A295707 -id:A295707 - cp's OEIS frontend

A059100 a(n) = n^2 + 2.

2, 3, 6, 11, 18, 27, 38, 51, 66, 83, 102, 123, 146, 171, 198, 227, 258, 291, 326, 363, 402, 443, 486, 531, 578, 627, 678, 731, 786, 843, 902, 963, 1026, 1091, 1158, 1227, 1298, 1371, 1446, 1523, 1602, 1683, 1766, 1851, 1938, 2027, 2118, 2211, 2306, 2403, 2502, 2603

Offset: 0

Henry Bottomley, Feb 13 2001

Let s(n) = Sum_{k>=1} 1/n^(2^k). Then I conjecture that the maximum element in the continued fraction for s(n) is n^2 + 2. - Benoit Cloitre, Aug 15 2002
Binomial transformation yields A081908, with A081908(0)=1 dropped. - R. J. Mathar, Oct 05 2008
1/a(n) = R(n)/r with R(n) the n-th radius of the Pappus chain of the symmetric arbelos with semicircle radii r, r1 = r/2 = r2. See the MathWorld link for Pappus chain (there are two of them, a left and a right one. In this case these two chains are congruent). - Wolfdieter Lang, Mar 01 2013
a(n) is the number of election results for an election with n+2 candidates, say C1, C2, ..., and C(n+2), and with only two voters (each casting a single vote) that have C1 and C2 receiving the same number of votes. See link below. - Dennis P. Walsh, May 08 2013
This sequence gives the set of values such that for sequences b(k+1) = a(n)*b(k) - b(k-1), with initial values b(0) = 2, b(1) = a(n), all such sequences are invariant under this transformation: b(k) = (b(j+k) + b(j-k))/b(j), except where b(j) = 0, for all integer values of j and k, including negative values. Examples are: at n=0, b(k) = 2 for all k; at n=1, b(k) = A005248; at n=2, b(k) = 2*A001541; at n=3, b(k)= A057076; at n=4, b(k) = 2*A023039. This b(k) family are also the transformation results for all related b'(k) (i.e., those with different initial values) including non-integer values. Further, these b(k) are also the bisections of the transformations of sequences of the form G(k+1) = n * G(k) + G(k-1), and those bisections are invariant for all initial values of g(0) and g(1), including non-integer values. For n = 1 this g(k) family includes Fibonacci and Lucas, where the invariant bisection is b(k) = A005248. The applicable bisection for this transformation of g(k) is for the odd values of k, and applies for all n. Also see A000032 for a related family of sequences. - Richard R. Forberg, Nov 22 2014
Also the number of maximum matchings in the n-gear graph. - Eric W. Weisstein, Dec 31 2017
Also the Wiener index of the n-dipyramidal graph. - Eric W. Weisstein, Jun 14 2018
Numbers of the form n^2+2 have no factors that are congruent to 7 (mod 8). - Gordon E. Michaels, Sep 12 2019
For n >= 1, the continued fraction expansion of sqrt(a(n)) is [n; {n, 2n}]. - Magus K. Chu, Sep 10 2022

			For n = 2, a(2) = 6 since there are 6 election results in a 4-candidate, 2-voter election that have candidates c1 and c2 tied. Letting <i,j> denote voter 1 voting for candidate i and voter 2 voting for candidate j, the six election results are <1,2>, <2,1>, <3,3>, <3,4>, <4,3>, and <4,4>. - _Dennis P. Walsh_, May 08 2013

