A225606 -id:A225606 - cp's OEIS frontend

A187397 Expansion of -2x^4 (3x^13 +2x^12 +x^11 -6x^10 -10x^9 -6x^8 +x^7 +7x^6 +5x^5 -x^4 -8x^3 -11x^2 -8x -5) / ((x -1)^4 (x +1)^2 (x^2 +1)^2 *(x^2 +x +1)^2).

0, 0, 0, 0, 10, 16, 22, 36, 54, 66, 92, 122, 156, 196, 240, 288, 366, 426, 490, 590, 698, 780, 904, 1036, 1176, 1326, 1484, 1650, 1874, 2060, 2254, 2512, 2782, 3006, 3300, 3606, 3924, 4256, 4600, 4956, 5398, 5782, 6178, 6666, 7170, 7608, 8144

Offset: 0

Sean A. Irvine, Mar 23 2011

In contrast, the number of distinct lines passing through 4 or more points in an n X n grid is given by 0, 0, 0, 10, 16, 22, 44, 74, 92, 154, 232, 326, 436, 562, 704, 998, 1268,.. = A018808(n) -A018809(n) -A018810(n) = A225606(n) -A018810(n). - David W. Wilson, Aug 05 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 2, 2, 0, -1, -4, -1, 0, 2, 2, 0, 0, -1).

Cf. A178465, A187062.

CoefficientList[ Series[ 2x^4 (5 + 8x + 11x^2 + 8x^3 + x^4 - 5x^5 - 7x^6 - x^7 + 6x^8 + 10x^9 + 6x^10 - x^11 - 2x^12 - 3x^13)/((-1 + x)^4 (1 + x)^2 (1 + x^2)^2 (1 + x + x^2)^2), {x, 0, 43}], x] (* or *) LinearRecurrence[{0, 0, 2, 2, 0, -1, -4, -1, 0, 2, 2, 0, 0, -1}, {10, 16, 22, 36, 54, 66, 92, 122, 156, 196, 240, 288, 366, 426}, 40] (* Robert G. Wilson v, Feb 17 2014 *)

Definition replaced with Colin Barker's g.f. by R. J. Mathar, Aug 06 2013

Offset changed from 1 to 0 and a(0)=0 added by Vincenzo Librandi, Feb 19 2014

A234248 Number of distinct lines passing through at least three points in a triangular grid of side n.

Original entry on oeis.org

3, 6, 12, 21, 36, 57, 90, 129, 186, 261, 354, 465, 612, 783, 990, 1233, 1524, 1863, 2262, 2703, 3216, 3801, 4458, 5187, 6024, 6951, 7986, 9129, 10392, 11775, 13302, 14943, 16746, 18711, 20844, 23145, 25668, 28377, 31296, 34425, 37782, 41367, 45210, 49287

Offset: 3

Heinrich Ludwig, Jan 18 2014

			     a
    b c
   d e f
  g h i j
In this triangle grid of side 4, there are a(4) = 6 distinct lines passing through at least 3 points: ag, gj, ja, ch, df, ib.

Jon E. Schoenfield, Table of n, a(n) for n = 3..10000

Cf. A225606 (analogous problem for square grids).

g(n) = if (n>0, n*(n+1)/2, 0);
a(n) = my(k=3); 3*sum(j=1, (n-1)\(k-1), eulerphi(j) * (g(n-(k-1)*j) - g(n-k*j))); \\ Michel Marcus, Aug 19 2014

a(n) = 3*Sum_{j=1..floor((n-1)/(k-1))} EulerPhi(j) * (g(n-(k-1)*j) - g(n-k*j)) where k = 3 (the minimum required number of points) and g(i) = A000217(i) (i.e., the i-th triangular number) if i > 0, otherwise 0. - Jon E. Schoenfield, Aug 17 2014

More terms from Jon E. Schoenfield, Aug 17 2014

A187062 Expansion of 2x^2 (4 +7x +5x^2 -x^3 -4x^4 +6x^6 +4x^7 -x^8 -2x^9) / ((1+x)^2 (1+x+x^2)^2 (1-x)^4) .

Original entry on oeis.org

0, 0, 8, 14, 26, 42, 64, 90, 134, 172, 232, 300, 378, 464, 584, 690, 834, 990, 1160, 1342, 1574, 1784, 2048, 2328, 2626, 2940, 3320, 3670, 4090, 4530, 4992, 5474, 6038, 6564, 7176, 7812, 8474, 9160, 9944, 10682, 11522, 12390, 13288, 14214

Offset: 1

Sean A. Irvine, Mar 21 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,2,-1,-4,-1,2,2,0,-1).

Cf. A000755, A178465, A187397, A225606.

CoefficientList[Series[ 2x^2 (4 + 7x + 5x^2 - x^3 - 4x^4 + 6x^6 + 4x^7 - x^8 - 2x^9)/((1 + x)^2 (1 + x + x^2)^2 (x - 1)^4), {x, 0, 43}], x]  (* or *) LinearRecurrence[ {0, 2, 2, -1, -4, -1, 2, 2, 0, -1}, {8, 14, 26, 42, 64, 90, 134, 172, 232, 300}, 42] (* Robert G. Wilson v, Feb 17 2014 *)

a(n) = 2*a(n-2) + 2*a(n-3) - a(n-4) - 4*a(n-5) - a(n-6) + 2*a(n-7) + 2*a(n-8) - a(n-10) .

Name replaced by L. Edson Jeffery's definition. R. J. Mathar, Aug 06 2013

cp's OEIS Frontend

A187397 Expansion of -2x^4 (3x^13 +2x^12 +x^11 -6x^10 -10x^9 -6x^8 +x^7 +7x^6 +5x^5 -x^4 -8x^3 -11x^2 -8x -5) / ((x -1)^4 (x +1)^2 (x^2 +1)^2 *(x^2 +x +1)^2).

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A234248 Number of distinct lines passing through at least three points in a triangular grid of side n.

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A187062 Expansion of 2x^2 (4 +7x +5x^2 -x^3 -4x^4 +6x^6 +4x^7 -x^8 -2x^9) / ((1+x)^2 (1+x+x^2)^2 (1-x)^4) .

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cp's OEIS Frontend

A187397 Expansion of -2*x^4 *(3*x^13 +2*x^12 +x^11 -6*x^10 -10*x^9 -6*x^8 +x^7 +7*x^6 +5*x^5 -x^4 -8*x^3 -11*x^2 -8*x -5) / ((x -1)^4 *(x +1)^2 *(x^2 +1)^2 *(x^2 +x +1)^2).

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A234248 Number of distinct lines passing through at least three points in a triangular grid of side n.

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Extensions

A187062 Expansion of 2*x^2 *(4 +7*x +5*x^2 -x^3 -4*x^4 +6*x^6 +4*x^7 -x^8 -2*x^9) / ((1+x)^2 *(1+x+x^2)^2 *(1-x)^4) .

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A187397 Expansion of -2x^4 (3x^13 +2x^12 +x^11 -6x^10 -10x^9 -6x^8 +x^7 +7x^6 +5x^5 -x^4 -8x^3 -11x^2 -8x -5) / ((x -1)^4 (x +1)^2 (x^2 +1)^2 *(x^2 +x +1)^2).

A187062 Expansion of 2x^2 (4 +7x +5x^2 -x^3 -4x^4 +6x^6 +4x^7 -x^8 -2x^9) / ((1+x)^2 (1+x+x^2)^2 (1-x)^4) .