A047858 -id:A047858 - cp's OEIS frontend

A047859 a(n) = T(2, n), where T is the array given by A047858.

1, 4, 11, 27, 63, 143, 319, 703, 1535, 3327, 7167, 15359, 32767, 69631, 147455, 311295, 655359, 1376255, 2883583, 6029311, 12582911, 26214399, 54525951, 113246207, 234881023, 486539263, 1006632959, 2080374783, 4294967295, 8858370047, 18253611007, 37580963839

Offset: 0

Clark Kimberling

n-th difference of a(n), a(n-1), ..., a(0) is (3, 4, 5, ...).
From Gus Wiseman, Oct 14 2022: (Start)
Also the number of compositions of 2*(n+1) whose maximum part is n+1. These are compositions of 2*(n+1) whose maximum part equals the sum of their remaining parts. For example, the a(0) = 1 through a(2) = 11 compositions are:
  (1,1)  (2,2)    (3,3)
         (1,1,2)  (1,2,3)
         (1,2,1)  (1,3,2)
         (2,1,1)  (2,1,3)
                  (2,3,1)
                  (3,1,2)
                  (3,2,1)
                  (1,1,1,3)
                  (1,1,3,1)
                  (1,3,1,1)
                  (3,1,1,1)
For length instead of maximum we have A001700.
These compositions are ranked by A357708. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

A011782 counts compositions.

Cf. A000120, A001511, A001700, A029931, A045623, A047858, A056239, A070939, A357708.

[(n+4)*2^(n-1)-1: n in [0..30]]; // Vincenzo Librandi, Sep 28 2011

Vec((1-x-x^2)/((1-x)*(1-2*x)^2) + O(x^40)) \\ Colin Barker, Aug 24 2016

Main diagonal of the array defined by: T(0, j) = j + 1 for j >= 0, T(i, 0) = i + 1 for i >= 0, T(i, j)= T(i-1, j-1) + T(i-1, j) + 1. a(n) = (n + 4)*2^(n-1) - 1. - Benoit Cloitre, Jun 17 2003
a(0) = 1, a(1) = 4, a(2) = 11, a(n) = 5*a(n-1) - 8*a(n-2) + 4*a(n-3). - Vincenzo Librandi, Sep 28 2011
G.f.: (1 - x - x^2)/((1 - x)*(1 - 2*x)^2). - Colin Barker, Aug 24 2016
a(n) = A045623(n) - 1. - Gus Wiseman, Oct 14 2022
E.g.f.: exp(x)*(exp(x)*(2 + x) - 1). - Stefano Spezia, Jan 02 2023

A047860 a(n) = T(3,n), array T given by A047858.

Original entry on oeis.org

1, 5, 14, 34, 78, 174, 382, 830, 1790, 3838, 8190, 17406, 36862, 77822, 163838, 344062, 720894, 1507326, 3145726, 6553598, 13631486, 28311550, 58720254, 121634814, 251658238, 520093694, 1073741822, 2214592510, 4563402750

Offset: 0

Clark Kimberling

The Wikipedia article on L-system Example 2 is "Pythagoras Tree" given by the axiom: 0 and rules: 1 -> 11, 0 -> 1[0]0. The length of the n-th string of symbols is a(n). This interpretation leads to a matrix power formula for a(n). - Michael Somos, Jan 12 2015

			G.f. = 1 + 5*x + 14*x^2 + 34*x^3 + 78*x^4 + 174*x^5 + 382*x^6 + 830*x^7 + ...
Using the Pythagoras Tree L-system, a(0) = #0 = 1, a(1) = #1[0]0 = 5, a(2) = #11[1[0]0]1[0]0 = 14. - _Michael Somos_, Jan 12 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).
Wikipedia, L-system Example 2: Pythagoras Tree

n-th difference of a(n), a(n-1), ..., a(0) is (4, 5, 6, ...).

First differences of A027993.

