A131246 -id:A131246 - cp's OEIS frontend

A131244 Row sums of triangle A131243.

1, 3, 6, 14, 30, 67, 146, 322, 705, 1549, 3396, 7453, 16346, 35861, 78659, 172549, 378487, 830234, 1821136, 3994730, 8762543, 19220904, 42161568, 92482585, 202863051, 444985664, 976088107, 2141075804, 4696507779

Offset: 0

Gary W. Adamson, Jun 22 2007

A131246 is a companion sequence.

			a(4) = 30 = sum of row 4 terms of A131243: (8 + 7 + 10 + 4 + 1).

Index entries for linear recurrences with constant coefficients, signature (2,2,-3,-1).

Cf. A131243, A131245, A131246.

A065941 := proc(n,k) binomial(n-floor((k+1)/2),floor(k/2)) ; end proc:
A131243 := proc(n,k) local a,j ; a := 0 ; for j from k to n do a := a+ A065941(n,j)*A065941(j,k) ; end do: a ; end proc:
A131244 := proc(n) add(A131243(n,k),k=0..n) ; end proc:
seq(A131244(n),n=0..50) ; # R. J. Mathar, Jan 29 2011

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

G.f. ( 1+x-x^3-2*x^2 ) / ( 1-2*x-2*x^2+3*x^3+x^4 ). - R. J. Mathar, Jan 29 2011

A131243 A065941^2 as an infinite lower triangular matrix.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 4, 4, 1, 8, 7, 10, 4, 1, 13, 12, 23, 12, 6, 1, 21, 20, 48, 29, 21, 6, 1, 34, 33, 96, 64, 62, 24, 8, 1, 55, 54, 185, 132, 160, 74, 36, 8, 1, 89, 88, 348, 261, 382, 200, 130, 40, 10, 1, 144, 143, 642, 500, 859, 492, 400, 150, 55, 10, 1

Offset: 0

Gary W. Adamson, Jun 22 2007

Left border, Fibonacci numbers; next border to the right, (Fibonacci numbers - 1).

			First few rows of the triangle:
   1;
   2,  1;
   3,  2,   1;
   5,  4,   4,   1;
   8,  7,  10,   4,   1;
  13, 12,  23,  12,   6,   1;
  21, 20,  48,  29,  21,   6,   1;
  34, 33,  96,  64,  62,  24,   8,  1;
  55, 54, 185, 132, 160,  74,  36,  8,  1;
  89, 88, 348, 261, 382, 200, 130, 40, 10, 1;

Hakan Icoz, Rows 0..100, flattened

Cf. A065941, A131244 (row sums), A131245, A131246.

Data and Example corrected by Jon E. Schoenfield, Feb 26 2022

More terms from Hakan Icoz, Jan 23 2023

A131245 A046854^2 as an infinite lower triangular matrix.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 5, 2, 1, 8, 9, 7, 2, 1, 13, 19, 13, 9, 2, 1, 21, 33, 34, 17, 11, 2, 1, 34, 65, 61, 53, 21, 13, 2, 1, 55, 111, 141, 97, 76, 25, 15, 2, 1, 89, 210, 248, 257, 141, 103, 29, 17, 2, 1, 144, 355, 534, 461, 421, 193, 134, 33, 19, 2, 1

Offset: 0

Gary W. Adamson, Jun 22 2007

Left border = Fibonacci numbers.
Row sums = A131246.
A131243 is the square of the reflection triangle to A046854: A065941.
Row sums of A131243 = (1, 3, 6, 14, 30, 67, 146, 322, 705, 1549, ...).

			First few rows of the triangle:
   1;
   2,  1;
   3,  2,  1;
   5,  5,  2,  1;
   8,  9,  7,  2,  1;
  13, 19, 13,  9,  2,  1;
  21, 33, 34, 17, 11,  2,  1;
  ...

Cf. A131243, A131244, A131246, A046854, A065941.

T(n, k) = binomial((n+k)\2, k);
row(n) = my(m=matrix(n+1, n+1, i, j, T(i-1,j-1))); vector(n+1, i, (m^2)[n+1,i]);
lista(nn) = for (n=0, nn, my(v=row(n)); for (i=1, #v, print1(v[i], ", "));); \\ Michel Marcus, Feb 28 2022

More terms from Michel Marcus, Feb 28 2022

A131344 A046854 * A065941.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 4, 3, 1, 8, 7, 8, 3, 1, 13, 12, 18, 9, 4, 1, 21, 20, 38, 21, 14, 4, 1, 34, 33, 76, 47, 39, 15, 5, 1, 55, 54, 147, 97, 100, 43, 21, 5, 1, 89, 88, 277, 194, 236, 115, 69, 22, 6, 1

Offset: 0

Gary W. Adamson, Jun 30 2007

Left border = Fibonacci numbers starting with F(2). Row sums = A131246: (1, 3, 6, 13, 27, 57,...). A131345 = A065941 * A046854.

			First few rows of the triangle are:
1;
2, 1;
3, 2, 1;
5, 4, 3, 1;
8, 7, 8, 3, 1;
13, 12, 18, 9, 4, 1;
...

Cf. A046854, A065941, A131246, A131345.

A046854 * A065941 as infinite lower triangular matrices.

A183314 Number of n X 2 binary arrays with an element zero only if there are an even number of ones to its left and an even number of ones above it.

Original entry on oeis.org

3, 6, 13, 27, 57, 119, 250, 523, 1097, 2297, 4815, 10086, 21137, 44283, 92793, 194419, 407378, 853559, 1788481, 3747361, 7851867, 16451910, 34471669, 72228171, 151339401, 317100335, 664418698, 1392152131, 2916968489, 6111905849

Offset: 1

R. H. Hardin, Jan 03 2011

nonn

Column 2 of A183322.

Is this related to A131246?

			Some solutions for 5 X 2.
..0..0....1..1....1..1....1..1....0..1....0..0....0..0....0..1....0..0....0..0
..0..0....1..1....1..1....1..1....0..1....0..1....0..1....0..1....0..0....0..1
..1..1....0..1....0..0....0..0....0..0....0..1....0..1....0..1....1..1....0..1
..1..1....0..1....0..1....0..0....1..1....0..1....0..0....0..1....1..1....1..1
..1..1....0..0....0..1....0..0....1..1....1..1....1..1....0..0....0..1....1..1

R. H. Hardin, Table of n, a(n) for n = 1..200

Cf. A131246, A183322.

Empirical: a(n) = a(n-1) + 3*a(n-2) - a(n-3) - a(n-4).

Empirical g.f.: x*(3 + 3*x - 2*x^2 - x^3) / (1 - x - 3*x^2 + x^3 + x^4). - Colin Barker, Mar 27 2018

cp's OEIS Frontend

A131244 Row sums of triangle A131243.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

A131243 A065941^2 as an infinite lower triangular matrix.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A131245 A046854^2 as an infinite lower triangular matrix.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions

A131344 A046854 * A065941.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A183314 Number of n X 2 binary arrays with an element zero only if there are an even number of ones to its left and an even number of ones above it.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula