A099003 -id:A099003 - cp's OEIS frontend

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).

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

A139634 a(n) = 10*2^(n-1) - 9.

Original entry on oeis.org

1, 11, 31, 71, 151, 311, 631, 1271, 2551, 5111, 10231, 20471, 40951, 81911, 163831, 327671, 655351, 1310711, 2621431, 5242871, 10485751, 20971511, 41943031, 83886071, 167772151, 335544311, 671088631, 1342177271, 2684354551

Offset: 1

Gary W. Adamson, Apr 29 2008

Binomial transform of [1, 10, 10, 10,...].
A007318 * [1, 10, 10, 10,...].
The binomial transform of [1, c, c, c,...] has the terms a(n)=1-c+c*2^(n-1) if the offset 1 is chosen. The o.g.f. of the a(n) is x{1+(c-2)x}/{(2x-1)(x-1)}. This applies to A139634 with c=10, to A139635 with c=11, to A139697 with c=12, to A139698 with c=25 and to A099003, A139700, A139701 accordingly. - R. J. Mathar, May 11 2008

			a(4) = 71 = (1, 3, 3, 1) dot (1, 10, 10, 10) = (1 + 30 + 30 + 10).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A007318.

[10*2^(n-1)-9: n in [1..50]]; // Vincenzo Librandi, Mar 30 2014

A139634:=n->10*2^(n-1)-9; seq(A139634(n), n=1..30); # Wesley Ivan Hurt, Mar 26 2014

a=1; lst={a}; k=10; Do[a+=k; AppendTo[lst, a]; k+=k, {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 17 2008 *)
CoefficientList[Series[(8 x + 1)/((x - 1) (2 x - 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 30 2014 *)
LinearRecurrence[{3,-2},{1,11},30] (* Harvey P. Dale, Feb 19 2023 *)

a(n)=10*2^(n-1)-9 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 2*a(n-1) + 9, with n>1, a(1)=1. - Vincenzo Librandi, Nov 24 2010
From Colin Barker, Oct 10 2013: (Start)
a(n) = 3*a(n-1) - 2*a(n-2).
G.f.: x*(8*x+1) / ((x-1)*(2*x-1)). (End)

More terms from Vladimir Joseph Stephan Orlovsky, Dec 17 2008

Simpler definition from Jon E. Schoenfield, Jun 23 2010

A139697 Binomial transform of [1, 12, 12, 12, ...].

Original entry on oeis.org

1, 13, 37, 85, 181, 373, 757, 1525, 3061, 6133, 12277, 24565, 49141, 98293, 196597, 393205, 786421, 1572853, 3145717, 6291445, 12582901, 25165813, 50331637, 100663285, 201326581, 402653173, 805306357, 1610612725, 3221225461, 6442450933, 12884901877

Offset: 1

Gary W. Adamson, Apr 29 2008

The binomial transform of [1, c, c, c, ...] has the terms a(n) = 1 - c + c*2^(n-1) if the offset 1 is chosen. The o.g.f. of the a(n) is x*(1+(c-2)*x)/((2x-1)*(x-1)). This applies to A139634 with c=10, to A139635 with c=11, to A139697 with c=12, to A139698 with c=25 and to A099003, A139700, A139701 accordingly. - R. J. Mathar, May 11 2008

			a(4) = 85 = (1, 3, 3, 1) dot (1, 12, 12, 12) = (1 + 36 + 36 + 12).

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

Cf. A139634, A139635.

seq(12*2^(n-1)-11,n=1..25); # Emeric Deutsch, May 05 2008

a=1; lst={a}; k=12; Do[a+=k; AppendTo[lst, a]; k+=k, {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 17 2008 *)

A007318 * [1, 12, 12, 12, ...].
a(n) = 12*2^(n-1) - 11. - Emeric Deutsch, May 05 2008
a(n) = 2*a(n-1) + 11 (with a(1)=1). - Vincenzo Librandi, Nov 24 2010
From Colin Barker, Oct 10 2013: (Start)
a(n) = 3*2^(n+1) - 11.
a(n) = 3*a(n-1) - 2*a(n-2).
G.f.: x*(10*x+1) / ((x-1)*(2*x-1)). (End)

More terms from Emeric Deutsch, May 05 2008

More terms from Colin Barker, Oct 10 2013

A139635 Binomial transform of [1, 11, 11, 11, ...].

Original entry on oeis.org

1, 12, 34, 78, 166, 342, 694, 1398, 2806, 5622, 11254, 22518, 45046, 90102, 180214, 360438, 720886, 1441782, 2883574, 5767158, 11534326, 23068662, 46137334, 92274678, 184549366, 369098742, 738197494, 1476394998, 2952790006, 5905580022, 11811160054

Offset: 1

Gary W. Adamson, Apr 29 2008

A007318 * [1, 11, 11, 11, ...].

The binomial transform of [1, c, c, c, ...] has the terms a(n) = 1 - c + c*2^(n-1) if the offset 1 is chosen. The o.g.f. of the a(n) is x*(1+(c-2)*x)/((2x-1)*(x-1)). This applies to A139634 with c=10, to A139635 with c=11, to A139697 with c=12, to A139698 with c=25 and to A099003, A139700, A139701 accordingly. - R. J. Mathar, May 11 2008

			a(4) = 78 = (1, 3, 3, 1) dot (1, 11, 11, 11) = (1 + 33 + 33 + 11).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A139634.

seq(11*2^(n-1)-10,n=1.. 25); # Emeric Deutsch, May 03 2008

a=1; lst={a}; k=11; Do[a+=k; AppendTo[lst, a]; k+=k, {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 17 2008 *)
CoefficientList[Series[(9 x + 1)/((x - 1) (2 x - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 13 2014 *)
LinearRecurrence[{3,-2},{1,12},40] (* Harvey P. Dale, Oct 26 2015 *)

Vec(x*(9*x+1)/((x-1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Mar 11 2014

a(n) = 11*2^(n-1) - 10. - Emeric Deutsch, May 03 2008
a(n) = 2*a(n-1) + 10, with n > 1, a(1)=1. - Vincenzo Librandi, Nov 24 2010
From Colin Barker, Mar 11 2014: (Start)
a(n) = 3*a(n-1) - 2*a(n-2).
G.f.: x*(9*x+1) / ((x-1)*(2*x-1)). (End)

More terms from Emeric Deutsch, May 03 2008

More terms from Colin Barker, Mar 11 2014

A139698 Binomial transform of [1, 25, 25, 25, ...].

Original entry on oeis.org

1, 26, 76, 176, 376, 776, 1576, 3176, 6376, 12776, 25576, 51176, 102376, 204776, 409576, 819176, 1638376, 3276776, 6553576, 13107176, 26214376, 52428776, 104857576, 209715176, 419430376, 838860776, 1677721576, 3355443176, 6710886376, 13421772776, 26843545576

Offset: 1

Gary W. Adamson, Apr 29 2008

The binomial transform of [1, c, c, c, ...] has the terms a(n)=1-c+c*2^(n-1) if the offset 1 is chosen. The o.g.f. of the a(n) is x{1+(c-2)x}/{(2x-1)(x-1)}. This applies to A139634 with c=10, to A139635 with c=11, to A139697 with c=12, to A139698 with c=25 and to A099003, A139700, A139701 accordingly. - R. J. Mathar, May 11 2008

			a(3) = 76 = (1, 2, 1) dot (1, 25, 25) = (1 + 50 + 25).

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A139634, A139635, A139697.

[25*2^(n-1)-24 : n in [1..40]]; // Wesley Ivan Hurt, Jan 17 2017

seq(25*2^(n-1)-24,n=1..25); # Emeric Deutsch, May 03 2008

LinearRecurrence[{3,-2},{1,26},40] (* Harvey P. Dale, Jul 25 2021 *)

Vec(x*(23*x+1)/((x-1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Mar 11 2014

A007318 * [1, 25, 25, 25, ...].
a(n) = 25*2^(n-1)-24. - Emeric Deutsch, May 03 2008
a(n) = 2*a(n-1) + 24 (with a(1)=1). - Vincenzo Librandi, Nov 24 2010
a(n) = 3*a(n-1)-2*a(n-2). G.f.: x*(23*x+1) / ((x-1)*(2*x-1)). - Colin Barker, Mar 11 2014

More terms from Emeric Deutsch, May 03 2008

More terms from Colin Barker, Mar 11 2014

A139700 Binomial transform of [1, 30, 30, 30, ...].

Original entry on oeis.org

1, 31, 91, 211, 451, 931, 1891, 3811, 7651, 15331, 30691, 61411, 122851, 245731, 491491, 983011, 1966051, 3932131, 7864291, 15728611, 31457251, 62914531, 125829091, 251658211, 503316451, 1006632931, 2013265891, 4026531811, 8053063651, 16106127331

Offset: 1

Gary W. Adamson, Apr 29 2008

The binomial transform of [1, c, c, c, ...] has the terms a(n) = 1 - c + c*2^(n-1) if the offset 1 is chosen. The o.g.f. of the a(n) is x{1+(c-2)x}/{(2x-1)(x-1)}. This applies to A139634 with c=10, to A139635 with c=11, to A139697 with c=12, to A139698 with c=25 and to A099003, A139700, A139701 accordingly. - R. J. Mathar, May 11 2008

			a(3) = 91 = (1, 2, 1) dot (1, 30, 30) = (1 + 60 + 30).

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

Cf. A139699, A139698, A139697, A139635, A139634.

seq(30*2^(n-1)-29,n=1..27); # Emeric Deutsch, May 07 2008

LinearRecurrence[{3,-2},{1,31},30] (* Harvey P. Dale, Apr 18 2018 *)

Vec(x*(28*x+1)/((x-1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Mar 11 2014

A007318 * [1, 30, 30, 30, ...].
a(n) = 30*2^(n-1) - 29. - Emeric Deutsch, May 07 2008
a(n) = 2*a(n-1) + 29 (with a(1)=1). - Vincenzo Librandi, Nov 24 2010
From Colin Barker, Mar 11 2014: (Start)
a(n) = 3*a(n-1) - 2*a(n-2).
G.f.: x*(28*x+1) / ((x-1)*(2*x-1)). (End)

More terms from Emeric Deutsch, May 07 2008

More terms from Colin Barker, Mar 11 2014

A139701 Binomial transform of [1, 100, 100, 100, ...].

Original entry on oeis.org

1, 101, 301, 701, 1501, 3101, 6301, 12701, 25501, 51101, 102301, 204701, 409501, 819101, 1638301, 3276701, 6553501, 13107101, 26214301, 52428701, 104857501, 209715101, 419430301, 838860701, 1677721501, 3355443101, 6710886301, 13421772701, 26843545501

Offset: 1

Gary W. Adamson, Apr 29 2008

The binomial transform of [1, c, c, c, ...] has the terms a(n)=1-c+c*2^(n-1) if the offset 1 is chosen. The o.g.f. of a(n) is x{1+(c-2)x}/{(2x-1)(x-1)}. This applies to A139634 with c=10, to A139635 with c=11, to A139697 with c=12, to A139698 with c=25 and to A099003, A139700, A139701 accordingly. - R. J. Mathar, May 11 2008

			a(3) = 301 = (1, 2, 1) dot (1, 100, 100) = (1 + 200 + 100).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A007318, A099003, A139634, A139635, A139697, A139698, A139700, A139701.

[100*2^(n-1)-99 : n in [1..30]]; // Wesley Ivan Hurt, Aug 16 2016

a:=proc(n) options operator, arrow: 100*2^(n-1)-99 end proc: seq(a(n), n=1.. 30); # Emeric Deutsch, May 03 2008

100*2^(Range[30] - 1) - 99 (* Wesley Ivan Hurt, Aug 16 2016 *)
LinearRecurrence[{3, -2}, {1, 101}, 40] (* Vincenzo Librandi, Aug 17 2016 *)

Vec(x*(98*x+1)/((x-1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Mar 11 2014

A007318 * [1, 100, 100, 100, ...].
a(n) = 100*2^(n-1)-99. - Emeric Deutsch, May 03 2008
a(n) = 2*a(n-1)+99 for n > 1. [Vincenzo Librandi, Nov 24 2010]
a(n) = 3*a(n-1) - 2*a(n-2) for n > 2. G.f.: x*(98*x+1) / ((x-1)*(2*x-1)). - Colin Barker, Mar 11 2014

More terms from Emeric Deutsch, May 03 2008

More terms from Colin Barker, Mar 11 2014

cp's OEIS Frontend

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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A139634 a(n) = 10*2^(n-1) - 9.

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A139697 Binomial transform of [1, 12, 12, 12, ...].

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A139635 Binomial transform of [1, 11, 11, 11, ...].

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A139698 Binomial transform of [1, 25, 25, 25, ...].

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A139700 Binomial transform of [1, 30, 30, 30, ...].

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