A187062 -id:A187062 - 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

A178465 Expansion of -2x^2(-3-2x+x^2-x^3-2x^4+x^5) / ( (1+x)^2*(x-1)^4 ).

Original entry on oeis.org

0, 0, 6, 16, 36, 66, 114, 176, 264, 370, 510, 672, 876, 1106, 1386, 1696, 2064, 2466, 2934, 3440, 4020, 4642, 5346, 6096, 6936, 7826, 8814, 9856, 11004, 12210, 13530, 14912, 16416, 17986, 19686, 21456, 23364, 25346, 27474, 29680, 32040, 34482

Offset: 0

Sean A. Irvine, Mar 23 2011

nonn

Cf. A187062, A187397, A061804, A018808.

CoefficientList[ Series[ 2x^2 (3 + 2x - x^2 + x^3 + 2x^4 - x^5)/((1 + x)^2 (x - 1)^4), {x, 0, 42}], x] (* Robert G. Wilson v, Feb 17 2014 *)

def A178465(n): return n+(m:=n&1)+(n*(n**2-m)>>1) if n != 1 else 0 # Chai Wah Wu, Aug 30 2022

For n even, a(n) = n*(2+n^2)/2 = A061804(n/2). For n>1 and odd, a(n)=(n+1)*(n^2-n+2)/2 = 2*A212133((n+1)/2).

a(n) = (2-2*(-1)^n+(3+(-1)^n)*n+2*n^3)/4 for n>1. [Colin Barker, Feb 18 2013]

Discrepancy with A018808 resolved. David W. Wilson, Aug 05 2013
First line of formulas corrected. R. J. Mathar, Aug 05 2013
Prepended a(0)=0, Joerg Arndt, Feb 19 2014

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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A178465 Expansion of -2x^2(-3-2x+x^2-x^3-2x^4+x^5) / ( (1+x)^2*(x-1)^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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A178465 Expansion of -2*x^2*(-3-2*x+x^2-x^3-2*x^4+x^5) / ( (1+x)^2*(x-1)^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).

A178465 Expansion of -2x^2(-3-2x+x^2-x^3-2x^4+x^5) / ( (1+x)^2*(x-1)^4 ).