A071244 -id:A071244 - cp's OEIS frontend

A227327 Number of non-equivalent ways to choose two points in an equilateral triangle grid of side n.

0, 1, 4, 10, 22, 41, 72, 116, 180, 265, 380, 526, 714, 945, 1232, 1576, 1992, 2481, 3060, 3730, 4510, 5401, 6424, 7580, 8892, 10361, 12012, 13846, 15890, 18145, 20640, 23376, 26384, 29665, 33252, 37146, 41382, 45961, 50920, 56260, 62020, 68201, 74844

Offset: 1

Heinrich Ludwig, Jul 07 2013

The sequence is an alternating composition of A178073 and A071244: a(n) = 2*A071244((n+1)/2) if n is odd, otherwise a(n) = A178073(n/2).

			for n = 3 there are the following 4 choices of 2 points (X) (rotations and reflections being ignored):
     X         X         X         .
    X .       . .       . .       X X
   . . .     X . .     . X .     . . .

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Corresponding questions about the number of ways in a square grid are treated by A083374 (2 points) and A178208 (3 points).

Cf. A178073, A071244.

Table[b = n^4 + 2*n^3 + 8*n^2; If[EvenQ[n], c = b - 8*n, c = b - 2*n - 9]; c/48, {n, 43}] (* T. D. Noe, Jul 09 2013 *)
CoefficientList[Series[-x (x^3 - x^2 + x + 1) / ((x - 1)^5  (x + 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 02 2013 *)
LinearRecurrence[{3,-1,-5,5,1,-3,1},{0,1,4,10,22,41,72},50] (* Harvey P. Dale, May 11 2019 *)

a(n) = (n^4 + 2*n^3 + 8*n^2 - 8*n    )/48; if n even.
a(n) = (n^4 + 2*n^3 + 8*n^2 - 2*n - 9)/48; if n odd.
G.f.: -x^2*(x^3-x^2+x+1) / ((x-1)^5*(x+1)^2). - Colin Barker, Jul 12 2013

A071245 a(n) = n(n-1)(2*n^2 + 1)/6.

Original entry on oeis.org

0, 0, 3, 19, 66, 170, 365, 693, 1204, 1956, 3015, 4455, 6358, 8814, 11921, 15785, 20520, 26248, 33099, 41211, 50730, 61810, 74613, 89309, 106076, 125100, 146575, 170703, 197694, 227766, 261145, 298065, 338768, 383504, 432531, 486115, 544530, 608058

Offset: 0

N. J. A. Sloane, Jun 12 2002

The first differences are given in A277228. - J. M. Bergot, Sep 14 2016

T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A071238, A071244, A277228 (first differences).

[n*(n-1)*(2*n^2+1)/6: n in [0..40]]; // Vincenzo Librandi, Jun 14 2011

Table[n (n - 1) (2 n^2 + 1)/6, {n, 0, 37}] (* or *)
CoefficientList[Series[(-3 x^2 - 4 x^3 - x^4)/(-1 + x)^5, {x, 0, 37}], x] (* Michael De Vlieger, Sep 14 2016 *)

a(n)=n*(n-1)*(2*n^2+1)/6; \\ Joerg Arndt, Sep 04 2013

def A071245(n): return binomial(n,2)*(2*n^2+1)//3
[A071245(n) for n in range(41)] # G. C. Greubel, Aug 07 2024

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(0)=0, a(1)=0, a(2)=3, a(3)=19, a(4)=66. - Yosu Yurramendi, Sep 03 2013
G.f.: x^2*(3 + 4*x + x^2)/(1-x)^5. - Michael De Vlieger, Sep 14 2016
E.g.f.: (1/6)*x^2*(9 + 10*x + 2*x^2)*exp(x). - G. C. Greubel, Sep 23 2016

A071246 a(n) = n(n - 1)(2*n^2 + n + 2)/6.

