A092353 -id:A092353 - cp's OEIS frontend

A262612 Triangle read by rows T(n,k) in which column k lists the partial sums of the k-th column of triangle A236104.

1, 5, 14, 1, 30, 2, 55, 6, 91, 10, 1, 140, 19, 2, 204, 28, 3, 285, 44, 7, 385, 60, 11, 1, 506, 85, 15, 2, 650, 110, 24, 3, 819, 146, 33, 4, 1015, 182, 42, 8, 1240, 231, 58, 12, 1, 1496, 280, 74, 16, 2, 1785, 344, 90, 20, 3, 2109, 408, 115, 29, 4, 2470, 489, 140, 38, 5, 2870, 570, 165, 47, 9, 3311, 670, 201, 56, 13, 1

Offset: 1

Omar E. Pol, Nov 03 2015

Alternating sum of row n equals A175254(n), i.e., Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = A175254(n), which is also the volume (or the total number of units cubes) in the first n levels of the stepped pyramid described in A245092.

Row n has length A003056(n) hence the first element of column k is in row A000217(k).

			Triangle begins:
     1;
     5;
    14,    1;
    30,    2;
    55,    6;
    91,   10,    1;
   140,   19,    2;
   204,   28,    3;
   285,   44,    7;
   385,   60,   11,    1;
   506,   85,   15,    2;
   650,  110,   24,    3;
   819,  146,   33,    4;
  1015,  182,   42,    8;
  1240,  231,   58,   12,    1;
  1496,  280,   74,   16,    2;
  1785,  344,   90,   20,    3;
  2109,  408,  115,   29,    4;
  2470,  489,  140,   38,    5;
  2870,  570,  165,   47,    9;
  3311,  670,  201,   56,   13,    1;
  3795,  770,  237,   72,   17,    2;
  4324,  891,  273,   88,   21,    3;
  4900, 1012,  322,  104,   25,    4;
  ...
For n = 6 we have that A175254(6) = [1] + [1 + 3] + [1 + 3 + 4] + [1 + 3 + 4 + 7] + [1 + 3 + 4 + 7 + 6] + [1 + 3 + 4 + 7 + 6 + 12] = 1 + 4 + 8 + 15 + 21 + 33 = 82. On the other hand the alternating sum of the 6th row of the triangle is 91 - 10 + 1 = 82, equaling A175254(6).

Cf. A000203, A000217, A003056, A024916, A175254, A196020, A235791, A236104, A237048, A237591, A237593, A237270, A237271, A245092, A261699, A262626.

Column 1 gives A000330, n >= 1. Column 2 is A005993. It appears that column 3 is A092353.

A287195 Independence and clique covering number of the n-triangular honeycomb acute knight graph.

Original entry on oeis.org

1, 3, 3, 5, 9, 9, 12, 18, 18, 22, 30, 30, 35, 45, 45, 51, 63, 63, 70, 84, 84, 92, 108, 108, 117, 135, 135, 145, 165, 165, 176, 198, 198, 210, 234, 234, 247, 273, 273, 287, 315, 315, 330, 360, 360, 376, 408, 408, 425, 459, 459, 477, 513, 513, 532, 570, 570

Offset: 1

Eric W. Weisstein, May 21 2017

a(n) is also the length of row n in A244500.

Colin Barker, Table of n, a(n) for n = 1..1000
Stan Wagon, Graph Theory Problems from Hexagonal and Traditional Chess, The College Mathematics Journal, Vol. 45, No. 4, September 2014, pp. 278-287.
Eric Weisstein's World of Mathematics, Clique Covering Number.
Eric Weisstein's World of Mathematics, Independence Number.
Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Cf. A244500.

Cf. A066377, A092353.

LinearRecurrence[{1, 0, 2, -2, 0, -1, 1}, {1, 3, 3, 5, 9, 9, 12}, 50]
Table[1/18 ((n + 3) (3 n + 2) - 2 (n + 3) Cos[2 n Pi/3] - 2 Sqrt[3] (n + 1) Sin[2 n Pi/3]), {n, 50}]
Table[Piecewise[{{n (n + 3), Mod[n, 3] == 0}, {(n + 1) (n + 2), Mod[n, 3] == 1}, {(n + 1) (n + 4), Mod[n, 3] == 2}}]/6, {n, 50}]

Vec(x*(1 + 2*x) / ((1 - x)^3*(1 + x + x^2)^2) + O(x^60)) \\ Colin Barker, Jul 15 2017

From Colin Barker, Jul 15 2017: (Start)
G.f.: x*(1 + 2*x) / ((1 - x)^3*(1 + x + x^2)^2).
a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7) for n>7. (End)
From Ridouane Oudra, Jun 23 2024: (Start)
a(n) = Sum_{i=1..n+3} (i mod 3)*floor(i/3);
a(n) = (1/2)*(n^2 + n + (n^2 - 5*n)*t -(6*n - 9)*t^2 + 9*t^3), where t = floor(n/3);
a(n) = A066377(n+1) - A092353(n). (End)
E.g.f.: exp(-x/2)*(exp(3*x/2)*(6 + 14*x + 3*x^2) - 2*(3 + x)*cos(sqrt(3)*x/2) - 2*sqrt(3)*(1 - x)*sin(sqrt(3)*x/2))/18. - Stefano Spezia, Jun 23 2024

A115265 Correlation triangle for floor((n+3)/3).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 3, 3, 2, 2, 4, 4, 4, 4, 2, 3, 4, 5, 7, 5, 4, 3, 3, 5, 6, 8, 8, 6, 5, 3, 3, 6, 7, 9, 11, 9, 7, 6, 3, 4, 6, 8, 12, 12, 12, 12, 8, 6, 4, 4, 7, 9, 13, 15, 15, 15, 13, 9, 7, 4

Offset: 0

Paul Barry, Jan 18 2006

Row sums are A115266. Diagonal sums are A115267.

T(2n,n) is A092353. T(2n,n)-T(2n,n+1)=A087508(n+1).

			Triangle begins
1;
1,1;
1,2,1;
2,2,2,2;
2,3,3,3,2;
2,4,4,4,4,2;
3,4,5,7,5,4,3;
3,5,6,8,8,6,5,3;
3,6,7,9,11,9,7,6,3;

T[n_, k_] := Sum[Boole[j <= k] * Floor[(k - j + 3)/3] * Boole[j <= n-k] * Floor[(n - k - j + 3)/3], {j, 0, n}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 15 2017 *)

G.f.: (1+x+x^2)(1+xy+x^2*y^2)/((1-x^3)^2*(1-x^3*y^3)^2*(1-x^2*y)).

T(n, k) = sum{j=0..n, [j<=k]*floor((k-j+3)/3)*[j<=n-k]*floor((n-k-j+3)/3)}.

cp's OEIS Frontend

A262612 Triangle read by rows T(n,k) in which column k lists the partial sums of the k-th column of triangle A236104.

Views

Author

Keywords

Comments

Examples

Crossrefs

A287195 Independence and clique covering number of the n-triangular honeycomb acute knight graph.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A115265 Correlation triangle for floor((n+3)/3).

Views

Author

Keywords

Comments

Examples

Programs

Mathematica

Formula