A176145 -id:A176145 - cp's OEIS frontend

A117662 a(n) = n(n-1)(n-2)*(n+3)/12.

0, 0, 0, 3, 14, 40, 90, 175, 308, 504, 780, 1155, 1650, 2288, 3094, 4095, 5320, 6800, 8568, 10659, 13110, 15960, 19250, 23023, 27324, 32200, 37700, 43875, 50778, 58464, 66990, 76415, 86800, 98208, 110704, 124355, 139230, 155400, 172938, 191919

Offset: 0

Roger L. Bagula, Apr 11 2006

Also, the number of external intersections of the diagonals of a general n-gon = (A176145) - (A000332). - Michel Lagneau, Apr 21 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Aram Bingham, Lisa Johnston, Colin Lawson, Rosa Orellana, Jianping Pan, and Chelsea Sato, The Chromatic Symmetric Function for Unicyclic Graphs, arXiv:2505.06486 [math.CO], 2025. See p. 12.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Essentially the same as A050297 and A005701.

Cf. A000332, A176145.

[n*(n-1)*(n-2)*(n+3)/12: n in [0..50]]; // Vincenzo Librandi, Oct 10 2013

seq(n*(n-1)*(n-2)*(n+3)/12, n=0..40); # Wesley Ivan Hurt, Oct 10 2013

Table[n(n-1)(n-2)(n+3)/12, {n,0,100}] (* Wesley Ivan Hurt, Sep 26 2013 *)
CoefficientList[Series[x^3 (3 - x)/(1 - x)^5, {x, 0, 80}], x] (* Vincenzo Librandi, Oct 10 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{0,0,0,3,14},80] (* Harvey P. Dale, Jan 01 2025 *)

G.f.: x^3*(3-x)/(1-x)^5. - Colin Barker, Jan 31 2012
From Amiram Eldar, May 17 2025: (Start)
Sum_{n>=3} 1/a(n) = 137/300.
Sum_{n>=3} (-1)^(n+1)/a(n) = 32*log(2)/5 - 1247/300. (End)

Edited by N. J. A. Sloane, Apr 23 2006

A211380 Number of pairs of intersecting diagonals in the interior and exterior of a regular n-gon.

Original entry on oeis.org

0, 1, 5, 15, 42, 94, 189, 340, 572, 903, 1365, 1981, 2790, 3820, 5117, 6714, 8664, 11005, 13797, 17083, 20930, 25386, 30525, 36400, 43092, 50659, 59189, 68745, 79422, 91288, 104445, 118966, 134960, 152505, 171717, 192679, 215514, 240310, 267197, 296268, 327660

Offset: 3

Martin Renner, Feb 07 2013

Eric Weisstein, Regular Polygon Division by Diagonals (MathWorld).
Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Cf. A176145, A211379.

a:=n->piecewise(n mod 2 = 0,1/8*n*(n^3-11*n^2+43*n-58),n mod 2 = 1,1/8*n*(n-3)*(n^2-8*n+19),0);

Drop[CoefficientList[Series[x^4(2x^5-3x^4-7x^3-x^2-2x-1)/((x-1)^5(x+1)^2),{x,0,50}],x],3] (* or *) LinearRecurrence[{3,-1,-5,5,1,-3,1},{0,1,5,15,42,94,189},50] (* Harvey P. Dale, Dec 03 2022 *)

def A211380(n): return n*(n*(n*(n-11)+43)-58+(n&1))>>3 # Chai Wah Wu, Nov 22 2023

a(n) = 1/8*n*(n^3-11*n^2+43*n-58) for n even;
a(n) = 1/8*n*(n-3)*(n^2-8*n+19) for n odd.
a(n) = A176145(n) - A211379(n).
G.f.: x^4*(2*x^5-3*x^4-7*x^3-x^2-2*x-1) / ((x-1)^5*(x+1)^2). [Colin Barker, Feb 14 2013]

A294259 a(n) = n(n^3 + 2n^2 - 5*n + 10)/8.

Original entry on oeis.org

0, 1, 4, 15, 43, 100, 201, 364, 610, 963, 1450, 2101, 2949, 4030, 5383, 7050, 9076, 11509, 14400, 17803, 21775, 26376, 31669, 37720, 44598, 52375, 61126, 70929, 81865, 94018, 107475, 122326, 138664, 156585, 176188, 197575, 220851, 246124, 273505, 303108, 335050, 369451

Offset: 0

Bruno Berselli, Oct 30 2017

a(n) is even for n in A047481.

Also, a(n) is divisible by 5 if and only if n belongs to A047218.

			After 0:
1   =                     -(0) + (1);
4   =                 -(0 + 1) + (2 + 2*3/2);
15  =             -(0 + 1 + 2) + (3 + 4 + 5 + 3*4/2);
43  =         -(0 + 1 + 2 + 3) + (4 + 5 + 6 + 7 + 8 + 9 + 4*5/2);
100 =     -(0 + 1 + 2 + 3 + 4) + (5 + 6 + 7 + 8 + ... + 14 + 5*6/2);
201 = -(0 + 1 + 2 + 3 + 4 + 5) + (6 + 7 + 8 + 9 + ... + 20 + 6*7/2), etc.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A002817, A176145.
Cf. A101374: the sums in the Example section end in squares.
Subsequence of A047207.