Original entry on oeis.org

0, 0, 4, 23, 76, 190, 400, 749, 1288, 2076, 3180, 4675, 6644, 9178, 12376, 16345, 21200, 27064, 34068, 42351, 52060, 63350, 76384, 91333, 108376, 127700, 149500, 173979, 201348, 231826, 265640, 303025, 344224, 389488, 439076, 493255, 552300, 616494

Offset: 0

N. J. A. Sloane, Jun 12 2002

T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A071239, A071244, A071245.

[n*(n-1)*(2*n^2+n+2)/6: n in [0..40]]; // Vincenzo Librandi, Jun 14 2011

CoefficientList[Series[-(4x^2 + 3x^3 + x^4)/(x - 1)^5, {x, 0, 50}], x] (* or *) Table[n*(n - 1)*(2*n^2 + n + 2)/6, {n, 0, 50}] (* Indranil Ghosh, Apr 05 2017 *)

a(n) = n*(n - 1)*(2*n^2 + n + 2)/6; \\ Indranil Ghosh, Apr 05 2017

def a(n): return n*(n - 1)*(2*n**2 + n + 2)/6 # Indranil Ghosh, Apr 05 2017

def A071246(n): return binomial(n,2)*(2+n+2*n^2)//3
[A071246(n) for n in range(41)] # G. C. Greubel, Aug 07 2024

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), n > 4, a(0)=0, a(1)=0, a(2)=4, a(3)=23, a(4)=76. - Yosu Yurramendi, Sep 03 2013
From Indranil Ghosh, Apr 05 2017: (Start)
G.f.: x^2*(4 + 3*x + x^2)/(1 - x)^5.
E.g.f.: exp(x)*x^2*(4+x)*(3+2*x)/6. (End)

A071252 a(n) = n(n - 1)(n^2 + 1)/2.

Original entry on oeis.org

0, 0, 5, 30, 102, 260, 555, 1050, 1820, 2952, 4545, 6710, 9570, 13260, 17927, 23730, 30840, 39440, 49725, 61902, 76190, 92820, 112035, 134090, 159252, 187800, 220025, 256230, 296730, 341852, 391935, 447330, 508400, 575520, 649077, 729470, 817110, 912420

Offset: 0

N. J. A. Sloane, Jun 12 2002

T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A002378, A002522, A071239, A071244, A071245, A071246.

[n*(n-1)*(n^2+1)/2: n in [0..40]]; // Vincenzo Librandi, Jun 14 2011

f[n_] := n (n - 1) (n^2 + 1)/2 (* Or *) f[n_] := Floor[n^5/(n + 1)]/2; Array[f, 38, 0] (* Robert G. Wilson v, Apr 01 2012 *)

a(n)=n*(n-1)*(n^2+1)/2; \\ Joerg Arndt, Sep 04 2013

def a(n): return  n*(n - 1)*(n**2 + 1)/2 # Indranil Ghosh, Apr 05 2017

def A071252(n): return binomial(n,2)*(1+n^2)
[A071252(n) for n in range(41)] # G. C. Greubel, Aug 07 2024

a(n) = floor(n^5/(n+1))/2. - Gary Detlefs, Mar 31 2011
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) n>4, a(0)=0, a(1)=0, a(2)=5, a(3)=30, a(4)=102. - Yosu Yurramendi, Sep 03 2013
G.f.:  x^2*(5+5*x+2*x^2)/(1-x)^5. - Joerg Arndt, Sep 04 2013
From Indranil Ghosh, Apr 05 2017: (Start)
a(n) = A002378(n) * A002522(n) / 2.
E.g.f.: exp(x)*x^2*(5 + 5*x + x^2)/2.
(End)

cp's OEIS Frontend

A227327 Number of non-equivalent ways to choose two points in an equilateral triangle grid of side n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A071245 a(n) = n*(n-1)*(2*n^2 + 1)/6.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A071246 a(n) = n*(n - 1)*(2*n^2 + n + 2)/6.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

SageMath

Formula

A071252 a(n) = n*(n - 1)*(n^2 + 1)/2.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

SageMath

Formula

A071245 a(n) = n(n-1)(2*n^2 + 1)/6.

A071246 a(n) = n(n - 1)(2*n^2 + n + 2)/6.

A071252 a(n) = n(n - 1)(n^2 + 1)/2.