A131830 -id:A131830 - cp's OEIS frontend

A128715 A131830 + A103451 - A000012 as infinite lower triangular matrices.

1, 2, 2, 3, 1, 3, 4, 2, 2, 4, 5, 3, 5, 3, 5, 6, 4, 9, 9, 4, 6, 7, 5, 14, 19, 14, 5, 7, 8, 6, 20, 34, 34, 20, 6, 8, 9, 7, 27, 55, 69, 55, 27, 7, 9, 10, 8, 35, 83, 125, 125, 83, 35, 8, 10

Offset: 0

Gary W. Adamson, Aug 09 2007

Row sums are (apart from the first row) given by A005126.

			First few rows of the triangle are:
1;
2, 2;
3, 1, 3;
4, 2, 2, 4;
5, 3, 5, 3, 5;
6, 4, 9, 9, 4, 6;
7, 5, 14, 19, 14, 5, 7;
8, 6, 20, 34, 34, 20, 6, 8;
9, 7, 27, 55, 69, 55, 27, 7, 9;
...

Cf. A131830, A103451, A000012, A005126.

A131832 2*(A131830) - A000012.

Original entry on oeis.org

1, 3, 3, 5, 3, 5, 7, 5, 5, 7, 9, 7, 11, 7, 9, 11, 9, 19, 19, 9, 11, 13, 11, 29, 39, 29, 11, 13, 15, 13, 41, 69, 69, 41, 13, 15, 17, 15, 55, 111, 139, 111, 55, 15, 17, 19, 17, 71, 167, 251, 251, 167, 71, 17, 19

Offset: 0

Gary W. Adamson, Jul 20 2007

Row sums = A131833: (1, 6, 13, 24, 43, 78, ...).

			First few rows of the triangle:
   1;
   3,  3;
   5,  3,  5;
   7,  5,  5,  7;
   9,  7, 11,  7,  9;
  11,  9, 19, 19,  9, 11;
  13, 11, 29, 39, 29, 11, 13;
  15, 13, 41, 69, 69, 41, 13, 15;
  ...

Cf. A131830, A131833.

2*(A131810) - A000012 as infinite lower triangular matrices.

A131821 Triangle read by rows: row n consists of n followed by (n-2) ones then n.

Original entry on oeis.org

1, 2, 2, 3, 1, 3, 4, 1, 1, 4, 5, 1, 1, 1, 5, 6, 1, 1, 1, 1, 6, 7, 1, 1, 1, 1, 1, 7, 8, 1, 1, 1, 1, 1, 1, 8, 9, 1, 1, 1, 1, 1, 1, 1, 9, 10, 1, 1, 1, 1, 1, 1, 1, 1, 10, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 14, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 14

Offset: 1

Gary W. Adamson, Jul 19 2007

			First few rows of the triangle are:
  1;
  2, 2;
  3, 1, 3;
  4, 1, 1, 4;
  5, 1, 1, 1, 5;
  6, 1, 1, 1, 1, 6;
  7, 1, 1, 1, 1, 1, 7;
  ...

Row sums: A016777.

Cf. A130296, A051340, A016777, A131830.

A131821 := proc(n,c) if c=1 or c=n then n ; else 1 ; fi ; end: for n from 1 to 16 do for c from 1 to n do printf("%d,",A131821(n,c)) ; od: od: # R. J. Mathar, May 30 2008

T[n_, k_] := If[k == 1 || k == n, n, 1];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 05 2023 *)

T(n,k) := if k = 1 or k = n then n else 1$
create_list(T(n, k), n, 1, 12, k, 1, n); /* Franck Maminirina Ramaharo, Dec 19 2018 */

from math import isqrt
def A131821(n): return m+(k>r) if 0<=(k:=n<<1)-(r:=(m:=isqrt(k))*(m+1))<=2 or n<=2 else 1 # Chai Wah Wu, Nov 07 2024

T(n,k) = A130296(n,k) + A051340(k-1,n-k) - 1.
T(n,1) = T(n,n) = n for n >= 1, and T(n,k) = 1 for 2 <= k <= n - 1, n >= 3.
From Franck Maminirina Ramaharo, Dec 19 2018: (Start)
T(n,k) = A130296(n,k) + A130296(n,n-k+1) - 1.
G.f.: y*(x - 3*x^2*y^2 + (x^2 + x^3)*y^3)/((1 - y)^2*(1 - x*y)^2).
E.g.f.: (1 - x^2 +(x^2 + (x - x^2)*y)*exp(y) - (1 - (x - x^2)*y)*exp(x*y))/(1 - x). (End)

More terms from R. J. Mathar, May 30 2008

A100314 Number of 2 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (01;1).

Original entry on oeis.org

1, 4, 8, 14, 24, 42, 76, 142, 272, 530, 1044, 2070, 4120, 8218, 16412, 32798, 65568, 131106, 262180, 524326, 1048616, 2097194, 4194348, 8388654, 16777264, 33554482, 67108916, 134217782, 268435512, 536870970, 1073741884, 2147483710, 4294967360, 8589934658

Offset: 0

Sergey Kitaev, Nov 13 2004

An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1 < i2, j1 < j2 and these elements are in the same relative order as those in the triple (x,y,z). In general, the number of m X n 0-1 matrices in question is given by 2^m + 2^n + 2*(n*m-n-m).

Arthur H. Stroud, Approximate calculation of multiple integrals, Prentice-Hall, 1971.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Ronald Cools, Encyclopaedia of Cubature Formulas
S. Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. this sequence (m=2), A100315 (m=3), A100316 (m=4).
Cf. A000051, A000079, A001787, A002064, A005126, A005843.
Cf. A052548, A058331, A099003, A132750, A176691.
Row sums of A131830.

List([0..40],n->2^n+2*n); # Muniru A Asiru, Dec 21 2018

[2^n+2*n: n in [1..40]]; // Vincenzo Librandi, Oct 22 2011

a:= proc(n) 2^n + 2*n: end: seq(a(n),n=0..50); # Gary W. Adamson, Jul 20 2007

LinearRecurrence[{4,-5,2}, {1,4,8}, 34] (* Jean-François Alcover, Mar 19 2020 *)

makelist(2^n + 2*n, n, 0, 50); /* Franck Maminirina Ramaharo, Dec 19 2018 */

[2^n +2*n for n in range(41)] # G. C. Greubel, Feb 01 2023

a(n) = 2^n + 2*n.
From Gary W. Adamson, Jul 20 2007: (Start)
Binomial transform of (1, 3, 1, 1, 1, ...).
For n > 0, a(n) = 2*A005126(n-1). (End)
From R. J. Mathar, Jun 13 2008: (Start)
G.f.: 1 + 2*x*(2 -4*x +x^2)/((1-x)^2*(1-2*x)).
a(n+1)-a(n) = A052548(n). (End)
From Colin Barker, Oct 16 2013: (Start)
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
G.f.: (1 - 3*x^2)/((1-x)^2*(1-2*x)). (End)
E.g.f.: exp(2*x) + 2*x*exp(x). - Franck Maminirina Ramaharo, Dec 19 2018
a(n) = A000079(n) + A005843(n). - Muniru A Asiru, Dec 21 2018

a(0)=1 prepended by Alois P. Heinz, Dec 21 2018

A321123 a(n) = 2^n + 2n^2 + 2n + 1.

Original entry on oeis.org

2, 7, 17, 33, 57, 93, 149, 241, 401, 693, 1245, 2313, 4409, 8557, 16805, 33249, 66081, 131685, 262829, 525049, 1049417, 2098077, 4195317, 8389713, 16778417, 33555733, 67110269, 134219241, 268437081, 536872653, 1073743685, 2147485633, 4294969409, 8589936837

Offset: 0

Franck Maminirina Ramaharo, Dec 17 2018

For n >= 2, a(n) is the number of evaluation points on the n-dimensional cube in Genz and Malik's degree 7 cubature rule.

Marius A. Burtea, Table of n, a(n) for n = 0..201
Ronald Cools, Encyclopaedia of Cubature Formulas
Ronald Cools and Philip Rabinowitz, Monomial cubature rules since "Stroud": a compilation, Journal of Computational and Applied Mathematics Vol. 48 (1993), 309-326.
Alan C. Genz and Awais A. Malik, Remarks on algorithm 006: An adaptive algorithm for numerical integration over an N-dimensional rectangular region, Journal of Computational and Applied Mathematics Vol. 6 (1980), 295-302.
Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Cf. A000051, A000079, A001787, A001844, A002064, A005126, A058331, A100314, A131830, A100314, A132750, A176691, A322593.

[2^n + 2*n^2 + 2*n + 1: n in [0..33]]; // Marius A. Burtea, Dec 28 2018

Table[2^n + 2*n^2 + 2*n + 1, {n, 0, 50}]

makelist(2^n + 2*n^2 + 2*n + 1, n, 0, 50);

a(n) = A000079(n) + A001844(n).
a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4), n >= 4.
G.f.: (2 - 3*x - 3*x^3)/((1 - 2*x)*(1 - x)^3).
E.g.f.: exp(2*x) + (1 + 4*x + 2*x^2)*exp(x).

A322593 a(n) = 2^n + 2*n^2 + 1.

Original entry on oeis.org

2, 5, 13, 27, 49, 83, 137, 227, 385, 675, 1225, 2291, 4385, 8531, 16777, 33219, 66049, 131651, 262793, 525011, 1049377, 2098035, 4195273, 8389667, 16778369, 33555683, 67110217, 134219187, 268437025, 536872595, 1073743625, 2147485571, 4294969345, 8589936771

Offset: 0

Franck Maminirina Ramaharo, Dec 18 2018

For n = 3..7, a(n) is the number of evaluating points on the n-dimensional sphere (also n-space with weight function exp(-r^2) or exp(-r)) in a degree 7 cubature rule.

Arthur H. Stroud, Approximate calculation of multiple integrals, Prentice-Hall, 1971.

Marius A. Burtea, Table of n, a(n) for n = 0..200
Ronald Cools, Encyclopaedia of Cubature Formulas
Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Cf. A000051, A000079, A001580, A001787, A002064, A005126, A058331, A100314, A131830, A100314, A132750, A176691, A321123.

[2^n + 2*n^2 + 1: n in [0..33]]; // Marius A. Burtea, Dec 28 2018

Table[2^n + 2*n^2 + 1, {n, 0, 50}]
LinearRecurrence[{5,-9,7,-2},{2,5,13,27},50] (* Harvey P. Dale, Mar 23 2021 *)

Maxima
```
makelist(2^n + 2*n^2 + 1, n, 0, 50);
```

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n >= 4.
a(n) = a(n-1) + A100315(n-1), n >= 2.
G.f.: (2 - 5*x + 6*x^2 - 7*x^3)/((1 - 2*x)*(1 - x)^3)
E.g.f.: exp(2*x) + (1 + 2*x + 2*x^2)*exp(x).

cp's OEIS Frontend

A128715 A131830 + A103451 - A000012 as infinite lower triangular matrices.

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A131832 2*(A131830) - A000012.

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A131821 Triangle read by rows: row n consists of n followed by (n-2) ones then n.

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A100314 Number of 2 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (01;1).

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A321123 a(n) = 2^n + 2*n^2 + 2*n + 1.

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A322593 a(n) = 2^n + 2*n^2 + 1.

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A321123 a(n) = 2^n + 2n^2 + 2n + 1.