Harry J. Smith, Table of n, a(n) for n = 0..1000
Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 52, 56.
Hesam Dashti, A New Upper Bound on the Length of Shortest Permutation Strings; An Algorithm for Generating Permutation Strings, arXiv:1009.5053 [math.CO], 2010. - _Jonathan Vos Post_, Sep 28 2010
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Alexander Soifer, Coffee Hour and the Conway-Soifer Cover-Up, In: How Does One Cut a Triangle? (2009), pp. 147-156. See also here
Dennis P. Walsh, Notes on a tied election.
Eric Weisstein's World of Mathematics, Dipyramidal Graph.
Eric Weisstein's World of Mathematics, Gear Graph.
Eric Weisstein's World of Mathematics, Matching.
Eric Weisstein's World of Mathematics, Maximum Independent Edge Set.
Eric Weisstein's World of Mathematics, Near-Square Prime.
Eric Weisstein's World of Mathematics, Pappus Chain.
Eric Weisstein's World of Mathematics, Wiener Index.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000032, A000196, A000290, A001541, A002522, A005248, A008865, A056105, A056109, A056899, A057076, A069987, A081908, A114964, A156798, A166464.
Apart from initial terms, same as A010000.
2nd row/column of A295707.

a059100 = (+ 2) . (^ 2)
a059100_list = scanl (+) (2) [1, 3 ..]
-- Reinhard Zumkeller, Feb 09 2015

with(combinat, fibonacci):seq(fibonacci(3, i)+1, i=0..49); # Zerinvary Lajos, Mar 20 2008

Table[n^2 + 2, {n, 0, 50}] (* Vladimir Joseph Stephan Orlovsky, Dec 15 2008 *)
LinearRecurrence[{3, -3, 1}, {2, 3, 6}, 50] (* Vincenzo Librandi, Feb 15 2012 *)
Range[0, 20]^2 + 2 (* Eric W. Weisstein, Dec 31 2017 *)
CoefficientList[Series[(-2 + 3 x - 3 x^2)/(-1 + x)^3, {x, 0, 20}], x] (* Eric W. Weisstein, Dec 31 2017 *)

a(n) = { n^2+2 } \\ Harry J. Smith, Jun 24 2009

[lucas_number1(3,n,-2) for n in range(0, 50)] # Zerinvary Lajos, May 16 2009

G.f.: (2 - 3*x + 3*x^2)/(1 - x)^3. - R. J. Mathar, Oct 05 2008
a(n) = ((n - 2)^2 + 2*(n + 1)^2)/3. - Reinhard Zumkeller, Feb 13 2009
a(n) = A000196(A156798(n) - A000290(n)). - Reinhard Zumkeller, Feb 16 2009
a(n) = 2*n + a(n-1) - 1 with a(0) = 2. - Vincenzo Librandi, Aug 07 2010
a(n+3) = (A166464(n+5) - A166464(n))/20. - Paul Curtz, Nov 07 2012
From Paul Curtz, Nov 07 2012: (Start)
a(3*n) mod 9 = 2.
a(3*n+1) = 3*A056109(n).
a(3*n+2) = 3*A056105(n+1). (End)
Sum_{n >= 1} 1/a(n) = Pi * coth(sqrt(2)*Pi) / 2^(3/2) - 1/4. - Vaclav Kotesovec, May 01 2018
From Amiram Eldar, Jan 29 2021: (Start)
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(2)*Pi*(csch(sqrt(2)*Pi)))/4.
Product_{n>=0} (1 + 1/a(n)) = sqrt(3/2)*csch(sqrt(2)*Pi)*sinh(sqrt(3)*Pi).
Product_{n>=0} (1 - 1/a(n)) = csch(sqrt(2)*Pi)*sinh(Pi)/sqrt(2). (End)
E.g.f.: exp(x)*(2 + x + x^2). - Stefano Spezia, Aug 07 2024

A107348 Triangle read by rows: T(m,n) = number of different lines in a rectangular m X n array of points with integer coordinates (x,y): 0 <= x <= m, 0 <= y <= n.

Original entry on oeis.org

0, 1, 6, 1, 11, 20, 1, 18, 35, 62, 1, 27, 52, 93, 140, 1, 38, 75, 136, 207, 306, 1, 51, 100, 181, 274, 405, 536, 1, 66, 131, 238, 361, 534, 709, 938, 1, 83, 164, 299, 454, 673, 894, 1183, 1492, 1, 102, 203, 370, 563, 836, 1111, 1470, 1855, 2306

Offset: 0

Dan Dima, May 23 2005

We may assume n <= m since T(m,n)=T(n,m).

