A338041 -id:A338041 - cp's OEIS frontend

A082111 a(n) = n^2 + 5*n + 1.

1, 7, 15, 25, 37, 51, 67, 85, 105, 127, 151, 177, 205, 235, 267, 301, 337, 375, 415, 457, 501, 547, 595, 645, 697, 751, 807, 865, 925, 987, 1051, 1117, 1185, 1255, 1327, 1401, 1477, 1555, 1635, 1717, 1801, 1887, 1975, 2065, 2157, 2251, 2347, 2445, 2545, 2647

Offset: 0

Paul Barry, Apr 04 2003

From Gary W. Adamson, Jul 29 2009: (Start)
Let (a,b) = roots to x^2 - 5*x + 1 = 0 = 4.79128... and 0.208712...
Then a(n) = (n + a) * (n + b). Example: a(5) = 51 = (5 + 4.79128...) * (5 + 0.208712...) (End)
For n > 0: a(n) = A176271(n+2,n). - Reinhard Zumkeller, Apr 13 2010
a(n-2) = n*(n+1) - 5, n >= 0, with a(-2) = -5 and a(-1) = -3, gives the values for a*c of indefinite binary quadratic forms [a, b, c] of discriminant D = 21 for b = 2*n + 1. In general D = b^2 - 4*a*c > 0 and the form [a, b, c] is a*x^2 + b*x*y + c*y^2. - Wolfdieter Lang, Aug 15 2013
Numbers m > 0 such that 4m+21 is a square. - Bruce J. Nicholson, Jul 19 2017
Numbers represented as 151 in number base B. If 'digits' from B upwards are allowed then 151(2)=15, 151(3)=25, 151(4)=37, 151(5)=51 also. - Ron Knott, Nov 14 2017
If A and B are sequences satisfying the recurrence t(n) = 5*t(n-1) - t(n-2) with initial values A(0) = 1, A(1) = n+5 and B(0) = -1, B(1) = n, then a(n) = A(i)^2 - A(i-1)*A(i+1) = B(j)^2 - B(j-1)*B(j+1) for i, j > 0. - Klaus Purath, Oct 18 2020
The prime terms in this sequence are listed in A089376. The prime factors are given in A038893. With the exception of 3 and 7, each prime factor p divides exactly 2 out of any p consecutive terms. If a(i) and a(k) form such a pair that are divisible by p, then i + k == -5 (mod p). - Klaus Purath, Nov 24 2020

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

First row of A082110.

Cf. A002522, A014209, A028387, A028872, A338041.

Table[n^2 + 5*n + 1,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 19 2011 *)
LinearRecurrence[{3,-3,1},{1,7,15},80] (* Harvey P. Dale, Apr 22 2012 *)

a(n)=n^2+5*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 2*n + a(n-1) + 4 (with a(0)=1). - Vincenzo Librandi, Aug 08 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=7, a(2)=15. - Harvey P. Dale, Apr 22 2012
Sum_{n>=0} 1/a(n) = 8/15 + Pi*tan(sqrt(21)*Pi/2)/sqrt(21) = 1.424563592286456286... . - Vaclav Kotesovec, Apr 10 2016
From G. C. Greubel, Jul 19 2017: (Start)
G.f.: (1 + 4*x - 3*x^2)/(1 - x)^3.
E.g.f.: (x^2 + 6*x + 1)*exp(x). (End)
a(n) = A014209(n+1) - 2 = A338041(2*n+1). - Hugo Pfoertner, Oct 08 2020
a(n) = A249547(n+1) - A024206(n-4), n >= 5. - Klaus Purath, Nov 24 2020

New title (using given formula) from Hugo Pfoertner, Oct 08 2020

A338122 Place three points evenly spaced around a circle, draw n evenly spaced rays from each of the points, a(n) is the number of regions thus created. See Comments for details.

Original entry on oeis.org

1, 6, 10, 18, 31, 30, 58, 60, 73, 90, 118, 72, 160, 168, 187, 204, 262, 240, 325, 306, 358, 396, 457, 324, 535, 546, 580, 594, 709, 666, 808, 780, 859, 918, 1012, 780, 1126, 1140, 1189, 1212, 1372, 1308, 1507, 1458, 1576, 1656, 1783, 1464, 1933, 1950, 2014, 2034

Offset: 1

Lars Blomberg, Oct 11 2020

nonn

The first ray from each point goes opposite to the direction to the center of the circle. Should a ray hit another point it is terminated there.

To produce the illustrations below, all pairwise intersections between the rays are calculated and the maximum distance to the center, incremented by 20%, is taken as radius of a circle. Then all intersections between the rays and the circle defines a polygon which is used as limit.

			For n=1 there are three rays that do not intersect, so a(1)=1.

Lars Blomberg, Table of n, a(n) for n = 1..800
Lars Blomberg, Illustration for n=3
Lars Blomberg, Illustration for n=9
Lars Blomberg, Illustration for n=15
Lars Blomberg, Illustration for n=24
Lars Blomberg, Illustration for n=27
Lars Blomberg, Illustration for n=30
Lars Blomberg, Illustration for n=45

Cf. A338041 (two start points), A338123 (vertices), A338124 (edges).

