A031940 -id:A031940 - cp's OEIS frontend

A204556 Left edge of the triangle A045975.

1, 2, 9, 24, 45, 90, 133, 224, 297, 450, 561, 792, 949, 1274, 1485, 1920, 2193, 2754, 3097, 3800, 4221, 5082, 5589, 6624, 7225, 8450, 9153, 10584, 11397, 13050, 13981, 15872, 16929, 19074, 20265, 22680, 24013, 26714, 28197, 31200, 32841, 36162, 37969, 41624

Offset: 1

Reinhard Zumkeller, Jan 18 2012

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Haskell
```
a204556 = head . a045975_row
```

[n*(2*n^2-3*n+(-1)^n*(n-3)+3)/4: n in [1..50]]; // G. C. Greubel, Jun 15 2018

Table[n*(2*n^2 - 3*n + (-1)^n*(n - 3) + 3)/4, {n, 1, 50}] (* G. C. Greubel, Jun 15 2018 *)

Vec(x*(1+x+4*x^2+12*x^3+3*x^4+3*x^5)/((1+x)^3*(x-1)^4) + O(x^99)) \\ Charles R Greathouse IV, Jun 12 2015

for(n=1, 50, print1(n*(2*n^2-3*n+(-1)^n*(n-3)+3)/4, ", ")) \\ G. C. Greubel, Jun 15 2018

a(n) = A045975(n,1);
a(n) = A031940(n-1) * n for n > 1;
a(n) = A204557(n) - A045895(n).
G.f.: x*(1+x+4*x^2+12*x^3+3*x^4+3*x^5) / ((1+x)^3*(x-1)^4). - R. J. Mathar, Aug 13 2012
From Colin Barker, Jan 28 2016: (Start)
a(n) = n*(2*n^2-3*n+(-1)^n*(n-3)+3)/4.
a(n) = (n^3-n^2)/2 for n even.
a(n) = (n^3-2*n^2+3*n)/2 for n odd.
(End)

A031878 Maximal number of edges in Hamiltonian path in complete graph on n nodes.

Original entry on oeis.org

0, 1, 3, 5, 10, 13, 21, 25, 36, 41, 55, 61, 78, 85, 105, 113, 136, 145, 171, 181, 210, 221, 253, 265, 300, 313, 351, 365, 406, 421, 465, 481, 528, 545, 595, 613, 666, 685, 741, 761, 820, 841, 903, 925, 990, 1013, 1081, 1105, 1176, 1201, 1275, 1301, 1378

Offset: 1

Colin Mallows

Given a regular polygon with n sides, a(n) is the number of circles that have an edge of the polygon as a diameter (5 for n=4, 10 for n=5, 13 for n=6, ...). - Ahmet Arduç, Jan 28 2017

Quasipolynomial of order 2. [Charles R Greathouse IV, Dec 07 2011]

			E.g. for n=4 [1:2][2:3][3:1][1:4][4:2], so a(4) = 5.

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A031940.

LinearRecurrence[{1,2,-2,-1,1},{0,1,3,5,10},60] (* Harvey P. Dale, Mar 14 2015 *)
CoefficientList[ Series[-x (x^3 + 2x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 52}], x]  (* Robert G. Wilson v, Jul 30 2018 *)

a(n)=if(n%2,n^2-n,n^2-2*n+2)/2  \\ Charles R Greathouse IV, Dec 07 2011

a(n) = C(n, 2) if n odd, a(n) = C(n, 2)-n/2+1 if n even.
G.f.: x^2*(1+2*x+x^3)/((1-x)*(1-x^2)).
a(n) = ( n*n +n -(n-1)*(n mod 2) )/2. [Frank Ellermann]

A299807 Rectangular array read by antidiagonals: T(n,k) is the number of distinct sums of k complex n-th roots of 1, n >= 1, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 9, 10, 5, 1, 1, 6, 15, 16, 15, 6, 1, 1, 7, 19, 35, 25, 21, 7, 1, 1, 8, 28, 37, 70, 36, 28, 8, 1, 1, 9, 33, 84, 61, 126, 49, 36, 9, 1, 1, 10, 45, 96, 210, 91, 210, 64, 45, 10, 1, 1, 11, 51, 163, 225, 462, 127, 330, 81, 55, 11, 1, 1, 12, 66, 180, 477, 456, 924, 169, 495, 100, 66

Offset: 1

Max Alekseyev, Feb 24 2018

			Array starts:
  n=1:  1,  1,  1,   1,   1,    1,    1,    1,     1,     1,     1, ...
  n=2:  1,  2,  3,   4,   5,    6,    7,    8,     9,    10,    11, ...
  n=3:  1,  3,  6,  10,  15,   21,   28,   36,    45,    55,    66, ...
  n=4:  1,  4,  9,  16,  25,   36,   49,   64,    81,   100,   121, ...
  n=5:  1,  5, 15,  35,  70,  126,  210,  330,   495,   715,  1001, ...
  n=6:  1,  6, 19,  37,  61,   91,  127,  169,   217,   271,   331, ...
  n=7:  1,  7, 28,  84, 210,  462,  924, 1716,  3003,  5005,  8008, ...
  n=8:  1,  8, 33,  96, 225,  456,  833, 1408,  2241,  3400,  4961, ...
  n=9:  1,  9, 45, 163, 477, 1197, 2674, 5454, 10341, 18469, 31383, ...
  n=10: 1, 10, 51, 180, 501, 1131, 2221, 3951,  6531, 10201, 15231, ...
  ...

Max Alekseyev, Table of n, a(n) for n = 1..351

Rows: A000012 (n=1), A000027 (n=2), A000217 (n=3), A000290 (n=4), A000332 (n=5), A354343 (n=6), A000579 (n=7), A014820 (n=8).
Columns: A000012 (k=0), A000027 (k=1), A031940 (k=2).
Diagonal: A299754 (n=k).
Cf. A103314, A107754, A107861, A108380, A107848, A107753, A108381, A143008.

From Chai Wah Wu, May 28 2018: (Start)
The following are all conjectures.
For m >= 0, the 2^(m+1)-th row are the figurate numbers based on the 2^m-dimensional regular convex polytope with g.f.: (1+x)^(2^m-1)/(1-x)^(2^m+1).
For p prime, the n=p row corresponds to binomial(k+p-1,p-1) for k = 0,1,2,3,..., which is the maximum possible (i.e., the number of combinations with repetitions of k choices from p categories) with g.f.: 1/(1-x)^p.
(End)

Row 6 corrected by Max Alekseyev, Aug 14 2022

cp's OEIS Frontend

A204556 Left edge of the triangle A045975.

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A031878 Maximal number of edges in Hamiltonian path in complete graph on n nodes.

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A299807 Rectangular array read by antidiagonals: T(n,k) is the number of distinct sums of k complex n-th roots of 1, n >= 1, k >= 0.

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