A319840 -id:A319840 - cp's OEIS frontend

A330287 Permanent of the n-th principal submatrix M(n) of A319840.

1, 1, 8, 208, 11488, 1093056, 158972160, 32734095360, 9049229328384, 3230305304002560, 1445344680438005760, 791762592707031859200, 521023492500173338705920, 405448567547957922512240640, 368210800911998093644372377600, 385879616532879866123928993792000, 462151848929747968377341029122048000

Offset: 0

Stefano Spezia, Dec 11 2019

nonn

The matrix M(n) is defined as M[i,j,n] = i*j if i < 3 or j < 3 and M[i,j,n] = 2*(i + j) - 4 otherwise.
det(M(0)) = det(M(1)) = 1 and det(M(n)) = 0 for n > 1.
For n > 0, the trace of the matrix M(n) is A001844(n-1).
For n > 0, the antitrace of the matrix M(n) is A005893(n-1).
For n > 1, the super- and subdiagonal sum is A001105(n-1).

			For n = 1 the matrix M(1) is
  1
with permanent a(1) = 1.
For n = 2 the matrix M(2) is
  1, 2
  2, 4
with permanent a(2) = 8.
For n = 3 the matrix M(3) is
  1,  2,  3
  2,  4,  6
  3,  6,  8
with permanent a(3) = 208.

Vaclav Kotesovec, Table of n, a(n) for n = 0..35

Cf. A001105, A001844, A005893, A319840.

tm(n) = matrix(n, n, i, j, if ((i<3) || (j<3), i*j, 2*(i+j)-4));
a(n) = matpermanent(tm(n));

a(n) ~ c * A238261^n * n!^2 / sqrt(n), where c = 0.0286685259829... - Vaclav Kotesovec, Aug 19 2021

A370753 Antidiagonal products of A319840.

Original entry on oeis.org

1, 1, 4, 36, 576, 12800, 360000, 12192768, 481890304, 21743271936, 1101996057600, 61952000000000, 3824628881965056, 257164113195565056, 18704075505689706496, 1462975070062038220800, 122444006400000000000000, 10918111308394619734065152, 1033255398127440061257744384

Offset: 0

Stefano Spezia, Jun 22 2024

nonn

a(n) has trailing zeros iff n is congruent to 0 or 1 mod 5. Cf. A008851.
a(n) is a square iff n = 1 or congruent to {1, 3, 4} mod 5. Cf. A047206.
It appears that: (Start)
a(n) is a cube iff n = 0, 1, or is of the form (3*m - 4)^3 with m > 1 (A016791);
the only fourth powers in the sequence are 1 and a(9) = 21743271936 = 384^4;
the only fifth powers in the sequence are 1 and a(32) = 227200942336^5;
a(n) is a sixth power iff n = 0, 1, or is of the form (6*m - 10)^3 with m > 1;
the only seventh powers in the sequence are 1 and a(128) = 77458109039896212820250015287665035595218944^7. (End)

Cf. A000079, A000169, A000290, A008851, A016791, A047206, A319840.

a[0]=a[1]=1; a[n_]:=n^2*2^(n-2)*(n-1)^(n-2); Array[a,19,0]

a(0) = a(1) = 1, and a(n) = n^2*2^(n-2)*(n - 1)^(n-2) for n > 1.

A319702 Filter sequence for sequences that are constant for all even terms >= 2.

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 2, 10, 2, 11, 2, 12, 2, 13, 2, 14, 2, 15, 2, 16, 2, 17, 2, 18, 2, 19, 2, 20, 2, 21, 2, 22, 2, 23, 2, 24, 2, 25, 2, 26, 2, 27, 2, 28, 2, 29, 2, 30, 2, 31, 2, 32, 2, 33, 2, 34, 2, 35, 2, 36, 2, 37, 2, 38, 2, 39, 2, 40, 2, 41, 2, 42, 2, 43, 2, 44, 2, 45, 2, 46, 2, 47, 2, 48, 2, 49, 2, 50, 2, 51, 2, 52, 2, 53, 2, 54, 2, 55, 2, 56, 2, 57, 2, 58, 2, 59, 2, 60, 2, 61, 2

Offset: 1

Antti Karttunen, Oct 02 2018

Restricted growth sequence transform of A141310.

For n > 2, a(n-1) is the number of occurrences of n in A319840. - Stefano Spezia, Apr 07 2023

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A141310, A319701, A319840.

A319702(n) = if(n<=2, n, if(!(n%2), 2, (n+3)/2));

a(1) = 1, and for n > 1, if n is even, a(n) = 2, otherwise a(n) = (n+3)/2.
From Stefano Spezia, Apr 07 2023: (Start)
O.g.f.: x*(1 + 2*x + x^2 - 2*x^3 - x^4)/((1 - x)^2*(1 + x)^2).
E.g.f.: ((4 + x)*cosh(x) + 3*sinh(x) - 2*(2 + x))/2. (End)

A330700 a(n) = (n - 1)n(2n^2 + 4n - 1)/6.

