A134238 -id:A134238 - cp's OEIS frontend

A203552 a(n) = n(5n^2 - 3*n + 4) / 6.

0, 1, 6, 20, 48, 95, 166, 266, 400, 573, 790, 1056, 1376, 1755, 2198, 2710, 3296, 3961, 4710, 5548, 6480, 7511, 8646, 9890, 11248, 12725, 14326, 16056, 17920, 19923, 22070, 24366, 26816, 29425, 32198, 35140, 38256, 41551, 45030, 48698

Offset: 0

Michael Somos, Jan 02 2012

			G.f. = x + 6*x^2 + 20*x^3 + 48*x^4 + 95*x^5 + 166*x^6 + 266*x^7 + 400*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Philip Boalch, Counting the fission trees and nonabelian Hodge graphs, arXiv:2410.23358 [math.AG], 2024. See p. 11.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000125, A134238, A203551.

I:=[0, 1, 6, 20]; [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..50]]; // Vincenzo Librandi, Jan 07 2012

LinearRecurrence[{4,-6,4,-1},{0,1,6,20},40] (* Vincenzo Librandi, Jan 07 2012 *)

PARI
```
{a(n) = n * (5*n^2 - 3*n + 4) / 6};
```

a(n) = Sum_{k = 1..n} A(k-1, n-k) where A(i, j) = i^2 + i*j + j^2 + i + j + 1.
G.f.: x * (1 + 2*x + 2*x^2) / (1 - x)^4.
a( n) = -A203551(-n) for all n in Z.
a(n)-a(n-1) = A134238(n). - Bruno Berselli, Jan 03 2012
a(n) = 4*A000125(n) + 2*A000125(n+1) - A000125(n+3). - Ivan N. Ianakiev, Aug 21 2013
E.g.f.: x*(5*x^2 + 12*x + 6)*exp(x)/6. - G. C. Greubel, Aug 12 2018

A134237 Triangle read by rows, a(1) = 1, n-th row n terms of: (2n-1, 2n, 2n+1, ..., followed by n).

Original entry on oeis.org

1, 3, 2, 5, 6, 3, 7, 8, 9, 4, 9, 10, 11, 12, 5, 11, 12, 13, 14, 15, 6, 13, 14, 15, 16, 17, 18, 7, 15, 16, 17, 18, 19, 20, 21, 8, 17, 18, 19, 20, 21, 22, 23, 24, 9, 19, 20, 21, 22, 23, 24, 25, 26, 27, 10, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 11, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 12

Offset: 1

Gary W. Adamson, Oct 14 2007

Row sums = A134238: (1, 5, 14, 28, 47, 71, 100, 134, ...).

			First few rows of the triangle:
   1;
   3,  2;
   5,  6,  3;
   7,  8,  9,  4;
   9, 10, 11, 12,  5;
  11, 12, 13, 14, 15,  6;
  13, 14, 15, 16, 17, 18,  7;
  ...

Cf. A134238.

T(n, k) = if (k < n, (2*n-1) + (k-1), n);
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 13 2017

Corrected and extended by Michel Marcus, Jul 13 2017

A364352 a(n) is the number of regions into which the plane is divided by n lines parallel to each edge of an equilateral triangle with side n such that the lines extend the parallel edge and divide the other edges into unit segments.

Original entry on oeis.org

7, 16, 30, 49, 73, 102, 136, 175, 219, 268, 322, 381, 445, 514, 588, 667, 751, 840, 934, 1033, 1137, 1246, 1360, 1479, 1603, 1732, 1866, 2005, 2149, 2298, 2452, 2611, 2775, 2944, 3118, 3297, 3481, 3670, 3864, 4063, 4267, 4476, 4690, 4909, 5133, 5362, 5596, 5835, 6079, 6328

Offset: 1

Nicolay Avilov, Jul 20 2023

Detailed instructions for drawing the lines. Along the edges of an equilateral triangle with side n, points are marked that divide the edges into unit segments. Draw all infinite straight lines that connect those points and are parallel to the edges of the triangle. For n = 1..5, the link shows the construction of these lines.

			a(1) = 1 + 3 + 3 = 7;
a(2) = 2^2 + 3*3 + 3 = 16;
a(5) = 5^2 + 3*9 + 3*6 + 3 = 73.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Nicolay Avilov, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A134238, A147875, A177862, A343755, A364401.

LinearRecurrence[{3,-3,1},{7,16,30},100] (* Paolo Xausa, Oct 16 2023 *)

a(n) = n*(5*n + 3)/2 + 3;
a(n) = A147875(n) + 3 = A134238(n+1) + 2.
From Stefano Spezia, Nov 23 2023: (Start)
O.g.f.: x*(7 - 5*x + 3*x^2)/(1 - x)^3.
E.g.f.: exp(x)*(3 + 4*x + 5*x^2/2) - 3. (End)

Edited by Peter Munn, Sep 02 2023

cp's OEIS Frontend

A203552 a(n) = n(5n^2 - 3*n + 4) / 6.

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A134237 Triangle read by rows, a(1) = 1, n-th row n terms of: (2n-1, 2n, 2n+1, ..., followed by n).

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A364352 a(n) is the number of regions into which the plane is divided by n lines parallel to each edge of an equilateral triangle with side n such that the lines extend the parallel edge and divide the other edges into unit segments.

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cp's OEIS Frontend

A203552 a(n) = n*(5*n^2 - 3*n + 4) / 6.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A134237 Triangle read by rows, a(1) = 1, n-th row n terms of: (2n-1, 2n, 2n+1, ..., followed by n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions

A364352 a(n) is the number of regions into which the plane is divided by n lines parallel to each edge of an equilateral triangle with side n such that the lines extend the parallel edge and divide the other edges into unit segments.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A203552 a(n) = n(5n^2 - 3*n + 4) / 6.