A080838 -id:A080838 - cp's OEIS frontend

A274228 Triangle read by rows: T(n,k) (n>=3, 0<=k<=n-3) = number of n-sequences of 0's and 1's with exactly one pair of adjacent 0's and exactly k pairs of adjacent 1's.

2, 3, 2, 4, 4, 2, 5, 8, 5, 2, 6, 12, 12, 6, 2, 7, 18, 21, 16, 7, 2, 8, 24, 36, 32, 20, 8, 2, 9, 32, 54, 60, 45, 24, 9, 2, 10, 40, 80, 100, 90, 60, 28, 10, 2, 11, 50, 110, 160, 165, 126, 77, 32, 11, 2, 12, 60, 150, 240, 280, 252, 168, 96, 36, 12, 2, 13, 72, 195, 350, 455, 448, 364, 216, 117, 40, 13, 2

Offset: 3

Jeremy Dover, Jun 14 2016

			n=3 => 100, 001 -> T(3,0) = 2.
n=4 => 0010, 0100, 1001 -> T(4,0) = 3; 0011, 1100 -> T(4,1) = 2.
Triangle starts:
2,
3, 2,
4, 4, 2,
5, 8, 5, 2,
6, 12, 12, 6, 2,
7, 18, 21, 16, 7, 2,
8, 24, 36, 32, 20, 8, 2,
9, 32, 54, 60, 45, 24, 9, 2,
10, 40, 80, 100, 90, 60, 28, 10, 2,
11, 50, 110, 160, 165, 126, 77, 32, 11, 2,
12, 60, 150, 240, 280, 252, 168, 96, 36, 12, 2,
13, 72, 195, 350, 455, 448, 364, 216, 117, 40, 13, 2,
...

Row sums give A001629.
Cf. A073044.
Columns of table:
  T(n,0)=A000027(n-1)
  T(n,1)=A007590(n-1)
  T(n,2)=A080838(n-1)
  T(n,3)=A032091(n)

Table[(k + 1) (Binomial[Floor[(n + k - 2)/2], k + 1] + Binomial[Floor[(n + k - 3)/2], k + 1]) + 2 Binomial[Floor[(n + k - 3)/2], k], {n, 3, 14}, {k, 0, n - 3}] // Flatten (* Michael De Vlieger, Jun 16 2016 *)

T(n,k) = (k+1)*(binomial((n+k-2)\2,k+1)+binomial((n+k-3)\2,k+1))+2*binomial((n+k-3)\2,k); \\ Michel Marcus, Jun 17 2016

T(n,k) = (k+1)*(binomial(floor((n+k-2)/2),k+1)+binomial(floor((n+k-3)/2),k+1))+2*binomial(floor((n+k-3)/2),k).
T(n,k) = (k+1)*A073044(n-2,k+1) + 2*A046854(n-3,k).
T(n,k) = A274742(n,k)+A274742(n-1,k)+A046854(n-3,k).

A060423 Number of obtuse triangles made from vertices of a regular n-gon.

Original entry on oeis.org

0, 0, 0, 0, 0, 5, 6, 21, 24, 54, 60, 110, 120, 195, 210, 315, 336, 476, 504, 684, 720, 945, 990, 1265, 1320, 1650, 1716, 2106, 2184, 2639, 2730, 3255, 3360, 3960, 4080, 4760, 4896, 5661, 5814, 6669, 6840, 7790, 7980, 9030, 9240, 10395, 10626

Offset: 0

Sen-Peng Eu, Apr 05 2001

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Léo Ducas, Kissing Number of Craig's Lattice and Spherical Decoding, Bachelor's seminar AGM Spring 2025, Leiden Univ. (Netherlands, 2024). See p. 2.
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A000330, A007290, A046092, A080838.

[n*(2*n-3-(-1)^n)*(2*n-7-(-1)^n)/32 : n in [0..60]]; // Wesley Ivan Hurt, Apr 14 2017

A060423:=n->n*(2*n-3-(-1)^n)*(2*n-7-(-1)^n)/32; seq(A060423(n), n=0..100); # Wesley Ivan Hurt, Dec 31 2013

Table[n(2n-3-(-1)^n)(2n-7-(-1)^n)/32, {n, 0, 100}] (* Wesley Ivan Hurt, Dec 31 2013 *)
Table[If[EvenQ[n],(n(n-2)(n-4))/8,(n(n-1)(n-3))/8],{n,0,50}] (* Harvey P. Dale, Sep 18 2018 *)
LinearRecurrence[{1, 3, -3, -3, 3, 1, -1}, {0, 0, 0, 0, 0, 5, 6}, 51] (* Mike Sheppard, Feb 17 2025 *)

a(n)=polcoeff(x^5*(5+x)/(1-x)/(1-x^2)^3+x*O(x^n),n)

a(n) = n*(n-1)*(n-3)/8 when n odd; n*(n-2)*(n-4)/8 when n even.
G.f.: x^5*(x+5)/((1-x)(1-x^2)^3). - Michael Somos, Jan 30 2004
For n odd, a(n) = A080838(n). - Gerald McGarvey, Sep 14 2008
a(n) = n*(2*n-3-(-1)^n)*(2*n-7-(-1)^n)/32. - Wesley Ivan Hurt, Dec 31 2013
E.g.f.: x*((x - 3)*x*cosh(x) + (x^2 - x + 3)*sinh(x))/8. - Stefano Spezia, May 28 2022
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7). - Mike Sheppard, Feb 17 2025

cp's OEIS Frontend

A274228 Triangle read by rows: T(n,k) (n>=3, 0<=k<=n-3) = number of n-sequences of 0's and 1's with exactly one pair of adjacent 0's and exactly k pairs of adjacent 1's.

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