/(1+k-kp)!+Sum[la[kp]^b/b! alfa[k-b,kp-1],{b,0,k-kp}];
S[n_]:=Sum[la[i],{i,1,n}];
Table[Denominator@S[n],{n,1,10}]

A354556 Numerators of a sequence related to the Secretary Problem with Multiple Stoppings.

Original entry on oeis.org

1, 3, 47, 2761, 4162637, 380537052235603, 705040594914523588948186792543, 302500210177484374840641189918370275991590974715547528765249, 49554292678269029432299170288905873298367846539726510384850403192729912522937262239403638817695466470734534217406992001
Offset: 1

Views

Author

José María Grau Ribas, May 28 2022

Keywords

Comments

exp(-A354556(n)/A354557(n)) is the asymptotic value of the n-th threshold in the Secretary Problem with n Stoppings.

Examples

			1, 3/2, 47/24, 2761/1152, 4162637/1474560, 380537052235603/117413668454400, 705040594914523588948186792543/193003573558876719588311040000

Links

Crossrefs

Cf. A354557 (denominators).

Programs

  • Mathematica
    la[1]=1;alfa[k_,1]=1/k!;alfa[k_,k_]:=1;la[k_]:=la[k]=1-alfa[k,k-1];
alfa[k_,kp_]:=alfa[k,kp]= la[kp]^(1+k-kp)/(1+k-kp)!+Sum[la[kp]^b/b! alfa[k-b,kp-1],{b,0,k-kp}];
S[n_]:=Sum[la[i],{i,1,n}];
Table[Numerator@S[n],{n,1,10}]

A354946 a(n) = gcd(A001923(n),n^n).

Original entry on oeis.org

1, 1, 1, 32, 1, 1, 1, 4, 1, 1, 1, 8, 1, 1, 9, 4, 17, 3, 19, 16, 7, 1, 1, 4, 1, 1, 1, 8, 1, 1, 1, 4, 3, 17, 1, 768, 1, 1, 1, 4, 1, 1, 1, 8, 1, 1, 1, 4, 1, 125, 3, 16, 1, 9, 1, 4, 1, 1, 1, 8, 1, 1, 1, 4, 5, 1, 1, 32, 9, 1, 1, 12, 1, 1, 5, 8, 1, 1, 1, 4, 1, 1
Offset: 1

Views

Author

José María Grau Ribas, Jun 12 2022

Keywords

Links

Crossrefs

Cf. A000312, A001923.

Programs

  • Mathematica
    A001923[n_]:=Sum[k^k,{k,1,n}]; A[n_]:=GCD[A001923[n],n^n]; Table[A[n],{n,1,33}]
  • PARI
    a(n) = gcd(sum(k=1, n, k^k), n^n); \\ Michel Marcus, Jul 11 2022

A352796 Numbers m such that {d + m/d : d | m } does not contain consecutive integers.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74
Offset: 1

Views

Author

José María Grau Ribas, Jun 08 2022

Keywords

Comments

Conjecture: Complement of A072389.

Links

Crossrefs

Cf. A072389, A335572.

Programs

  • Mathematica
    S[n_]:=Divisors[n]+n/Divisors[n]//Union; Test[n_]:= {aux=S[n];Union[ {False},Table[aux[[i+1]]-aux[[i]] ==1,{i,Length[aux]-1}]]}[[1]]   //Last; Select[Range[1000],Test[#]&]
  • PARI
    isok(m) = my(list=List()); fordiv(m, d, listput(list, d+m/d)); my(w = Set(vector(#list-1, k, list[k+1]-list[k]))); #select(x->(x==1), w) == 0; \\ Michel Marcus, Jun 09 2022
  • Python
    from sympy import divisors
def ok(n):
    s = sorted(set(d + n//d for d in divisors(n)))
    return 1 not in set(s[i+1]-s[i] for i in range(len(s)-1))
print([k for k in range(1, 75) if ok(k)]) # Michael S. Branicky, Jul 10 2022

A346557 9-Sondow numbers: numbers k such that p^s divides k/p + 9 for every prime power divisor p^s of k.

Original entry on oeis.org

1, 2, 5, 22, 54, 378, 16254, 423522, 19930521798, 472458569418
Offset: 1

Views

Author

José María Grau Ribas, Feb 04 2022

Keywords

Comments

Numbers k such that A235137(k) == 9 (mod k).
A positive integer k is a 9-Sondow number if satisfies any of the following equivalent properties:
1) p^s divides k/p + 9 for every prime power divisor p^s of k.
2) 9/k + Sum_{prime p|k} 1/p is an integer.
3) 9 + Sum_{prime p|k} k/p == 0 (mod k).
4) Sum_{i=1..k} i^phi(k) == 9 (mod k).
Other numbers in the sequence: 19930521798, 472458569418, 76413794252037195696360941349774

Links

Crossrefs

Cf. A054377, A007850, A235137, A348058, A348059.
(-1) and (-2) -Sondow numbers: A326715, A330069.
1-Sondow to 9-Sondow numbers: A349193, A330068, A346551, A346552, A346553, A346554, A346555, A346556, this sequence.

Programs

  • Mathematica
    Sondow[mu_][n_]:=Sondow[mu][n]=Module[{fa=FactorInteger[n]},IntegerQ[mu/n+Sum[1/fa[[i,1]],{i,Length[fa]}]]]
Select[Range[1000000],Sondow[9][#]&]

Extensions

a(9)-a(10) verified by Martin Ehrenstein, Feb 04 2022

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