A131110 -id:A131110 - cp's OEIS frontend

A033484 a(n) = 3*2^n - 2.

1, 4, 10, 22, 46, 94, 190, 382, 766, 1534, 3070, 6142, 12286, 24574, 49150, 98302, 196606, 393214, 786430, 1572862, 3145726, 6291454, 12582910, 25165822, 50331646, 100663294, 201326590, 402653182, 805306366, 1610612734, 3221225470

Offset: 0

N. J. A. Sloane

Number of nodes in rooted tree of height n in which every node (including the root) has valency 3.
Pascal diamond numbers: reflect Pascal's n-th triangle vertically and sum all elements. E.g., a(3)=1+(1+1)+(1+2+1)+(1+1)+1. - Paul Barry, Jun 23 2003
Number of 2 X n binary 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 i1 < i2 and j1 < j2 and these elements are in the same relative order as those in the triple (x,y,z). - Sergey Kitaev, Nov 11 2004
Binomial and inverse binomial transform are in A001047 (shifted) and A122553. - R. J. Mathar, Sep 02 2008
a(n) = (Sum_{k=0..n-1} a(n)) + (2*n + 1); e.g., a(3) = 22 = (1 + 4 + 10) + 7. - Gary W. Adamson, Jan 21 2009
Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if either 0) x is a proper subset of y or y is a proper subset of x and x and y are disjoint, or 1) x equals y. Then a(n) = |R|. - Ross La Haye, Mar 19 2009
Equals the Jacobsthal sequence A001045 convolved with (1, 3, 4, 4, 4, 4, 4, ...). - Gary W. Adamson, May 24 2009
Equals the eigensequence of a triangle with the odd integers as the left border and the rest 1's. - Gary W. Adamson, Jul 24 2010
An elephant sequence, see A175655. For the central square four A[5] vectors, with decimal values 58, 154, 178 and 184, lead to this sequence. For the corner squares these vectors lead to the companion sequence A097813. - Johannes W. Meijer, Aug 15 2010
a(n+2) is the integer with bit string "10" * "1"^n * "10".
a(n) = A027383(2n). - Jason Kimberley, Nov 03 2011
a(n) = A153893(n)-1 = A083416(2n+1). - Philippe Deléham, Apr 14 2013
a(n) = A082560(n+1,A000079(n)) = A232642(n+1,A128588(n+1)). - Reinhard Zumkeller, May 14 2015
a(n) is the sum of the entries in the n-th and (n+1)-st rows of Pascal's triangle minus 2. - Stuart E Anderson, Aug 27 2017
Also the number of independent vertex sets and vertex covers in the complete tripartite graph K_{n,n,n}. - Eric W. Weisstein, Sep 21 2017
Apparently, a(n) is the least k such that the binary expansion of A000045(k) ends with exactly n+1 ones. - Rémy Sigrist, Sep 25 2021
a(n) is the number of root ancestral configurations for a pair consisting of a matching gene tree and species tree with the modified lodgepole shape and n+1 cherry nodes. - Noah A Rosenberg, Jan 16 2025

			Binary: 1, 100, 1010, 10110, 101110, 1011110, 10111110, 101111110, 1011111110, 10111111110, 101111111110, 1011111111110, 10111111111110,
G.f. = 1 + 4*x + 10*x^2 + 22*x^3 + 46*x^4 + 94*x^5 + 190*x^6 + 382*x^7 + ...

J. Riordan, Series-parallel realization of the sum modulo 2 of n switching variables, in Claude Elwood Shannon: Collected Papers, edited by N. J. A. Sloane and A. D. Wyner, IEEE Press, NY, 1993, pp. 877-878.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, The Triple Riordan Group, arXiv:2412.05461 [math.CO], 2024. See pp. 3, 10.
Dennis E. Davenport, Shakuan K. Frankson, Louis W. Shapiro, and Leon C. Woodson, An Invitation to the Riordan Group, Enum. Comb. Appl. (2024) Vol. 4, No. 3, Art. #S2S1. See p. 22.
Erik D. Demaine et al., Picture-Hanging Puzzles, arXiv:1203.3602 [cs.DS], 2012, 2014. See p. 8, actually length(Sn) is 2^n+2^(n-1)-2, that is, a(n-1).
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Egor Lappo and Noah A. Rosenberg, A lattice structure for ancestral configurations arising from the relationship between gene trees and species trees, Adv. Appl. Math. 343 (2024), 65-81.
Eric Weisstein's World of Mathematics, Complete Tripartite Graph
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Vertex Cover
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000045, A007283, A036563, A131110, A051597, A132776, A001045.
Cf. A000918.
Cf. A112468, A112739.
Cf. A082560, A000079, A232642, A128588.