			Triangle begins
0,
1, 6,
1, 11, 20,
1, 18, 35, 62,
1, 27, 52, 93, 140,
1, 38, 75, 136, 207, 306,
1, 51, 100, 181, 274, 405, 536,
1, 66, 131, 238, 361, 534, 709, 938,
1, 83, 164, 299, 454, 673, 894, 1183, 1492,
1, 102, 203, 370, 563, 836, 1111, 1470, 1855, 2306,
...

M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh, On the minimal teaching sets of two-dimensional threshold functions, SIAM J. Disc. Math. 29(1), 2015, pp. 157-165.
Les Reid, Problem #7: How Many Lines Does the Lattice of Points Generate?, Problems from the 04-05 academic year, Challenge Archive, Missouri State University's Problem Corner.

Cf. A295707 (symmetric array), A018808 (diagonal). A160842 - A160850 (columns).

VR := proc(m,n,q) local a,i,j; a:=0;
for i from -m+1 to m-1 do for j from -n+1 to n-1 do
if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;
LL:=(m,n)->(VR(m,n,1)-VR(m,n,2))/2;
for m from 1 to 12 do lprint([seq(LL(m,n),n=1..m)]); od: # N. J. A. Sloane, Feb 10 2020

VR[m_, n_, q_] := Sum[If[GCD[i, j] == q, (m - Abs[i])(n - Abs[j]), 0], {i, -m + 1, m - 1}, {j, -n + 1, n - 1}];
LL[m_, n_] := (1/2)(VR[m, n, 1] - VR[m, n, 2]);
Table[LL[m, n], {m, 1, 10}, {n, 1, m}] // Flatten (* Jean-François Alcover, Jun 04 2023, after N. J. A. Sloane *)

T(0, 0) = 0; T(m, 0) = 1, m >= 1.
When both m,n -> +oo, T(m,n) / 2Cmn -> 9/(2*pi^2). - Dan Dima, Mar 18 2006
T(n,m) = A295707(n,m). - R. J. Mathar, Dec 17 2017

T(3,3) corrected and sequence extended by R. J. Mathar, Dec 17 2017

A160842 Number of lines through at least 2 points of a 2 X n grid of points.

Original entry on oeis.org

0, 1, 6, 11, 18, 27, 38, 51, 66, 83, 102, 123, 146, 171, 198, 227, 258, 291, 326, 363, 402, 443, 486, 531, 578, 627, 678, 731, 786, 843, 902, 963, 1026, 1091, 1158, 1227, 1298, 1371, 1446, 1523, 1602, 1683, 1766, 1851, 1938, 2027, 2118, 2211, 2306, 2403, 2502

Offset: 0

Seppo Mustonen, May 28 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
S. Mustonen, On lines and their intersection points in a rectangular grid of points
Seppo Mustonen, On lines and their intersection points in a rectangular grid of points [Local copy]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A295707, A107348.

[0,1] cat [n^2 + 2: n in [2..100]]; // G. C. Greubel, Apr 30 2018

a[n_]:=If[n<2,n,n^2+2] Table[a[n],{n,0,50}]
Join[{0,1},Range[2,50]^2+2] (* Harvey P. Dale, Feb 06 2015 *)

Vec(-x*(2*x^3-4*x^2+3*x+1) / (x-1)^3 + O(x^100)) \\ Colin Barker, May 24 2015

a(n) = n^2 + 2 = A059100(n) = A010000(n) for n > 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4. - Colin Barker, May 24 2015
G.f.: -x*(2*x^3 - 4*x^2 + 3*x + 1) / (x-1)^3. - Colin Barker, May 24 2015
Sum_{n>=1} 1/a(n) = Pi * coth(sqrt(2)*Pi) / 2^(3/2) - 1/4. - Vaclav Kotesovec, May 01 2018

A160850 Number of lines through at least 2 points of a 10 X n grid of points.

