A040002 -id:A040002 - cp's OEIS frontend

A278481 Number of neighbors of the n-th term in a full isosceles triangle read by rows.

2, 4, 4, 4, 6, 4, 4, 6, 6, 4, 4, 6, 6, 6, 4, 4, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 4

Offset: 1

Omar E. Pol, Nov 23 2016

Apart from the left border and the right border, the rest of the elements are 6's.

For the same idea but for a right triangle see A278480; for a square array see A278545, for a square spiral see A010731; and for a hexagonal spiral see A010722.

			The sequence written as an isosceles triangle begins:
.
.                     2;
.                   4,  4;
.                 4,  6,  4;
.               4,  6,  6,  4;
.             4,  6,  6,  6,  4;
.           4,  6,  6,  6,  6,  4;
.         4,  6,  6,  6,  6,  6,  4;
.       4,  6,  6,  6,  6,  6,  6,  4;
.     4,  6,  6,  6,  6,  6,  6,  6,  4;
.   4,  6,  6,  6,  6,  6,  6,  6,  6,  4;
...

Row sums give A016933.
Left border gives A040002, the same as the right border.
Middle column gives the elements > 1 of A134201, also twice A122553.
Cf. A010722, A010731, A275015, A278317, A278480, A278545.

A269457 a(n) = 5(n + 1)(n + 4)/2.

Original entry on oeis.org

10, 25, 45, 70, 100, 135, 175, 220, 270, 325, 385, 450, 520, 595, 675, 760, 850, 945, 1045, 1150, 1260, 1375, 1495, 1620, 1750, 1885, 2025, 2170, 2320, 2475, 2635, 2800, 2970, 3145, 3325, 3510, 3700, 3895, 4095, 4300, 4510, 4725, 4945, 5170, 5400, 5635

Offset: 0

Ilya Gutkovskiy, Feb 27 2016

More generally, the ordinary generating function for the sequences of the form k*(n + 1)*(n - 1 + k)/2 is (k*(k - 1)/2 + (k*(3 - k)/2)*x)/(1 - x)^3 (see links section).

			a(0) = 0 + 1 + 2 + 3 + 4 = 10;
a(1) = 0 + 1 + 2 + 3 + 4 + 1 + 2 + 3 + 4 + 5 = 25;
a(2) = 0 + 1 + 2 + 3 + 4 + 1 + 2 + 3 + 4 + 5 + 2 + 3 + 4 + 5 + 6 = 45, etc.

Ilya Gutkovskiy, Sequences of the form k*(n + 1)*(n - 1 + k)/2.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000096, A008587, A028895, A040002, A045943, A054000, A055998, A067728.

[5*(n+1)*(n+4)/2: n in [0..50]]; // Vincenzo Librandi, Feb 28 2016

Table[5 (n + 1) ((n + 4)/2), {n, 0, 45}]
Table[Sum[5 (k + 2), {k, 0, n}], {n, 0, 45}]
LinearRecurrence[{3, -3, 1}, {10, 25, 45}, 46]

a(n) = 5*(n + 1)*(n + 4)/2; \\ Michel Marcus, Feb 29 2016

Vec(5*(2-x)/(1-x)^3 + O(x^100)) \\ Altug Alkan, Mar 04 2016

G.f.: 5*(2 - x)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = Sum_{k=0..n} 5*(k + 2) = Sum_{k=0..n} A008587(k + 2).
Sum_{n>=0} 1/a(n) = 11/45 = 0.24444444444... = A040002.
a(n) = 5*A000096(n+1).
a(n) = A055998(2*n+2) + A055998(n+1). - Bruno Berselli, Sep 23 2016
E.g.f.: 5*exp(x)*(4 + 6*x + x^2)/2. - Elmo R. Oliveira, Dec 24 2024

A307508 Primes p for which the continued fraction expansion of sqrt(p) does not have a 1 in the second position.

Original entry on oeis.org

2, 5, 11, 17, 19, 29, 37, 41, 53, 67, 71, 83, 89, 101, 103, 107, 109, 127, 131, 149, 151, 173, 179, 181, 197, 199, 227, 229, 233, 239, 257, 263, 269, 271, 293, 331, 337, 367, 373, 379, 401, 409, 419, 443, 449, 457, 461, 487, 491, 499, 503, 541, 547, 577, 587, 593, 599

Offset: 1

Michel Marcus, Apr 11 2019

nonn

These are the primes that are located between a square number and the following oblong number. - Charles Kusniec, Apr 17 2020

Primes in A063656. - Charles Kusniec, Sep 04 2022

			For p = 2,  we have [1; 2, ...]; see A040000.
For p = 5,  we have [2; 4, ...]; see A040002.
For p = 11, we have [3; 3, ...]; see A040007.

Michel Marcus, Table of n, a(n) for n = 1..5000
Piotr Miska and Maciej Ulas, On consecutive 1's in continued fractions expansions of square roots of prime numbers, Experimental Mathematics, 31:1 (2022), 238-251. Also arXiv:1904.03404 [math.NT], 2019.
Index entries for continued fractions for constants

Cf. A040000, A040002, A040007, A067614, A307453, A063656.

Complement of A334163 with respect to the primes.

isok(p) = isprime(p) && contfrac(sqrt(p))[2] != 1;

A350549 a(n) is the permanent of a square matrix M(n) whose general element M_{i,j} is defined by floor((j - i + 1)/2).