List([0..35], n-> 3*2^n -2); # G. C. Greubel, Nov 18 2019

a033484 = (subtract 2) . (* 3) . (2 ^)
a033484_list = iterate ((subtract 2) . (* 2) . (+ 2)) 1
-- Reinhard Zumkeller, Apr 23 2013

[3*2^n-2: n in [1..36]]; // Vincenzo Librandi, Nov 22 2010

with(combinat):a:=n->stirling2(n,2)+stirling2(n+1,2): seq(a(n), n=1..35); # Zerinvary Lajos, Oct 07 2007
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=(a[n-1]+1)*2 od: seq(a[n], n=1..35); # Zerinvary Lajos, Feb 22 2008

Table[3 2^n - 2, {n, 0, 35}] (* Vladimir Joseph Stephan Orlovsky, Dec 16 2008 *)
(* Start from Eric W. Weisstein, Sep 21 2017 *)
3*2^Range[0, 35] - 2
LinearRecurrence[{3, -2}, {1, 4}, 36]
CoefficientList[Series[(1+x)/(1-3x+2x^2), {x, 0, 35}], x] (* End *)

a(n) = 3<Charles R Greathouse IV, Nov 02 2011

[3*2^n -2 for n in (0..35)] # G. C. Greubel, Nov 18 2019

G.f.: (1+x)/(1-3*x+2*x^2).
a(n) = 2*(a(n-1) + 1) for n>0, with a(0)=1.
a(n) = A007283(n) - 2.
G.f. is equivalent to (1-2*x-3*x^2)/((1-x)*(1-2*x)*(1-3*x)). - Paul Barry, Apr 28 2004
From Reinhard Zumkeller, Oct 09 2004: (Start)
A099257(a(n)) = A099258(a(n)) = a(n).
a(n) = 2*A055010(n) = (A068156(n) - 1)/2. (End)
Row sums of triangle A130452. - Gary W. Adamson, May 26 2007
Row sums of triangle A131110. - Gary W. Adamson, Jun 15 2007
Binomial transform of (1, 3, 3, 3, ...). - Gary W. Adamson, Oct 17 2007
Row sums of triangle A051597 (a triangle generated from Pascal's rule given right and left borders = 1, 2, 3, ...). - Gary W. Adamson, Nov 04 2007
Equals A132776 * [1/1, 1/2, 1/3, ...]. - Gary W. Adamson, Nov 16 2007
a(n) = Sum_{k=0..n} A112468(n,k)*3^k. - Philippe Deléham, Feb 23 2014
a(n) = -(2^n) * A036563(1-n) for all n in Z. - Michael Somos, Jul 04 2017
E.g.f.: 3*exp(2*x) - 2*exp(x). - G. C. Greubel, Nov 18 2019

A131115 Triangle read by rows: T(n,k) = 7*binomial(n,k) for 1 <= k <= n with T(n,n) = 1 for n >= 0.

Original entry on oeis.org

1, 7, 1, 7, 14, 1, 7, 21, 21, 1, 7, 28, 42, 28, 1, 7, 35, 70, 70, 35, 1, 7, 42, 105, 140, 105, 42, 1, 7, 49, 147, 245, 245, 147, 49, 1, 7, 56, 196, 392, 490, 392, 196, 56, 1, 7, 63, 252, 588, 882, 882, 588, 252, 63, 1, 7, 70, 315, 840, 1470, 1764, 1470, 840, 315, 70, 1

Offset: 0

Gary W. Adamson, Jun 15 2007

Row sums give A048489.
Non-diagonal entries of Pascal's triangle are multiplied by 7. - Emeric Deutsch, Jun 20 2007
The matrix inverse starts
            1;
           -7,         1;
           91,       -14,         1;
        -1771,       273,       -21,       1;
        45955,     -7084,       546,     -28,      1;
     -1490587,    229775,    -17710,     910,    -35,    1;
     58018051,  -8943522,    689325,  -35420,   1365,  -42,   1;
  -2634606331, 406126357, -31302327, 1608425, -61985, 1911, -49, 1;
  ... - R. J. Mathar, Mar 15 2013

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  7,  1;
  7, 14,  1;
  7, 21, 21,  1;
  7, 28, 42, 28,  1;
  7, 35, 70, 70, 35, 1;
  ...

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

Cf. A007318, A048489, A131110, A131111, A131112, A131113, A131114.

T:= function(n,k)
    if k=n then return 1;
    else return 7*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 7*Binomial(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 18 2019

T := proc (n, k) if k < n then 7*binomial(n, k) elif k = n then 1 else 0 end if end proc; for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form - Emeric Deutsch, Jun 20 2007

Table[If[k==n, 1, 7*Binomial[n, k]], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)

T(n,k)=if(k==n,1,7*binomial(n,k)) \\ Charles R Greathouse IV, Jan 16 2012

@CachedFunction
def T(n, k):
    if (k==n): return 1
    else: return 7*binomial(n, k)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 18 2019

G.f.: (1 + 6*x - t*x)/((1-t*x)*(1-x-t*x)). - Emeric Deutsch, Jun 20 2007

Corrected and extended by Emeric Deutsch, Jun 20 2007

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

cp's OEIS Frontend

A033484 a(n) = 3*2^n - 2.

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A131115 Triangle read by rows: T(n,k) = 7*binomial(n,k) for 1 <= k <= n with T(n,n) = 1 for n >= 0.

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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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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).