A131115 -id:A131115 - cp's OEIS frontend

A048489 a(n) = 7 * 2^n - 6.

1, 8, 22, 50, 106, 218, 442, 890, 1786, 3578, 7162, 14330, 28666, 57338, 114682, 229370, 458746, 917498, 1835002, 3670010, 7340026, 14680058, 29360122, 58720250, 117440506, 234881018, 469762042, 939524090, 1879048186

Offset: 0

Clark Kimberling

Number of 3 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (10;0) and (11;0). 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 i1Sergey Kitev, Nov 13 2004
Row sums of triangle A131115. - N. J. A. Sloane, Nov 10 2007
Equals binomial transform of [1, 7, 7, 7, ...]. - Gary W. Adamson, Apr 28 2008
Number of variations of a Componium barrel which produces n phrases. This sequence describes the variations produced by the Componium, a historical mechanical organ. Another way of describing it is: Number of base 8 n-digit numbers produced by repeating or advancing along this 14-step cycle: (0,1,2,3,4,5,6,7,6,5,4,3,2,1). Subset of A126362. - Jim Bumgardner, Dec 10 2013
a(n) = the sum of the terms in row(n) in a triangle with first column T(n,0)=
1+2*n and diagonal T(n,n)=1+4*n with T(i,j)=T(i-1,j-1) + T(i-1,j). - J. M. Bergot, May 11 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jim Bumgardner, Variations of the Componium, 2013
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 (3,-2).

a(n)=T(6, n), array T given by A048483.
n-th difference of a(n), a(n-1), ..., a(0) is (7, 7, 7, ...).
Cf. A131115.

A048489:=n->7*2^n-6: seq(A048489(n), n=0..40); # Wesley Ivan Hurt, Apr 18 2017

CoefficientList[Series[(1 + 5 x)/((2 x - 1) (x - 1)), {x, 0, 28}], x] (* Michael De Vlieger, May 22 2018 *)
7*2^Range[0,30]-6 (* or *) LinearRecurrence[{3,-2},{1,8},30] (* Harvey P. Dale, May 19 2019 *)

a(n)=7<Charles R Greathouse IV, Dec 10 2013

a(n) = A000079(n)*7-6 = A005009(n)-6. - Omar E. Pol, Dec 21 2008
a(n) = 2*a(n-1)+6 with n>0, a(0)=1. - Vincenzo Librandi, Aug 06 2010
G.f.: ( 1+5*x ) / ( (2*x-1)*(x-1) ). - R. J. Mathar, Oct 21 2012
a(n) = A063757(2*n). - Philippe Deléham, Apr 13 2013

A131112 T(n,k) = 4binomial(n,k) - 3I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

Original entry on oeis.org

1, 4, 1, 4, 8, 1, 4, 12, 12, 1, 4, 16, 24, 16, 1, 4, 20, 40, 40, 20, 1, 4, 24, 60, 80, 60, 24, 1, 4, 28, 84, 140, 140, 84, 28, 1, 4, 32, 112, 224, 280, 224, 112, 32, 1, 4, 36, 144, 336, 504, 504, 336, 144, 36, 1

Offset: 0

Gary W. Adamson, Jun 15 2007

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  4,  1;
  4,  8,  1;
  4, 12, 12,  1;
  4, 16, 24, 16,  1;
  4, 20, 40, 40, 20, 1;
  ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A007318, A036563, A131110, A131111, A131113, A131114, A131115.

T:= function(n,k)
    if k=n then return 1;
    else return 4*Binomial(n,k);
    fi;  end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 18 2019

[k eq n select 1 else 4*Binomial(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 18 2019

seq(seq(`if`(k=n, 1, 4*binomial(n,k)), k=0..n), n=0..10); # G. C. Greubel, Nov 18 2019

Table[If[k==n, 1, 4*Binomial[n, k]], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)

T(n,k) = if(k==n, 1, 4*binomial(n,k)); \\ G. C. Greubel, Nov 18 2019

def T(n, k):
    if (k==n): return 1
    else: return 4*binomial(n, k)
[[T(n, k) for k in (0..n)] for n in (0..10)]
# G. C. Greubel, Nov 18 2019

T(n,k) = 4*A007318(n,k) - 3*I(n,k), where A007318 = Pascal's triangle and I = Identity matrix.
n-th row sum = A036563(n+2) = 2^(n+2) - 3.
Bivariate o.g.f.: Sum_{n,k>=0} T(n,k)*x^n*y^k = (1 + 3*x - x*y)/((1 - x*y)*(1 - x - x*y)). - Petros Hadjicostas, Feb 20 2021

A131113 T(n,k) = 5binomial(n,k) - 4I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

Original entry on oeis.org

1, 5, 1, 5, 10, 1, 5, 15, 15, 1, 5, 20, 30, 20, 1, 5, 25, 50, 50, 25, 1, 5, 30, 75, 100, 75, 30, 1, 5, 35, 105, 175, 175, 105, 35, 1, 5, 40, 140, 280, 350, 280, 140, 40, 1, 5, 45, 180, 420, 630, 630, 420, 180, 45, 1

Offset: 0

Gary W. Adamson, Jun 15 2007

Row sums = A048487: (1, 6, 16, 36, 76, 156, ...).