Original entry on oeis.org

0, 1, 102, 203, 370, 563, 836, 1111, 1470, 1855, 2306, 2757, 3298, 3857, 4506, 5159, 5868, 6603, 7428, 8255, 9172, 10105, 11098, 12101, 13194, 14305, 15496, 16697, 17964, 19251, 20628, 21997, 23456, 24941, 26492, 28053, 29688, 31341, 33084

Offset: 0

Seppo Mustonen, May 28 2009

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000
S. Mustonen, On lines and their intersection points in a rectangular grid of points

Column k=10 of A295707.

m=10;
a[0]=0; a[1]=1;
a[2]=m^2+2;
a[3]=2*m^2+3-Mod[m,2];
a[n_]:=a[n]=2*a[n-1]-a[n-2]+2*p1[m,n]+2*p4[m,n]
p1[m_,n_]:=Sum[p2[m,n,y], {y,1,m-1}]
p2[m_,n_,y_]:=If[GCD[y,n-1]==1,m-y,0]
p[i_]:=If[i>0,i,0]
p2[m_,n_,x_,y_]:=p2[m,n,x,y]=(n-x)*(m-y)-p[n-2*x]*p[m-2*y]
p3[m_,n_,x_,y_]:=p2[m,n,x,y]-2*p2[m,n-1,x,y]+p2[m,n-2,x,y]
p4[m_,n_]:=p4[m,n]=If[Mod[n,2]==0,0,p42[m,n]]
p42[m_,n_]:=p42[m,n]=Sum[p43[m,n,y], {y,1,m-1}]
p43[m_,n_,y_]:=If[GCD[(n-1)/2,y]==1,p3[m,n,(n-1)/2,y],0]
Table[a[n],{n,0,39}]

a(n) = (1/2)*(f(m,n,1) - f(m,n,2)) where f(m,n,k) = Sum((n-|kx|)*(m-|ky|)); -n < kx < n, -m < ky < m, (x,y)=1, m=10.
For another more efficient formula, see Mathematica code below.
Conjectures from Colin Barker, Dec 25 2017: (Start)
G.f.: x*(1 + 102*x + 204*x^2 + 472*x^3 + 766*x^4 + 1205*x^5 + 1571*x^6 + 1999*x^7 + 2188*x^8 + 2334*x^9 + 2168*x^10 + 1959*x^11 + 1531*x^12 + 1165*x^13 + 746*x^14 + 462*x^15 + 214*x^16 + 112*x^17 + 21*x^18 + 10*x^19 + 10*x^20) / ((1 - x)^3*(1 + x)*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)).
a(n) =  -a(n-2) + a(n-5) + a(n-6) + 2*a(n-7) + a(n-8) + a(n-9) - a(n-11) - a(n-12) - 2*a(n-13) - a(n-14) - a(n-15) + a(n-18) + a(n-20) for n>21.
(End)

A160843 Number of lines through at least 2 points of a 3 X n grid of points.

Original entry on oeis.org

0, 1, 11, 20, 35, 52, 75, 100, 131, 164, 203, 244, 291, 340, 395, 452, 515, 580, 651, 724, 803, 884, 971, 1060, 1155, 1252, 1355, 1460, 1571, 1684, 1803, 1924, 2051, 2180, 2315, 2452, 2595, 2740, 2891, 3044, 3203, 3364, 3531, 3700, 3875, 4052, 4235, 4420

Offset: 0

Seppo Mustonen, May 28 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
S. Mustonen, On lines and their intersection points in a rectangular grid of points
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

3rd row/column of A107348, A295707.

