A154560 -id:A154560 - cp's OEIS frontend

A058794 Row 3 of A007754.

2, 18, 52, 110, 198, 322, 488, 702, 970, 1298, 1692, 2158, 2702, 3330, 4048, 4862, 5778, 6802, 7940, 9198, 10582, 12098, 13752, 15550, 17498, 19602, 21868, 24302, 26910, 29698, 32672, 35838, 39202, 42770, 46548, 50542, 54758, 59202, 63880

Offset: 0

Christian G. Bower, Dec 02 2000

For n >= 2, a(n) is the number of ways a triangle with side length n-1 can be completely surrounded by diamonds. See illustrations in Links. - Craig Knecht, Oct 08 2024

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Craig Knecht, Number of ways a single triangle can be surrounded by diamonds.
Craig Knecht, The 110 ways the T2 triangle can be surrounded by diamonds.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A007754, A154560.

List([0..40], n -> n^3+6*n^2+9*n+2); # G. C. Greubel, Nov 29 2018

[n^3+6*n^2+9*n+2: n in [0..40]]; // Vincenzo Librandi, Sep 22 2016

seq(sum(n^2-3, k=1..n), n=2..40); # Zerinvary Lajos, Jan 28 2008
seq ((n^3)-3*n, n=2..40); # Zerinvary Lajos, Jun 17 2008

LinearRecurrence[{4,-6,4,-1}, {2,18,52,110}, 40] (* Vladimir Joseph Stephan Orlovsky, Jan 15 2009 *)
Table[n^3 + 6 n^2 + 9 n + 2, {n, 0, 40}] (* Bruno Berselli, Jan 10 2015 *)

vector(40, n, n--; n^3+6*n^2+9*n+2) \\ G. C. Greubel, Nov 29 2018

[(n^3+6*n^2+9*n+2) for n in range(40)] # G. C. Greubel, Nov 29 2018

a(n) = n^3 + 6*n^2 + 9*n + 2.
G.f.: 2*(1 + 5*x - 4*x^2 + x^3)/(1-x)^4. - Colin Barker, Jan 10 2012
a(n) = (n + 2)*(n^2 + 4*n + 1) = 2*A154560(n). - Bruno Berselli, Jan 10 2015
E.g.f.: (2 + 16*x + 9*x^2 + x^3)*exp(x). - G. C. Greubel, Nov 29 2018

A152811 a(n) = 2(n^2 + 2n - 2).

Original entry on oeis.org

2, 12, 26, 44, 66, 92, 122, 156, 194, 236, 282, 332, 386, 444, 506, 572, 642, 716, 794, 876, 962, 1052, 1146, 1244, 1346, 1452, 1562, 1676, 1794, 1916, 2042, 2172, 2306, 2444, 2586, 2732, 2882, 3036, 3194, 3356, 3522, 3692, 3866, 4044, 4226, 4412, 4602, 4796, 4994

Offset: 1

Vincenzo Librandi, Dec 17 2008

Positive numbers k such that 2*k + 12 is a square. [Comment simplified by Zak Seidov, Jan 14 2009]
Sequence gives positive x values of solutions (x,y) to the Diophantine equation 2*x^3 + 12*x^2 = y^2. Corresponding y values are 8*A154560. There are three other solutions: (0,0), (-4,8) and (-6,0).
From a(2) onwards, third subdiagonal of triangle defined in A144562.
Also, nonnegative numbers of the form (m+sqrt(-3))^2 + (m-sqrt(-3))^2. - Bruno Berselli, Mar 13 2015
a(n-1) is the maximum Zagreb index over all maximal 2-degenerate graphs with n vertices.  The extremal graphs are 2-stars, so the bound also applies to 2-trees.  (The Zagreb index of a graph is the sum of the squares of the degrees over all vertices of the graph.) - Allan Bickle, Apr 11 2024

			a(4) = 2*(4^2 + 2*4 - 2) = 44 = 2*22 = 2*A028872(5); 2*44^3 + 12*44^2 = 193600 = 440^2 is a square.
The graph K_3 has 3 degree 2 vertices, so a(3-1) = 3*4 = 12.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs, Vol. 0(1) (2024), Article 5.
Allan Bickle, Zagreb Indices of Maximal k-degenerate Graphs, Australas. J. Combin., Vol. 89(1) (2024), pp. 167-178.
J. Estes and B. Wei, Sharp bounds of the Zagreb indices of k-trees, J. Comb. Optim., Vol. 27 (2014), pp. 271-291.
I. Gutman and K. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem., No. 50 (2004), pp. 83-92.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A028872 (n^2-3), A154560 ((n+3)^2*n/2+1), A144562 (triangle T(m,n) = 2m*n+m+n-1).