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  5,  1;
  5, 10,  1;
  5, 15, 15,  1;
  5, 20, 30,  20,  1;
  5, 25, 50,  50, 25,  1;
  5, 30, 75, 100, 75, 30, 1;
  ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A007318, A048487, A131110, A131112, A131114, A131115.

T:= function(n,k)
    if k=n then return 1;
    else return 5*Binomial(n,k);
    fi;  end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 18 2019

[k eq n select 1 else 5*Binomial(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 18 2019

seq(seq(`if`(k=n, 1, 5*binomial(n,k)), k=0..n), n=0..10); # G. C. Greubel, Nov 18 2019

Table[If[k==n, 1, 5*Binomial[n, k]], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)

T(n,k) = if(k==n, 1, 5*binomial(n,k)); \\ G. C. Greubel, Nov 18 2019

def T(n, k):
    if k == n: return 1
    else: return 5*binomial(n, k)
[[T(n, k) for k in (0..n)] for n in (0..10)]
# G. C. Greubel, Nov 18 2019

T(n,k) = 5*A007318(n,k) - 4*I(n,k), where A007318 = Pascal's triangle and I = Identity matrix.

Bivariate o.g.f.: Sum_{n,k>=0} T(n,k)*x^n*y^k = (1 + 4*x - x*y)/((1 - x*y)*(1 - x - x*y)). - Petros Hadjicostas, Feb 20 2021

A131114 T(n,k) = 6binomial(n,k) - 5I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

Original entry on oeis.org

1, 6, 1, 6, 12, 1, 6, 18, 18, 1, 6, 24, 36, 24, 1, 6, 30, 60, 60, 30, 1, 6, 36, 90, 120, 90, 36, 1, 6, 42, 126, 210, 210, 126, 42, 1, 6, 48, 168, 336, 420, 336, 168, 48, 1, 6, 54, 216, 504, 756, 756, 504, 216, 54, 1

Offset: 0

Gary W. Adamson, Jun 15 2007

Row sums give A048488.

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  6,  1;
  6, 12,  1;
  6, 18, 18,   1;
  6, 24, 36,  24,  1;
  6, 30, 60,  60, 30,  1;
  6, 36, 90, 120, 90, 36, 1;
  ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A007318, A048488, A131110, A131111, A131112, A131113, A131115.

T:= function(n,k)
    if k=n then return 1;
    else return 6*Binomial(n,k);
    fi;  end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 18 2019

[k eq n select 1 else 6*Binomial(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 18 2019

seq(seq(`if`(k=n, 1, 6*binomial(n,k)), k=0..n), n=0..10); # G. C. Greubel, Nov 18 2019

Table[If[k==n, 1, 6*Binomial[n, k]], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)

T(n,k) = if(k==n, 1, 6*binomial(n,k)); \\ G. C. Greubel, Nov 18 2019

def T(n, k):
    if (k==n): return 1
    else: return 6*binomial(n, k)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 18 2019

T(n,k) = 6*A007318(n,k) - 5*I(n,k), where A007318 = Pascal's triangle and I = Identity matrix.

Bivariate o.g.f.: Sum_{n,k>=0} T(n,k)*x^n*y^k = (1 + 5*x - x*y)/((1 - x*y)*(1 - x - x*y)).

A131111 T(n, k) = 3binomial(n,k) - 2I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

Original entry on oeis.org

1, 3, 1, 3, 6, 1, 3, 9, 9, 1, 3, 12, 18, 12, 1, 3, 15, 30, 30, 15, 1, 3, 18, 45, 60, 45, 18, 1, 3, 21, 63, 105, 105, 63, 21, 1, 3, 24, 84, 168, 210, 168, 84, 24, 1, 3, 27, 108, 252, 378, 378, 252, 108, 27, 1

Offset: 0

Gary W. Adamson, Jun 15 2007

Row sums = A033484: (1, 4, 10, 22, 46, ...) = 3*2^n - 2.

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  3,  1;
  3,  6,  1;
  3,  9,  9,  1;
  3, 12, 18, 12,  1;
  3, 15, 30, 30, 15,  1;
  3, 18, 45, 60, 45, 18, 1;
  ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A131110, A131112, A131113, A131114, A131115.

