A051895 -id:A051895 - cp's OEIS frontend

A045947 Triangles in open triangular matchstick arrangement (triangle minus one side) of side n.

0, 0, 2, 7, 17, 33, 57, 90, 134, 190, 260, 345, 447, 567, 707, 868, 1052, 1260, 1494, 1755, 2045, 2365, 2717, 3102, 3522, 3978, 4472, 5005, 5579, 6195, 6855, 7560, 8312, 9112, 9962, 10863, 11817, 12825, 13889, 15010, 16190, 17430, 18732, 20097, 21527

Offset: 0

R. K. Guy

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

First differences of A082289.

Cf. A000447, A002412, A051895.

[Floor((4*n^3+2*n^2-4*n)/16): n in [0..50]]; // Vincenzo Librandi, Aug 29 2011

LinearRecurrence[{3, -2, -2, 3, -1}, {0, 0, 2, 7, 17}, 45] (* Jean-François Alcover, Dec 12 2016 *)
CoefficientList[Series[(2x^2+x^3)/((1-x)^3(1-x^2)),{x,0,50}],x] (* Harvey P. Dale, Jun 26 2021 *)

PARI
```
a(n)=(4*n^3+2*n^2-4*n)\16
```

G.f.: (2*x^2+x^3)/((1-x)^3*(1-x^2)). - Michael Somos
a(n) = (1/16)*(2*n*(2*n^2+n-2)+(-1)^n-1). - Bruno Berselli, Aug 29 2011
a(2*n) = A000447(n)+A002412(n); a(2*n+1) = A051895(n). - J. M. Bergot, Apr 12 2018
E.g.f.: (x*(1 + 7*x + 2*x^2)*cosh(x) - (1 - x - 7*x^2 - 2*x^3)*sinh(x))/8. - Stefano Spezia, Aug 22 2023

A135713 a(n) = n(n+1)(4*n+1)/2.

Original entry on oeis.org

0, 5, 27, 78, 170, 315, 525, 812, 1188, 1665, 2255, 2970, 3822, 4823, 5985, 7320, 8840, 10557, 12483, 14630, 17010, 19635, 22517, 25668, 29100, 32825, 36855, 41202, 45878, 50895, 56265, 62000, 68112, 74613, 81515, 88830, 96570, 104747, 113373, 122460, 132020

Offset: 0

N. J. A. Sloane, Mar 05 2008

This sequence is related to A045944 by a(n) = n*A045944(n)-Sum_{i=0..n-1} A045944(i); this is the case d=6 in the identity n^2*(d*n+d-2)/2 - sum(k*(d*k+d-2)/2, k=0..n-1) = n*(n+1)*(2*d*n+d-3)/6 . - Bruno Berselli, Nov 19 2010

Bisection (even part) of A002717. See the Conway and Guy reference. - Wolfdieter Lang, Apr 16 2020

J. H. Conway and R. K. Guy, The Book of Numbers, p. 83.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
M. E. Larsen, The eternal triangle - a history of a counting problem, College Math. J., 20 (1989), 370-392.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Bisection of A002717 (even part).

Cf. A045944, A160378; A132124, A006331.

[n*(n+1)*(4*n+1)/2: n in [0..40]];  // Bruno Berselli, Aug 23 2011

LinearRecurrence[{4,-6,4,-1}, {0, 5, 27, 78}, 50] (* Vincenzo Librandi, Mar 01 2012 *)
Table[n*(n+1)*(4*n+1)/2,{n,0,25}] (* G. C. Greubel, Oct 29 2016 *)
Table[PolygonalNumber[n](4n+1),{n,0,40}] (* Harvey P. Dale, Apr 26 2025 *)

O.g.f.: x*(7*x+5)/(x-1)^4. - R. J. Mathar, Apr 22 2008.
a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4) for n>3. - Bruno Berselli, Nov 19 2010
a(-n-1) = -A051895(n).  - Bruno Berselli, Aug 23 2011
E.g.f.: (1/2)*x*(10 + 17*x + 4*x^2)*exp(x). - G. C. Greubel, Oct 29 2016
Sum_{n>=1} 1/a(n) = 2*(5 - 2*Pi/3 - 4*log(2)) = 0.26603235073404654... - Ilya Gutkovskiy, Oct 29 2016

A192796 Molecular topological indices of the crown graphs.

Original entry on oeis.org

0, 28, 132, 360, 760, 1380, 2268, 3472, 5040, 7020, 9460, 12408, 15912, 20020, 24780, 30240, 36448, 43452, 51300, 60040, 69720, 80388, 92092, 104880, 118800, 133900, 150228, 167832, 186760, 207060

Offset: 1

Eric W. Weisstein, Jul 10 2011

Crown graphs are defined for n>=3; extended to n=1 using the closed form.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Crown Graph
Eric Weisstein's World of Mathematics, Molecular Topological Index
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

I:=[0, 28, 132, 360]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 04 2012

CoefficientList[Series[4*x*(5*x+7)/(1-x)^4,{x,0,40}],x] (* Vincenzo Librandi, Jul 04 2012 *)
LinearRecurrence[{4,-6,4,-1},{0,28,132,360},30] (* Harvey P. Dale, Sep 04 2024 *)

a(n) = 2*n*(n-1)*(4*n-1) \\ Charles R Greathouse IV, Jul 10 2011

a(n) = 2*(n-1)*n*(4*n-1).
a(n) = 4*A051895(n).
G.f.: 4*x^2*(5*x+7)/(1-x)^4. - Vincenzo Librandi, Jul 04 2012
a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - Vincenzo Librandi, Jul 04 2012

A096200 a(n) = n(n-1)(n-2)(3n-2)/6.

Original entry on oeis.org

0, 0, 0, 7, 40, 130, 320, 665, 1232, 2100, 3360, 5115, 7480, 10582, 14560, 19565, 25760, 33320, 42432, 53295, 66120, 81130, 98560, 118657, 141680, 167900, 197600, 231075, 268632, 310590, 357280, 409045, 466240, 529232, 598400, 674135, 756840, 846930, 944832

Offset: 0

N. J. A. Sloane, Jul 27 2004

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A051895.

Table[n*(n-1)*(n-2)*(3*n-2)/6,{n,0,100}] (* Vladimir Joseph Stephan Orlovsky, Jun 03 2011 *)

From Stefano Spezia, Aug 31 2023: (Start)
G.f.: x^3*(7 + 5*x)/(1 - x)^5.
a(n+2) - a(n+1) = A051895(n). (End)

cp's OEIS Frontend

A045947 Triangles in open triangular matchstick arrangement (triangle minus one side) of side n.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A135713 a(n) = n*(n+1)*(4*n+1)/2.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A192796 Molecular topological indices of the crown graphs.

Views

Author

Keywords

Comments

Links

Programs

Magma

Mathematica

PARI

Formula

A096200 a(n) = n*(n-1)*(n-2)*(3*n-2)/6.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A135713 a(n) = n(n+1)(4*n+1)/2.

A096200 a(n) = n(n-1)(n-2)(3n-2)/6.