[0, 1] cat [2*n^2 + 3 - n mod 2: n in [2..100]]; // G. C. Greubel, Apr 30 2018

a[n_]:=If[n<2,n,2*n^2+3-Mod[n,2]] Table[a[n],{n,0,47}]
Join[{0, 1}, LinearRecurrence[{2, 0, -2, 1}, {11, 20, 35, 52}, 20]] (* G. C. Greubel, Apr 30 2018 *)

Vec(-x*(3*x^4-3*x^3-2*x^2+9*x+1)/((x-1)^3*(x+1)) + O(x^100)) \\ Colin Barker, May 24 2015

a(n) = 2*n^2 + 3 - n mod 2.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n > 5. - Colin Barker, May 24 2015
G.f.: -x*(3*x^4 - 3*x^3 - 2*x^2 + 9*x + 1) / ((x-1)^3*(x+1)). - Colin Barker, May 24 2015

A160844 Number of lines through at least 2 points of a 4 X n grid of points.

Original entry on oeis.org

0, 1, 18, 35, 62, 93, 136, 181, 238, 299, 370, 445, 532, 621, 722, 827, 942, 1061, 1192, 1325, 1470, 1619, 1778, 1941, 2116, 2293, 2482, 2675, 2878, 3085, 3304, 3525, 3758, 3995, 4242, 4493, 4756, 5021, 5298, 5579, 5870, 6165, 6472, 6781, 7102, 7427, 7762

Offset: 0

Seppo Mustonen, May 28 2009

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000
S. Mustonen, On lines and their intersection points in a rectangular grid of points
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

4th row/column of A107348, A295707.

I:=[18, 35, 62, 93, 136, 181]; [0,1] cat [n le 6 select I[n] else Self(n-1) +Self(n-2) -Self(n-4) -Self(n-5) +Self(n-6): n in [1..30]]; // G. C. Greubel, Apr 30 2018

a[0]=0; a[1]=1; a[2]=18; a[3]=35; a[n_]:=a[n]=a[n]=2*a[n-1]-a[n-2]+R[n] c4={10,4,12,2,12,4}; R[n_]:=c4[[Mod[n+2,6]+1]] Table[a[n],{n,0,46}]
Join[{0,1}, LinearRecurrence[{1,1,0,-1,-1,1}, {18, 35, 62, 93, 136, 181}, 50]] (* G. C. Greubel, Apr 30 2018 *)

my(x='x+O('x^30)); concat([0], Vec(x*(4*x^6-3*x^4+9*x^3+16*x^2+ 17*x+1 )/((1-x)^3*(x+1)*(x^2+x+1)))) \\ G. C. Greubel, Apr 30 2018

a(n) = 2*a(n-1) - a(n-2) + C(mod(n+2,6) + 1), C=(10,4,12,2,12,4), for n >= 4.
From Colin Barker, May 24 2015: (Start)
  a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6) for n > 5.
  G.f.: x*(4*x^6 - 3*x^4 + 9*x^3 + 16*x^2 + 17*x + 1) / ((1-x)^3*(x + 1)*(x^2 + x + 1)).
(End)

A160845 Number of lines through at least 2 points of a 5 X n grid of points.

Original entry on oeis.org

0, 1, 27, 52, 93, 140, 207, 274, 361, 454, 563, 676, 809, 944, 1099, 1258, 1433, 1614, 1815, 2016, 2237, 2464, 2707, 2954, 3221, 3490, 3779, 4072, 4381, 4696, 5031, 5366, 5721, 6082, 6459, 6840, 7241, 7644, 8067, 8494, 8937, 9386, 9855, 10324, 10813

Offset: 0

Seppo Mustonen, May 28 2009

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000
S. Mustonen, On lines and their intersection points in a rectangular grid of points

5th row/column of A107348, A295707.