Cf. A002378, A152811, A371912 (Zagreb indices of maximal k-degenerate graphs).

Magma
```
[ 2*(n^2+2*n-2) : n in [1..47] ];
```

Table[2*n*(n + 2) - 4, {n, 50}] (* Paolo Xausa, Aug 08 2024 *)

{m=4700; for(n=1, m, if(issquare(2*n^3+12*n^2), print1(n, ",")))}

G.f.: 2*x*(1 + 3*x - 2*x^2)/(1-x)^3. [corrected by Elmo R. Oliveira, Nov 17 2024]
a(n) = 2*A028872(n+1).
a(n) = a(n-1) + 4*n + 2 for n > 1, a(1)=2.
From Amiram Eldar, Mar 02 2023: (Start)
Sum_{n>=1} 1/a(n) = 1/3 - cot(sqrt(3)*Pi)*Pi/(4*sqrt(3)).
Sum_{n>=1} (-1)^(n+1)/a(n) = -(2 + sqrt(3)*Pi*cosec(sqrt(3)*Pi))/12. (End)
From Elmo R. Oliveira, Nov 17 2024: (Start)
E.g.f.: 2*(exp(x)*(x^2 + 3*x - 2) + 2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

Edited and extended by Klaus Brockhaus, Jan 12 2009

A177944 Array T(n,m) = 1/Beta(n+1, m+1) - n - m read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 16, 26, 16, 1, 1, 25, 55, 55, 25, 1, 1, 36, 99, 134, 99, 36, 1, 1, 49, 161, 273, 273, 161, 49, 1, 1, 64, 244, 496, 622, 496, 244, 64, 1, 1, 81, 351, 831, 1251, 1251, 831, 351, 81, 1

Offset: 0

Roger L. Bagula, May 15 2010

Antidiagonal sums are 1, 2, 6, 20, 60, 162, 406, 968, 2232, 5030, ... = (d+1)*(2^d-d).

The values appear to be related to binomial(n,m)^2.

			The array starts in row n=0, column m=0 as:
1,....1,....1,....1,.....1,.....1,.....1,.....1,
1,....4,....9,...16,....25,....36,....49,....64, A000290
1,....9,...26,...55,....99,...161,...244,...351, A154560
1,...16,...55,..134,...273,...496,...831,..1310,
1,...25,...99,..273,...622,..1251,..2300,..3949,
1,...36,..161,..496,..1251,..2762,..5533,.10284,
1,...49,..244,..831,..2300,..5533,.12000,.24011,
1,...64,..351,.1310,..3949,.10284,.24011,.51466,

Clear[t, n];
t[n_, m_] = 1/Beta[n + 1, m + 1] - n - m;
a = Table[Table[t[n, m], {m, 0, 10}], {n, 0, 10}];
Table[Table[a[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}];
Flatten[%]

T(n,m) = Gamma(n+m+2)/(Gamma(n+1)*Gamma(m+1)) - n - m = T(m,n).

Examples written in natural order, closed formula for antidiag. sum - The Assoc. Eds. of the OEIS, Nov 03 2010

cp's OEIS Frontend

A058794 Row 3 of A007754.

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A152811 a(n) = 2(n^2 + 2n - 2).

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A177944 Array T(n,m) = 1/Beta(n+1, m+1) - n - m read by antidiagonals.

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cp's OEIS Frontend

A058794 Row 3 of A007754.

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Author

Keywords

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Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A152811 a(n) = 2*(n^2 + 2*n - 2).

Views

Author

Keywords

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Programs

Magma

Mathematica

PARI

Formula

Extensions

A177944 Array T(n,m) = 1/Beta(n+1, m+1) - n - m read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Programs

Mathematica

Formula

Extensions

A152811 a(n) = 2(n^2 + 2n - 2).