Original entry on oeis.org

0, 0, 5, 29, 94, 230, 475, 875, 1484, 2364, 3585, 5225, 7370, 10114, 13559, 17815, 23000, 29240, 36669, 45429, 55670, 67550, 81235, 96899, 114724, 134900, 157625, 183105, 211554, 243194, 278255, 316975, 359600, 406384, 457589, 513485, 574350, 640470, 712139, 789659

Offset: 0

Stefano Spezia, Dec 26 2019

Conjectures: (Start)
For n > 1, a(n) is the absolute value of the trace of the 2nd exterior power of an n X n square matrix M(n) defined as M[i,j,n] = i*j if i < 3 or j < 3 and M[i,j,n] = 2*(i + j) - 4 otherwise (see A330287). Equivalently, a(n) is the absolute value of the coefficient of the term [x^(n-2)] in the characteristic polynomial of M(n), or the absolute value of the sum of all principal minors of M(n) of size 2.
For k > 2, the trace of the k-th exterior power of M(n) is equal to zero.
(End)

Wikipedia, Characteristic polynomial
Wikipedia, Exterior algebra
Wikipedia, Harmonic number
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A001105 (super- and subdiagonal sum of M(n)), A001844 (trace of M(n)), A005843 (antitrace of M(n)), A268581, A319840, A322844, A330287 (permanent of M(n)).

I:=[0, 0, 5, 29, 94]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]];

Mathematica
```
Table[(n-1)n(2n^2+4n-1)/6,{n,0,39}]
```

my(x='x + O('x^39)); concat([0, 0], Vec(serlaplace((1/6)*exp(x)*x^2*(15+14*x+2*x^2))))

(x^2*(5+4*x-x^2)/(1-x)^5).series(x, 40).coefficients(x, sparse=False)

O.g.f.: x^2*(5 + 4*x - x^2)/(1 - x)^5.
E.g.f.: exp(x)*x^2*(15 + 14*x + 2*x^2)/6.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4.
a(n) = A000217(n-1)*A268581(n-1)/3 for n > 0.
Sum_{k>=2} 1/a(k) = (1/5)*((18 + 7*sqrt(6))*H(2-sqrt(3/2)) + (18 - 7*sqrt(6))*H(2+sqrt(3/2)) - 30) = 0.254905801002729039998040617... where H(x) = Integral_{t=0..1} (1 - t^x)/(1 - t) dt is the function that interpolates the harmonic numbers.

A374402 Least number that is the lesser of two consecutive primes p and q whose binary expansions have the same length and agree at exactly n digit positions, or -1 if no such prime pair exists.

Original entry on oeis.org

2, 5, 23, 17, 41, 67, 137, 269, 521, 1049, 2081, 4111, 8233, 16417, 32771, 65537, 131113, 262147, 524309, 1048609, 2097257, 4194389, 8388617, 16777289, 33554501, 67109123, 134217929, 268435459, 536871017, 1073741827, 2147484041, 4294967497, 8589934627, 17179869731

Offset: 1

Jean-Marc Rebert, Jul 07 2024

			a(1) = 2 because 2 = 10_2 and 3 = 11_2 are two consecutive primes that, when written in base 2, both have 2 digits and agree at exactly 1 digit position (each has a 1 in its first digit position), and no earlier pair of consecutive primes has this property.
a(3) = 23 = 10111_2; the next prime is
       29 = 11101_2  (same number of binary digits),
            ^ ^ ^    and the digits agree at 3 digit positions,
  and no earlier pair of consecutive primes has this property.

Cf. A000040, A006879, A205510, A319840, A006879, A319840, A339080, A374176.

card(p)=my(u=binary(p),v=binary(nextprime(p+1))); if(#u!=#v,return(0)); sum(i=1,#u,u[i]==v[i])
a(n)=forprime(p=2^n,oo,if(card(p)==n,return(p)))

cp's OEIS Frontend

A330287 Permanent of the n-th principal submatrix M(n) of A319840.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A370753 Antidiagonal products of A319840.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A319702 Filter sequence for sequences that are constant for all even terms >= 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A330700 a(n) = (n - 1)*n*(2*n^2 + 4*n - 1)/6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A374402 Least number that is the lesser of two consecutive primes p and q whose binary expansions have the same length and agree at exactly n digit positions, or -1 if no such prime pair exists.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

A330700 a(n) = (n - 1)n(2n^2 + 4n - 1)/6.