m=5;
a[0]=0; a[1]=1;
a[2]=m^2+2;
a[3]=2*m^2+3-Mod[m,2];
a[n_]:=a[n]=2*a[n-1]-a[n-2]+2*p1[m,n]+2*p4[m,n]
p1[m_,n_]:=Sum[p2[m,n,y], {y,1,m-1}]
p2[m_,n_,y_]:=If[GCD[y,n-1]==1,m-y,0]
p[i_]:=If[i>0,i,0]
p2[m_,n_,x_,y_]:=p2[m,n,x,y]=(n-x)*(m-y)-p[n-2*x]*p[m-2*y]
p3[m_,n_,x_,y_]:=p2[m,n,x,y]-2*p2[m,n-1,x,y]+p2[m,n-2,x,y]
p4[m_,n_]:=p4[m,n]=If[Mod[n,2]==0,0,p42[m,n]]
p42[m_,n_]:=p42[m,n]=Sum[p43[m,n,y], {y,1,m-1}]
p43[m_,n_,y_]:=If[GCD[(n-1)/2,y]==1,p3[m,n,(n-1)/2,y],0]
Table[a[n],{n,0,44}]

a(n) = (1/2)*(f(m,n,1)-f(m,n,2)) where f(m,n,k) = Sum((n-|kx|)*(m-|ky|)); -n < kx < n, -m < ky < m, (x,y)=1, m=5.
For another more efficient formula, see Mathematica code below.
Conjectures from Colin Barker, May 24 2015: (Start)
a(n) = a(n-1) + a(n-3) - a(n-5) - a(n-7) + a(n-8) for n > 7.
G.f.: x*(5*x^8 + x^6 + 16*x^5 + 20*x^4 + 40*x^3 + 25*x^2 + 26*x + 1) / ((1 - x)^3*(x + 1)*(x^2 + 1)*(x^2 + x + 1)).
(End)

A160848 Number of lines through at least 2 points of an 8 X n grid of points.

Original entry on oeis.org

0, 1, 66, 131, 238, 361, 534, 709, 938, 1183, 1470, 1759, 2104, 2459, 2870, 3287, 3740, 4209, 4734, 5261, 5844, 6437, 7070, 7711, 8408, 9115, 9872, 10637, 11444, 12265, 13142, 14015, 14944, 15889, 16876, 17871, 18914, 19967, 21076, 22193, 23352

Offset: 0

Seppo Mustonen, May 28 2009

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000
S. Mustonen, On lines and their intersection points in a rectangular grid of points

Column k=8 of A295707.

m=8;
a[0]=0; a[1]=1;
a[2]=m^2+2;
a[3]=2*m^2+3-Mod[m,2];
a[n_]:=a[n]=2*a[n-1]-a[n-2]+2*p1[m,n]+2*p4[m,n]
p1[m_,n_]:=Sum[p2[m,n,y], {y,1,m-1}]
p2[m_,n_,y_]:=If[GCD[y,n-1]==1,m-y,0]
p[i_]:=If[i>0,i,0]
p2[m_,n_,x_,y_]:=p2[m,n,x,y]=(n-x)*(m-y)-p[n-2*x]*p[m-2*y]
p3[m_,n_,x_,y_]:=p2[m,n,x,y]-2*p2[m,n-1,x,y]+p2[m,n-2,x,y]
p4[m_,n_]:=p4[m,n]=If[Mod[n,2]==0,0,p42[m,n]]
p42[m_,n_]:=p42[m,n]=Sum[p43[m,n,y], {y,1,m-1}]
p43[m_,n_,y_]:=If[GCD[(n-1)/2,y]==1,p3[m,n,(n-1)/2,y],0]
Table[a[n],{n,0,40}]

a(n) = (1/2)*(f(m,n,1)-f(m,n,2)) where f(m,n,k)=Sum((n-|kx|)*(m-|ky|)); -n

For another more efficient formula, see Mathematica code below.

Conjectures from Colin Barker, Dec 25 2017: (Start)

G.f.: x*(1 + 66*x + 132*x^2 + 304*x^3 + 492*x^4 + 771*x^5 + 1003*x^6 + 1273*x^7 + 1390*x^8 + 1480*x^9 + 1374*x^10 + 1241*x^11 + 971*x^12 + 739*x^13 + 476*x^14 + 296*x^15 + 140*x^16 + 74*x^17 + 17*x^18 + 8*x^19 + 8*x^20) / ((1 - x)^3*(1 + x)*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)).

a(n) = -a(n-2) + a(n-5) + a(n-6) + 2*a(n-7) + a(n-8) + a(n-9) - a(n-11) - a(n-12) - 2*a(n-13) - a(n-14) - a(n-15) + a(n-18) + a(n-20) for n>21.