List([0..50], n -> n*(n^3+2*n^2-5*n+10)/8);

[n*(n^3+2*n^2-5*n+10)/8: n in [0..50]];

a := n -> n*(n*(n*(n+2)-5)+10)/8: seq(a(n),n=0..41); # Peter Luschny, Nov 06 2017

Table[n (n^3 + 2 n^2 - 5 n + 10)/8, {n, 0, 50}]
LinearRecurrence[{5,-10,10,-5,1},{0,1,4,15,43},50] (* Harvey P. Dale, Jan 08 2024 *)

makelist(n*(n^3+2*n^2-5*n+10)/8, n, 0, 50);

vector(50, n, n--; n*(n^3+2*n^2-5*n+10)/8)

[n*(n^3+2*n^2-5*n+10)/8 for n in range(50)]

O.g.f.: x*(1 - x + 5*x^2 - 2*x^3)/(1 - x)^5.
E.g.f.: x*(8 + 8*x + 8*x^2 + x^3)*exp(x)/8.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>4.
a(n) = 2*n + Sum_{i=0..n} i*(i^2 - 3)/2.

A211381 Number of pairs of intersecting diagonals in the exterior of a regular n-gon.

Original entry on oeis.org

0, 0, 0, 0, 7, 24, 63, 130, 242, 408, 650, 980, 1425, 2000, 2737, 3654, 4788, 6160, 7812, 9768, 12075, 14760, 17875, 21450, 25542, 30184, 35438, 41340, 47957, 55328, 63525, 72590, 82600, 93600, 105672, 118864, 133263, 148920, 165927, 184338, 204250, 225720

Offset: 3

Martin Renner, Feb 07 2013

Eric Weisstein's World of Mathematics, Regular Polygon Division by Diagonals.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Cf. A000332, A117662, A176145, A211380.

a:= n-> `if`(n mod 2 = 0, 1/24*n*(n-4)*(n-6)*(2*n-7), 1/24*n*(n-3)*(n-5)*(2*n-11)): seq (a(n), n=3..40);

a(n) = 1/24*n*(n-4)*(n-6)*(2*n-7) for n even.
a(n) = 1/24*n*(n-3)*(n-5)*(2*n-11) for n odd.
a(n) = A211380(n) - A000332(n).
G.f.: x^7*(2*x^2-3*x-7) / ((x-1)^5*(x+1)^2). [Colin Barker, Feb 14 2013]

A257925 a(n) = (n^2 - n + 1)*(n^2 + n - 1).

Original entry on oeis.org

1, 15, 77, 247, 609, 1271, 2365, 4047, 6497, 9919, 14541, 20615, 28417, 38247, 50429, 65311, 83265, 104687, 129997, 159639, 194081, 233815, 279357, 331247, 390049, 456351, 530765, 613927, 706497, 809159, 922621

Offset: 1

Matthew Ryan, Apr 17 2016

Subsequence of a(m,n)=(m^2 + n).(n^2 + m)/(m - n)^3 with m=n-1. Q N4 of the 2012 International Mathematical Olympiad paper poses the problem of proving more than 500 solutions exist below 2012 for the equation: a(m,n).(m - n)^3=(m^2 + n).(n^2 + m). Such solutions a(m,n) were called 'Friendly'. If m=2k-1 and n=k-1, solutions of the form a=4k-3 for some integer k, satisfy this requirement although others do exist for other (m,n) pairs e.g. if (m,n)=(1,2), a(m,n)=15.
If m=n-2, a(n)=(n^2 - 3*n + 4)*(n^2 + n - 2)/8. This is the sequence A176145 [t*(t-3)*(t^2-7*t+14)/8] with t=n+2.
Satisfies a linear recurrence having signature (5, -10, 10, -5, 1). - Harvey P. Dale, Apr 18 2019

			For n=1, a(1) = 1;
For n=2, a(2) = 15;
For n=3, a(3) = 77.

International Mathematical Olympiad 2012, Number Theory Question 4

Cf. A002061, A028387, A176145.

Table[(n^2-n+1)(n^2+n-1),{n,40}] (* Harvey P. Dale, Apr 18 2019 *)

a(n) = (n^2 - n + 1)*(n^2 + n - 1); \\ Michel Marcus, Apr 17 2016

a(n) = (n^2 - n + 1)*(n^2 + n - 1).

a(n) = A002061(n)*A028387(n-1). - Michel Marcus, Apr 17 2016

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A117662 a(n) = n*(n-1)*(n-2)*(n+3)/12.

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A211380 Number of pairs of intersecting diagonals in the interior and exterior of a regular n-gon.

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A294259 a(n) = n*(n^3 + 2*n^2 - 5*n + 10)/8.

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A211381 Number of pairs of intersecting diagonals in the exterior of a regular n-gon.

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A257925 a(n) = (n^2 - n + 1)*(n^2 + n - 1).

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A117662 a(n) = n(n-1)(n-2)*(n+3)/12.

A294259 a(n) = n(n^3 + 2n^2 - 5*n + 10)/8.