T:= function(n,k)
    if k=n then return 1;
    else return 3*Binomial(n,k);
    fi;  end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 18 2019

[k eq n select 1 else 3*Binomial(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 18 2019

seq(seq(`if`(k=n, 1, 3*binomial(n,k)), k=0..n), n=0..10); # G. C. Greubel, Nov 18 2019

Table[If[k==n, 1, 3*Binomial[n, k]], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)

T(n,k) = if(k==n, 1, 3*binomial(n,k)); \\ G. C. Greubel, Nov 18 2019

@CachedFunction
def T(n, k):
    if (k==n): return 1
    else: return 3*binomial(n, k)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 18 2019

T(n,k) = 3*A007318(n,k) - 2*I(n,k), where A007318 = Pascal's triangle and I = Identity matrix.

Bivariate o.g.f.: Sum_{n,k>=0} T(n,k)*x^n*y^k = (1 + 2*x - x*y)/((1 - x*y)*(1 - x - x*y)). - Petros Hadjicostas, Feb 20 2021

A168622 Triangle read by rows: T(n, k) = [x^k]( 7(1+x)^n - 6(1+x^n) ) with T(0, 0) = 1.

Original entry on oeis.org

1, 1, 1, 1, 14, 1, 1, 21, 21, 1, 1, 28, 42, 28, 1, 1, 35, 70, 70, 35, 1, 1, 42, 105, 140, 105, 42, 1, 1, 49, 147, 245, 245, 147, 49, 1, 1, 56, 196, 392, 490, 392, 196, 56, 1, 1, 63, 252, 588, 882, 882, 588, 252, 63, 1, 1, 70, 315, 840, 1470, 1764, 1470, 840, 315, 70, 1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Dec 01 2009

			Triangle begins as:
  1;
  1,  1;
  1, 14,   1;
  1, 21,  21,   1;
  1, 28,  42,  28,    1;
  1, 35,  70,  70,   35,    1;
  1, 42, 105, 140,  105,   42,    1;
  1, 49, 147, 245,  245,  147,   49,   1;
  1, 56, 196, 392,  490,  392,  196,  56,   1;
  1, 63, 252, 588,  882,  882,  588, 252,  63,  1;
  1, 70, 315, 840, 1470, 1764, 1470, 840, 315, 70, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A022090, A131115, A132047, A168620, A168623.

Columns (essentially): A008589 (k=1), A024966 (k=2).

A168622:= func< n,k | k eq 0 or k eq n select 1 else 7*Binomial(n,k) >;
[A168622(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 10 2025

(* First program *)
p[x_, n_]:= With[{m=3}, If[n==0, 1, (2*m+1)(1+x)^n - 2*m*(1+x^n)]];
Table[CoefficientList[p[x,n], x], {n,0,12}]//Flatten
(* Second program *)
A168622[n_, k_]:= If[k==0 || k==n, 1, 7*Binomial[n,k]];
Table[A168622[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 10 2025 *)

def A168622(n,k):
    if k==0 or k==n: return 1
    else: return 7*binomial(n,k)
print(flatten([[A168622(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Apr 10 2025

From G. C. Greubel, Apr 10 2025: (Start)
T(n, k) = 7*binomial(n, k), with T(n, 0) = T(n, n) = 1.
T(n, n-k) = T(n, k).
Sum_{k=0..n} T(n, k) = 2*A048489(n-1) + 6*[n=0].
Sum_{k=0..n} (-1)^k*T(n, k) = -6*(1 + (-1)^n) + 13*[n=0].
Sum_{k=0..floor(n/2)} T(n-k, k) = A022090(n+1) - 3*(3 + (-1)^n) + 6*[n=0].
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (14/sqrt(3))*(-1)^n*cos((4*n+1)*Pi/6) - 6*(1 + (-1)^n*cos(n*Pi/2)) + 6*[n=0]. (End)

cp's OEIS Frontend

A048489 a(n) = 7 * 2^n - 6.

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A131112 T(n,k) = 4*binomial(n,k) - 3*I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

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A131113 T(n,k) = 5*binomial(n,k) - 4*I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

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GAP

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A131114 T(n,k) = 6*binomial(n,k) - 5*I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

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GAP

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A131111 T(n, k) = 3*binomial(n,k) - 2*I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

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A168622 Triangle read by rows: T(n, k) = [x^k]( 7*(1+x)^n - 6*(1+x^n) ) with T(0, 0) = 1.

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A131112 T(n,k) = 4binomial(n,k) - 3I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

A131113 T(n,k) = 5binomial(n,k) - 4I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

A131114 T(n,k) = 6binomial(n,k) - 5I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

A131111 T(n, k) = 3binomial(n,k) - 2I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

A168622 Triangle read by rows: T(n, k) = [x^k]( 7(1+x)^n - 6(1+x^n) ) with T(0, 0) = 1.