(End)

A160849 Number of lines through at least 2 points of a 9 X n grid of points.

Original entry on oeis.org

0, 1, 83, 164, 299, 454, 673, 894, 1183, 1492, 1855, 2218, 2653, 3102, 3623, 4148, 4719, 5310, 5973, 6638, 7375, 8124, 8923, 9730, 10609, 11502, 12459, 13424, 14443, 15478, 16585, 17686, 18859, 20052, 21299, 22554, 23869, 25198, 26599, 28008

Offset: 0

Seppo Mustonen, May 28 2009

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000
S. Mustonen, On lines and their intersection points in a rectangular grid of points

Column k=9 of A295707.

m=9;
a[0]=0; a[1]=1;
a[2]=m^2+2;
a[3]=2*m^2+3-Mod[m,2];
a[n_]:=a[n]=2*a[n-1]-a[n-2]+2*p1[m,n]+2*p4[m,n]
p1[m_,n_]:=Sum[p2[m,n,y], {y,1,m-1}]
p2[m_,n_,y_]:=If[GCD[y,n-1]==1,m-y,0]
p[i_]:=If[i>0,i,0]
p2[m_,n_,x_,y_]:=p2[m,n,x,y]=(n-x)*(m-y)-p[n-2*x]*p[m-2*y]
p3[m_,n_,x_,y_]:=p2[m,n,x,y]-2*p2[m,n-1,x,y]+p2[m,n-2,x,y]
p4[m_,n_]:=p4[m,n]=If[Mod[n,2]==0,0,p42[m,n]]
p42[m_,n_]:=p42[m,n]=Sum[p43[m,n,y], {y,1,m-1}]
p43[m_,n_,y_]:=If[GCD[(n-1)/2,y]==1,p3[m,n,(n-1)/2,y],0]
Table[a[n],{n,0,39}]

a(n) = (1/2)*(f(m,n,1) - f(m,n,2)) where f(m,n,k) = Sum((n-|kx|)*(m-|ky|)); -n < kx < n, -m < ky < m, (x,y)=1, m=9.
For another more efficient formula, see Mathematica code below.
Conjectures from Colin Barker, Dec 25 2017: (Start)
G.f.: x*(1 + 83*x + 165*x^2 + 382*x^3 + 618*x^4 + 971*x^5 + 1264*x^6 + 1607*x^7 + 1756*x^8 + 1873*x^9 + 1738*x^10 + 1571*x^11 + 1228*x^12 + 935*x^13 + 600*x^14 + 373*x^15 + 174*x^16 + 92*x^17 + 19*x^18 + 9*x^19 + 9*x^20) / ((1 - x)^3*(1 + x)*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)).
a(n) = -a(n-2) + a(n-5) + a(n-6) + 2*a(n-7) + a(n-8) + a(n-9) - a(n-11) - a(n-12) - 2*a(n-13) - a(n-14) - a(n-15) + a(n-18) + a(n-20) for n>21.
(End)

Showing 1-9 of 9 results.

cp's OEIS Frontend

A059100 a(n) = n^2 + 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Sage

Formula

A107348 Triangle read by rows: T(m,n) = number of different lines in a rectangular m X n array of points with integer coordinates (x,y): 0 <= x <= m, 0 <= y <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A160842 Number of lines through at least 2 points of a 2 X n grid of points.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A160850 Number of lines through at least 2 points of a 10 X n grid of points.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A160843 Number of lines through at least 2 points of a 3 X n grid of points.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A160844 Number of lines through at least 2 points of a 4 X n grid of points.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A160845 Number of lines through at least 2 points of a 5 X n grid of points.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula