A131128 -id:A131128 - cp's OEIS frontend

A131130 Binomial transform of [1,1,7,1,7,1,7,1,...].

1, 2, 10, 26, 58, 122, 250, 506, 1018, 2042, 4090, 8186, 16378, 32762, 65530, 131066, 262138, 524282, 1048570, 2097146, 4194298, 8388602, 16777210, 33554426, 67108858, 134217722, 268435450, 536870906, 1073741818, 2147483642

Offset: 0

Gary W. Adamson, Jun 16 2007

For n >= 3, number of vertices of 4,4'-bipyridinium dendrimers (see the Arjomanfar and Gholami reference, p. 71). - Emeric Deutsch, Apr 12 2015

Number of ways to color a (2n-1) X (2n-1) chess board in a "balanced" way. A coloring is called balanced if, within every square subgrid made up of k^2 cells for 1 <= k <= 2*n-1, the number of black cells differs from the number of white cells by at most one. It is problem 3 from the British Maths Olympiad 2020. - Ruediger Jehn, Jan 27 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. Arjomanfar and N. Gholami, Computing the Szeged index of 4,4'-bipyridinium dendrimer, Iranian J. Math. Chem., 3, 2012, 67-72.
British Mathematical Olympiad, 2020 - Round 2, Problem 3.
Index to sequences related to Olympiads.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A095121 (bin transf 1,1,3,1,3,...), A131128 (bin transf 1,1,5,1,5,..), A131131.

Maple
```
1, seq(4*2^n -6, n = 1..30);
```

Join[{1},LinearRecurrence[{3,-2},{2,10},30]] (* Harvey P. Dale, Mar 07 2014 *)
CoefficientList[Series[(1 -x +6x^2)/(1 -3x +2x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 08 2014 *)

Row sums of triangle A131131.
a(n) = 4*2^n - 6 for n >= 1; a(0)=1.
From Philippe Deléham, Jan 04 2009: (Start)
a(n) = 3*a(n-1) - 2*a(n-2), n > 2; a(0)=1, a(1)=2, a(2)=10.
G.f.: (1-x+6*x^2) / (1-3*x+2*x^2). (End)
a(n) = 2*a(n-1) + 6 for n > 1, a(0)=1, a(1)=2. - Philippe Deléham, Sep 25 2009
E.g.f.: 3 - 6*exp(x) + 4*exp(2*x). - Stefano Spezia, Feb 05 2021

Edited by Emeric Deutsch, Jul 12 2007

A154117 Expansion of (1 - x + 3x^2)/((1-x)(1-2*x)).

Original entry on oeis.org

1, 2, 7, 17, 37, 77, 157, 317, 637, 1277, 2557, 5117, 10237, 20477, 40957, 81917, 163837, 327677, 655357, 1310717, 2621437, 5242877, 10485757, 20971517, 41943037, 83886077, 167772157, 335544317, 671088637, 1342177277, 2684354557

Offset: 0

Vladimir Joseph Stephan Orlovsky, Dec 15 2008

Binomial transform of 1,1,4,1,4,1,4,1,4,1,4,1,4,1,4,... - Philippe Deleham, Jan 05 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A094373, A000079, A083329, A095121, A131128, A131130.

[1] cat [5*2^n-3 : n in [0..30]]; // Vincenzo Librandi, Nov 11 2011

Join[{1}, Table[ 5*2^(n - 1) - 3, {n, 1, 10}]] (* or *) Join[{1, 2, 7}, LinearRecurrence[{3, -2}, {17, 37}, 10]] (* G. C. Greubel, Sep 02 2016 *)

a(n)=if(n, 5<<(n-1)-3, 1) \\ Charles R Greathouse IV, Sep 02 2016

From Philippe Deléham, Jan 05 2009: (Start)
a(n) = 3*a(n-1) - 2*a(n-2), n > 2.
a(n) = 2*a(n-1) + 3, n > 1.
a(n) = 5*2^(n-1) - 3, n >= 1. (End)
E.g.f.: (1/2)*(3 - 6*exp(x) + 5*exp(2*x)). - G. C. Greubel, Sep 02 2016

a(0) added by Philippe Deléham, Jan 05 2009

A132047 3A007318 - 2A103451 as infinite lower triangular matrices.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 9, 9, 1, 1, 12, 18, 12, 1, 1, 15, 30, 30, 15, 1, 1, 18, 45, 60, 45, 18, 1, 1, 21, 63, 105, 105, 63, 21, 1, 1, 24, 84, 168, 210, 168, 84, 24, 1, 1, 27, 108, 252, 378, 378, 252, 108, 27, 1, 1, 30, 135, 360, 630, 756, 630, 360, 135, 30, 1

Offset: 0

Gary W. Adamson, Aug 08 2007

			First few rows of the triangle are:
  1;
  1, 1;
  1, 6, 1;
  1, 9, 9, 1;
  1, 12, 18, 12, 1;
  1, 15, 30, 30, 15, 1;
  1, 18, 45, 60, 45, 18, 1;
  ...