a(n)=if( \
n%12==1,(3*n^2 + 11*n - 10)/4, \
n%12==2||n%12==10,(3*n^2 + 6*n)/4, \
n%12==3,(3*n^2 + 5*n - 2)/4, \
n%12==5,(3*n^2 + 11*n - 6)/4, \
n%12==6,(3*n^2 - 2*n + 24)/4, \
n%12==7,(3*n^2 + 11*n + 8)/4, \
n%12==9,(3*n^2 + 5*n + 4)/4, \
n%12==11,(3*n^2 + 11*n - 12)/4, \
n%24==4||n%24==20,(3*n^2 + 24)/4, \
n%24==8||n%24==16,(3*n^2 + 48)/4, \
n%120==0,(3*n^2 - 26*n)/4, \
n%120==12||n%120==36||n%120==84||n%120==108,(3*n^2 - 26*n + 168)/4, \
n%120==24||n%120==48||n%120==72||n%120==96,(3*n^2 - 26*n + 192)/4, \
n%120==60,(3*n^2 - 26*n - 24)/4, \
-1);
vector(800, n, a(n))

a(n) = 2160-a(n-4)+a(n-12)+a(n-16)+a(n-60)+a(n-64)-a(n-72)-a(n-76), n>78. (conjectured)
From Lars Blomberg, Oct 25 2020: (Start)
Conjectured for 1 <= n <= 800.
Select the row in the table below for which r = n mod m. Then a(n)=(a*n^2 + b*n + c)/d.
+===========================================+
|               r |   m | a |   b |   c | d |
+-------------------------------------------+
|               1 |  12 | 3 |  11 | -10 | 4 |
|           2, 10 |  12 | 3 |   6 |     | 4 |
|               3 |  12 | 3 |   5 |  -2 | 4 |
|               5 |  12 | 3 |  11 |  -6 | 4 |
|               6 |  12 | 3 |  -2 |  24 | 4 |
|               7 |  12 | 3 |  11 |   8 | 4 |
|               9 |  12 | 3 |   5 |   4 | 4 |
|              11 |  12 | 3 |  11 | -12 | 4 |
|           4, 20 |  24 | 3 |     |  24 | 4 |
|           8, 16 |  24 | 3 |     |  48 | 4 |
|               0 | 120 | 3 | -26 |     | 4 |
| 12, 36, 84, 108 | 120 | 3 | -26 | 168 | 4 |
|  24, 48, 72, 96 | 120 | 3 | -26 | 192 | 4 |
|              60 | 120 | 3 | -26 | -24 | 4 |
+===========================================+ (End)

A338042 Draw n rays from each of two distinct points in the plane; a(n) is the number of vertices thus created. See Comments for details.

Original entry on oeis.org

2, 2, 4, 2, 8, 4, 14, 8, 22, 14, 32, 22, 44, 32, 58, 44, 74, 58, 92, 74, 112, 92, 134, 112, 158, 134, 184, 158, 212, 184, 242, 212, 274, 242, 308, 274, 344, 308, 382, 344, 422, 382, 464, 422, 508, 464, 554, 508, 602, 554, 652, 602, 704, 652, 758, 704, 814, 758

Offset: 1

Lars Blomberg, Oct 08 2020

nonn

The rays are evenly spaced around each point. The first ray of one point goes opposite to the direction to the other point. Should a ray hit the other point it terminates there, that is, it is converted to a line segment.

See A338041 for illustrations.

			For n=1:    <-----x     x----->   so a(1)=2.
For n=2:    <-----x<--->x----->   so a(2)=2.

Cf. A338041 (regions), A338043 (edges), A008795.

a(n)=if(n%2==1,(n^2 + 7)/4,(n^2 - 6*n + 16)/4)
vector(200, n, a(n))

a(n) = (n^2 + 7)/4, n odd; (n^2 - 6*n + 16)/4, n even (conjectured).
Conjectured by Stefano Spezia, Oct 08 2020 after Lars Blomberg: (Start)
G.f.: 2*x*(1 - x^2 - x^3 + 2*x^4)/((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 5. (End)
Hugo Pfoertner, Oct 08 2020: Apparently a(n)=2*(A008795(n-3)+1).

A338043 Draw n rays from each of two distinct points in the plane; a(n) is the number of edges thus created. See Comments for details.

Original entry on oeis.org

2, 3, 10, 7, 22, 15, 38, 27, 58, 43, 82, 63, 110, 87, 142, 115, 178, 147, 218, 183, 262, 223, 310, 267, 362, 315, 418, 367, 478, 423, 542, 483, 610, 547, 682, 615, 758, 687, 838, 763, 922, 843, 1010, 927, 1102, 1015, 1198, 1107, 1298, 1203, 1402, 1303, 1510, 1407

Offset: 1

Lars Blomberg, Oct 08 2020

nonn

The rays are evenly spaced around each point. The first ray of one point goes in the opposite direction of the other point. Should a ray hit the other point, it terminates there, i.e., it is converted to a line segment.

See A338041 for illustrations.

			For n=1:    <-----x     x----->   so a(1)=2.
For n=2:    <-----x<--->x----->   so a(2)=3.

Cf. A338041 (regions), A338042 (vertices).

Cf. also A000096, A000124.

a(n)=if(n%2==1,(n^2 + 4*n - 1)/2,(n^2 - 2*n + 6)/2)
vector(200, n, a(n))

a(n) = (n^2 + 4*n - 1)/2, n odd; (n^2 - 2*n + 6)/2, n even (conjectured).
Conjectured by Stefano Spezia, Oct 08 2020 after Lars Blomberg: (Start)
G.f.: x*(2 + x + 3*x^2 - 5*x^3 + 3*x^4)/((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 5. (End)
Hugo Pfoertner, Oct 08 2020: It appears that even terms are 4*A000096+2, odd terms 4*A000124-1.

cp's OEIS Frontend

A082111 a(n) = n^2 + 5*n + 1.

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A338122 Place three points evenly spaced around a circle, draw n evenly spaced rays from each of the points, a(n) is the number of regions thus created. See Comments for details.

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A338042 Draw n rays from each of two distinct points in the plane; a(n) is the number of vertices thus created. See Comments for details.

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A338043 Draw n rays from each of two distinct points in the plane; a(n) is the number of edges thus created. See Comments for details.

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