Original entry on oeis.org

1, 0, 0, -1, 2, 20, -120, -4608, 41952, 2325024, -34876800, -3133087200, 66120252480, 8258565859200, -239533775631360, -40631838221721600, 1532513262269767680, 335620705700380262400, -16054693916748370329600, -4428138916386119015424000, 261291002534430572648448000

Offset: 0

Stefano Spezia, Jan 04 2022

sign

The matrix M(n) is the n-th principal submatrix of the array A010751.
In the n X n matrix M(n): the zero element appears 2*n - 1 times; the positive integers k appears iff 0 < k < floor(n/2), 2*n - 1 - A040002(k-1) times; the negative integer k appears iff -k < ceiling(n/2),  2*n - 5 + 4*(k + 1) times.
det(M(n)) = 0, except for n = 3 for which det(M(3)) = -1.
The trace and the subdiagonal sum of the matrix M(n) are zero.
The antitrace of the matrix M(n) is A142150(n+1).
The superdiagonal sum of the matrix M(n) is equal to n - 1.
The sum of the elements of the matrix M(n) is A002620(n).

			For n = 3 the matrix M(3) is
     0, 1, 1
     0, 0, 1
    -1, 0, 0
with permanent a(3) = -1.
For n = 4 the matrix M(4) is
    0,  1,  1,  2
    0,  0,  1,  1
   -1,  0,  0,  1
   -1, -1,  0,  0
with permanent a(4) = 2.

Cf. A002620, A004526, A010751, A040002, A060747, A110654, A142150.

a:= n-> `if`(n=0, 1, LinearAlgebra[Permanent](
         Matrix(n, (i, j)-> floor((j-i+1)/2)))):
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 19 2022

Join[{1},Table[Permanent[Table[Floor[(j-i+1)/2],{i,n},{j,n}]],{n,20}]]

a(n) = matpermanent(matrix(n, n, i, j, (j - i + 1)\2)); \\ Michel Marcus, Jan 04 2022

from sympy import Matrix
def A350549(n): return 1 if n == 0 else Matrix(n,n,lambda i,j:(j-i+1)//2).per() # Chai Wah Wu, Jan 12 2022

A064850 Period of continued fraction for sqrt(5)*n.

Original entry on oeis.org

1, 2, 6, 2, 5, 4, 10, 4, 2, 14, 12, 4, 5, 10, 28, 8, 1, 2, 4, 14, 6, 8, 6, 4, 31, 14, 10, 12, 12, 20, 8, 20, 20, 2, 52, 2, 19, 4, 28, 24, 18, 8, 50, 12, 28, 6, 10, 4, 70, 62, 8, 18, 7, 10, 6, 8, 8, 12, 72, 20, 3, 12, 8, 36, 41, 28, 86, 2, 6, 44, 84, 2, 43, 42, 120, 4, 52, 36, 28, 44, 38

Offset: 1

R. K. Guy, Oct 26 2001

nonn

			A040002 (cfrac for n=1) has period length 1, so a(1)=1. A040015 (cfrac for n=2) has period length 2, so a(2)=2. A010135 (cfrac for n=3) has period length 6, so a(3)=6. - _R. J. Mathar_, Feb 10 2016

Table[Length[ContinuedFraction[Sqrt[5]n][[2]]],{n,90}] (* Harvey P. Dale, Apr 13 2015 *)

A267649 a(0) = a(1) = 2 then a(n) = 4 for n>=2.

Original entry on oeis.org

2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

Natan Arie Consigli, Jan 19 2016

Decimal expansion of 101/450.
Also list of smallest n-composites.
A hyperoperator aggregation b[n]c is n-composite if b,c are positive non-right-identity elements.
The identity elements are:
Hyper-0 (zeration): none.
Hyper-1 (addition): 0.
Hyper-2 (multiplication): 1.
Hyper-3 (exponentiation): 1.
Hyper-n (n>2): 1.
For more information on hyperoperations see A054871.
Essentially the same as A255176, A151798, A123932, A113311, A040002 and A010709. - R. J. Mathar, May 25 2023
Continued fraction expansion of 2 + sqrt(1/5) = 2 + sqrt(5)/5. - Elmo R. Oliveira, Aug 06 2024

			a(0) = 2 because 1 is the smallest non-identity element in zeration and 1[0]1=2;
a(1) = 2 because 1 is the smallest non-identity element in addition and 1[1]1=2;
a(2) = 4 because 2 is the smallest non-identity element in multiplication and 2[2]2=4;
a(3) = 4 because 2 is the smallest non-identity element in exponentiation and 2[2]2=4;
a(4) = 4 because 2 is the smallest non-identity element in titration and 2[2]2=4;
Etc.

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000027 (1-composites), A002808 (composites), A267647 (3-composites), A097374 (4-composites).

a(n) = a[n]b where a,b are the positive smallest non-right-identity elements.
From Elmo R. Oliveira, Aug 06 2024: (Start)
G.f.: 4/(1 - x) - 2*(1 + x).
E.g.f.: 4*exp(x) - 2*(1 + x). (End)

cp's OEIS Frontend

A278481 Number of neighbors of the n-th term in a full isosceles triangle read by rows.

Views

Author

Keywords

Comments

Examples

Crossrefs

A269457 a(n) = 5*(n + 1)*(n + 4)/2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A307508 Primes p for which the continued fraction expansion of sqrt(p) does not have a 1 in the second position.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A350549 a(n) is the permanent of a square matrix M(n) whose general element M_{i,j} is defined by floor((j - i + 1)/2).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Python

A064850 Period of continued fraction for sqrt(5)*n.

Views

Author

Keywords

Examples

Programs

Mathematica

A267649 a(0) = a(1) = 2 then a(n) = 4 for n>=2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A269457 a(n) = 5(n + 1)(n + 4)/2.