Cf. A007318, A103451, A131128 (row sums).

T(n, k) = my(bnk = binomial(n, k)); 3*bnk - 2*(bnk==1); \\ Michel Marcus, Jun 16 2022

a(n) = 3*A007318(n) - 2*A103451(n).

T(n,k) = 3*C(n,k)-2*(C(n,k-n)+C(n,-k)-C(0,n+k)), 0<=k<=n. [Eric Werley, Jul 01 2011]

Corrected and extended by Roger L. Bagula, Nov 02 2008

A154251 Expansion of (1-x+7x^2)/((1-x)(1-2x)).

Original entry on oeis.org

1, 2, 11, 29, 65, 137, 281, 569, 1145, 2297, 4601, 9209, 18425, 36857, 73721, 147449, 294905, 589817, 1179641, 2359289, 4718585, 9437177, 18874361, 37748729, 75497465, 150994937, 301989881, 603979769, 1207959545, 2415919097

Offset: 0

Philippe Deléham, Jan 05 2009

Binomial transform of 1,1,8,1,8,1,8,1,8,1,8,1,8,1,8,...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf.: A094373, A000079, A083329, A095121, A154117, A131128, A154118, A131130

Join[{1},LinearRecurrence[{3,-2},{2,11}, 25]] (* or *) Join[{1},Table[9*2^(n-1) - 7, {n,1,25}]] (* G. C. Greubel, Sep 08 2016 *)

Vec((1-x+7*x^2)/((1-x)*(1-2*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = 3*a(n-1) - 2*a(n-2), n>2, with a(0)=1, a(1)=2, a(2)=11.
a(n) = 9*2^(n-1) - 7, n>0, with a(0)=1.
a(n) = 2*a(n-1) + 7, n>1, with a(0)=1, a(1)=2.
From G. C. Greubel, Sep 08 2016: (Start)
a(n) = 9*2^(n-1) - 7 for n >= 1.
E.g.f.: (1/2)*(9*exp(2*x) - 14*exp(x) + 7). (End)

A154252 Expansion of (1-x+8x^2)/((1-x)(1-2x)) .

Original entry on oeis.org

1, 2, 12, 32, 72, 152, 312, 632, 1272, 2552, 5112, 10232, 20472, 40952, 81912, 163832, 327672, 655352, 1310712, 2621432, 5242872, 10485752, 20971512, 41943032, 83886072, 167772152, 335544312, 671088632, 1342177272, 2684354552, 5368709112, 10737418232

Offset: 0

Philippe Deléham, Jan 05 2009

Binomial transform of 1,1,9,1,9,1,9,1,9,1,9,1,9,1,9,...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf.: A094373, A000079, A083329, A095121, A154117, A131128, A154118, A131130, A154251

Join[{1},LinearRecurrence[{3,-2},{2,12},40]]  (* Harvey P. Dale, Dec 30 2014 *)
Join[{1},Table[5*2^n - 8, {n,1,25}]] (* G. C. Greubel, Sep 08 2016 *)

Vec((1-x+8*x^2)/((1-x)*(1-2*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = 3*a(n-1) - 2*a(n-2), n>2, with a(0)=1, a(1)=2, a(2)=12.
a(n) = 2*a(n-1) + 8, n>1, with a(0)=1, a(1)=2.
a(n) = 10*2^(n-1) - 8, n>=1, with a(0)=1.
E.g.f.: 5*exp(2*x) - 8*exp(x) + 4. - G. C. Greubel, Sep 08 2016

Two terms corrected by Johannes W. Meijer, May 26 2011

A165751 a(n) = 4 - 3*2^n.

Original entry on oeis.org

1, -2, -8, -20, -44, -92, -188, -380, -764, -1532, -3068, -6140, -12284, -24572, -49148, -98300, -196604, -393212, -786428, -1572860, -3145724, -6291452, -12582908, -25165820, -50331644, -100663292, -201326588, -402653180, -805306364, -1610612732, -3221225468

Offset: 0

Philippe Deléham, Sep 26 2009

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A131128.

Table[4 - 3*2^n, {n, 0, 50}] (* or *) LinearRecurrence[{3,-2}, {1,-2}, 50] (* G. C. Greubel, Apr 07 2016 *)

my(x='x+O('x^99)); Vec((1-5*x)/(1-3*x+2*x^2)) \\ Altug Alkan, Apr 07 2016

a(n) = 2*a(n-1) - 4, a(0)=1.
a(n) = Sum_{0<=k<=n} A112555(n,k)*(-3)^(n-k).
G.f.: (1-5x)/(1-3x+2x^2).
From G. C. Greubel, Apr 07 2016: (Start)
a(n) = 3*a(n-1) - 2*a(n-2).
E.g.f.: 4*exp(x) - 3*exp(2*x). (End)
a(n) = -A131128(n) for n>=1. - R. J. Mathar, Feb 27 2019

A131129 3A007318 - 2A097806, where A007318 = Pascal's triangle and A097806 = the pairwise operator.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 3, 9, 7, 1, 3, 12, 18, 10, 1, 3, 15, 30, 30, 13, 1, 3, 18, 45, 60, 45, 16, 1, 3, 21, 63, 105, 105, 63, 19, 1, 3, 24, 84, 168, 210, 168, 84, 22, 1

Offset: 0

Gary W. Adamson, Jun 16 2007

Row sums = A131128: (1, 2, 8, 20, 44, 92, 188, 380, ...), the binomial transform of (1, 1, 5, 1, 5, 1, 5, ...). Triangle A131108 has row sums (1, 2, 6, 14, 30, 62, ...), the binomial transform of (1, 1, 3, 1, 3, 1, ...). Generalization: Given triangles generated from N*A007318 - (N-1)*A097806, row sums are binomial transforms of (1, 1, (2N-1), 1, (2N-1), 1, ...).

Triangle T(n,k), 0 <= k <= n, read by rows given by [1,2,-3,1,0,0,0,0,0,0,0,...] DELTA [1,0,0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 18 2007

			First few rows of the triangle:
  1;
  1,  1;
  3,  4,  1;
  3,  9,  7,  1;
  3, 12, 18, 10,  1;
  3, 15, 30, 30, 13,  1;
  ...

Cf. A097806, A131128, A095121.

G.f.: (1-x*y+2*x^2+2*x^2*y)/((-1+x+x*y)*(x*y-1)). - R. J. Mathar, Aug 12 2015

A131131 4A007318 - 3A097806.

Original entry on oeis.org

1, 1, 1, 4, 5, 1, 4, 12, 9, 1, 4, 16, 24, 13, 1, 4, 20, 40, 40, 17, 1, 4, 24, 60, 80, 60, 21, 1, 4, 28, 84, 140, 140, 84, 25, 1, 4, 32, 112, 224, 280, 224, 112, 29, 1

Offset: 0

Gary W. Adamson, Jun 16 2007

Row sums = A131130, (1, 2, 10, 26, 52, 98, 190, ...), the binomial transform of (1, 1, 7, 1, 7, 1, ...). Generally, triangles generated from N*A007318 - (N-1)*A097806 have row sums that are binomial transforms of (1, 1, (N-1), 1, (N-1), 1, ...). A095121 = (1, 2, 6, 14, 30, 62, ...), the binomial transform of (1, 1, 3, 1, 3, 1, ...) and = row sums of A131108.

Triangle T(n,k), 0 <= k <= n,read by rows given by [1,3,-4,1,0,0,0,0,0,0,0,...] DELTA [1,0,0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 18 2007

			First few rows of the triangle:
  1;
  1,  1;
  4,  5,  1;
  4, 12,  9,  1;
  4, 16, 24, 13,  1
  4, 20, 40, 40, 17,  1;
  ...

Cf. A131130, A131129, A131128, A131127, A046055, A131108, A095121, A097806.

4*A007318 - 3*A097806, where A007318 = Pascal's triangle and A097806 = the pairwise operator.

G.f.: (1-x*y+3*x^2+3*x^2*y)/((-1+x+x*y)*(x*y-1)). - R. J. Mathar, Aug 12 2015

A296954 Expansion of x(1 - x + 4x^2) / ((1 - x)(1 - 2x)).

Original entry on oeis.org

0, 1, 2, 8, 20, 44, 92, 188, 380, 764, 1532, 3068, 6140, 12284, 24572, 49148, 98300, 196604, 393212, 786428, 1572860, 3145724, 6291452, 12582908, 25165820, 50331644, 100663292, 201326588, 402653180, 805306364, 1610612732, 3221225468, 6442450940, 12884901884

Offset: 0

J. Devillet, Dec 22 2017

Number of bisymmetric, quasitrivial, and order-preserving binary operations on the n-element set {1,...,n} that have annihilator elements.

Apart from the offset the same as A131128. - R. J. Mathar, Jan 02 2018

Colin Barker, Table of n, a(n) for n = 0..1000
J. Devillet, Bisymmetric and quasitrivial operations: characterizations and enumerations, arXiv:1712.07856 [math.RA] (2017).
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A296953.

CoefficientList[Series[x (1 - x + 4 x^2)/((1 - x) (1 - 2 x)), {x, 0, 33}], x] (* Michael De Vlieger, Dec 23 2017 *)
LinearRecurrence[{3,-2},{0,1,2,8},40] (* Harvey P. Dale, Jun 05 2021 *)

concat(0, Vec(x*(1 - x + 4*x^2) / ((1 - x)*(1 - 2*x)) + O(x^40))) \\ Colin Barker, Dec 22 2017

a(n) = A296953(n)-2, a(0)=0, a(1)=1.
From Colin Barker, Dec 22 2017: (Start)
G.f.: x*(1 - x + 4*x^2) / ((1 - x)*(1 - 2*x)).
a(n) = 3*2^(n-1) - 4 for n>1.
a(n) = 3*a(n-1) - 2*a(n-2) for n>3.
(End)

G.f. in the name replaced by a better g.f. by Colin Barker, Dec 23 2017

A154312 Triangle T(n,k), 0<=k<=n, read by rows, given by [0,1/2,-1/2,0,0,0,0,0,0,0,...] DELTA [2,-1/2,-1/2,2,0,0,0,0,0,0,0 ...] where DELTA is the operator defined in A084938 .

Original entry on oeis.org

1, 0, 2, 0, 1, 3, 0, 0, 3, 5, 0, 0, 0, 7, 9, 0, 0, 0, 0, 15, 17, 0, 0, 0, 0, 0, 31, 33, 0, 0, 0, 0, 0, 0, 63, 65, 0, 0, 0, 0, 0, 0, 0, 127, 129, 0, 0, 0, 0, 0, 0, 0, 0, 255, 257, 0, 0, 0, 0, 0, 0, 0, 0, 0, 511, 513, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1023, 1025, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2047

Offset: 0

Philippe Deléham, Jan 07 2009

Column sums give A003945.

			Triangle begins:
1;
0, 2;
0, 1, 3;
0, 0, 3, 5;
0, 0, 0, 7, 9;
0, 0, 0, 0, 15, 17; ...

Cf. A000225, A094373

Sum_{k, 0<=k<=n}T(n,k)*x^(n-k)= A040000(n), A094373(n), A000079(n), A083329(n), A095121(n), A154117(n), A131128(n), A154118(n), A131130(n), A154251(n), A154252(n) for x = -1,0,1,2,3,4,5,6,7,8,9 respectively.
G.f.: (1-x*y+x^2*y-x^2*y^2)/(1-3*x*y+2*x^2*y^2). - Philippe Deléham, Nov 02 2013
T(n,k) = 3*T(n-1,k-1) - 2*T(n-2,k-2), T(0,0) = 1, T(1,0) = 0, T(1,1) = 2, T(2,0) = 0, T(2,1) = 1, T(2,2) = 3, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 02 2013

cp's OEIS Frontend

A131130 Binomial transform of [1,1,7,1,7,1,7,1,...].

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A154117 Expansion of (1 - x + 3*x^2)/((1-x)*(1-2*x)).

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A132047 3*A007318 - 2*A103451 as infinite lower triangular matrices.

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A154251 Expansion of (1-x+7x^2)/((1-x)(1-2x)).

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A154252 Expansion of (1-x+8x^2)/((1-x)(1-2x)) .

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A165751 a(n) = 4 - 3*2^n.

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A131129 3*A007318 - 2*A097806, where A007318 = Pascal's triangle and A097806 = the pairwise operator.

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A131131 4*A007318 - 3*A097806.

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A154117 Expansion of (1 - x + 3x^2)/((1-x)(1-2*x)).

A132047 3A007318 - 2A103451 as infinite lower triangular matrices.

A131129 3A007318 - 2A097806, where A007318 = Pascal's triangle and A097806 = the pairwise operator.

A131131 4A007318 - 3A097806.

A296954 Expansion of x(1 - x + 4x^2) / ((1 - x)(1 - 2x)).