[2^(n-1)*(n+6)-2: n in [0..30]]; // Vincenzo Librandi, Sep 28 2011

LinearRecurrence[{5,-8,4},{1,5,14},30] (* Harvey P. Dale, Sep 29 2012 *)

{a(n) = if( n<0, 0, [1, 1, 1, 1] * [2, 0, 0, 0; 1, 2, 0, 0; 1, 0, 1, 0; 1, 0, 0, 1]^n * [1, 0, 0, 0]~ )}; /* Michael Somos, Jan 12 2015 */

a(n)=([0,1,0; 0,0,1; 4,-8,5]^n*[1;5;14])[1,1] \\ Charles R Greathouse IV, Jul 19 2016

Main diagonal of the array defined by T(0, j)=j+1 j>=0, T(i, 0)=i+1 i>=0, T(i, j)=T(i-1, j-1)+T(i-1, j)+ 2; a(n)=2^(n-1)*(n+6)-2. - Benoit Cloitre, Jun 17 2003

a(0)=1, a(1)=5, a(2)=14, a(n) = 5*a(n-1)-8*a(n-2)+4*a(n-3). - Vincenzo Librandi, Sep 28 2011

A036542 a(n) = T(n, n), array T given by A047858.

Original entry on oeis.org

1, 3, 11, 34, 93, 236, 571, 1338, 3065, 6904, 15351, 33782, 73717, 159732, 344051, 737266, 1572849, 3342320, 7077871, 14942190, 31457261, 66060268, 138412011, 289406954, 603979753, 1258291176, 2617245671, 5435817958, 11274289125, 23353884644, 48318382051

Offset: 0

Clark Kimberling

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

LinearRecurrence[{6,-13,12,-4},{1,3,11,34},40] (* Harvey P. Dale, Jul 21 2024 *)

Vec((1-3*x+6*x^2-5*x^3)/((1-x)^2*(1-2*x)^2) + O(x^40)) \\ Colin Barker, Feb 20 2016

a(n) = 3*n * 2^(n-1) - n + 1.
From Colin Barker, Feb 20 2016: (Start)
a(n) = 6*a(n-1)-13*a(n-2)+12*a(n-3)-4*a(n-4) for n>3.
G.f.: (1-3*x+6*x^2-5*x^3) / ((1-x)^2*(1-2*x)^2).
(End)

A047861 a(n) = T(4,n), array T given by A047858.

Original entry on oeis.org

1, 6, 17, 41, 93, 205, 445, 957, 2045, 4349, 9213, 19453, 40957, 86013, 180221, 376829, 786429, 1638397, 3407869, 7077885, 14680061, 30408701, 62914557, 130023421, 268435453, 553648125, 1140850685, 2348810237, 4831838205, 9932111869, 20401094653, 41875931133

Offset: 0

Clark Kimberling

n-th difference of a(n), a(n-1), ..., a(0) is (5, 6, 7, ...).

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

[2^(n-1)*(n+8)-3: n in [0..30]]; // Vincenzo Librandi, Sep 28 2011

Vec((1+x-5*x^2)/((1-x)*(1-2*x)^2) + O(x^40)) \\ Colin Barker, Feb 17 2016

Main diagonal of the array defined by T(0, j)=j+1 j>=0, T(i, 0)=i+1 i>=0, T(i, j)=T(i-1, j-1)+T(i-1, j)+ 3; a(n)=2^(n-1)*(n+8)-3. - Benoit Cloitre, Jun 17 2003
a(0)=1, a(1)=6, a(2)=17, a(n) = 5*a(n-1) -8*a(n-2) +4*a(n-3). - Vincenzo Librandi, Sep 28 2011
G.f.: (1+x-5*x^2) / ((1-x)*(1-2*x)^2). - Colin Barker, Feb 17 2016

A047862 a(n) = T(5,n), array T given by A047858.

Original entry on oeis.org

1, 7, 20, 48, 108, 236, 508, 1084, 2300, 4860, 10236, 21500, 45052, 94204, 196604, 409596, 851964, 1769468, 3670012, 7602172, 15728636, 32505852, 67108860, 138412028, 285212668, 587202556, 1207959548, 2483027964, 5100273660, 10468982780, 21474836476

Offset: 0

Clark Kimberling

n-th difference of a(n), a(n-1), ..., a(0) is (6, 7, 8, ...).

More generally the main diagonal of the array defined by T(0,j) = j+1 with j>=0, T(i,0) = i+1 with i>=0, T(i,j) = T(i-1,j-1) + T(i-1,j) + A, is given by T(n,n) = 2^(n-1)*(n+2*A+2)-A. - Benoit Cloitre, Jun 17 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

[2^(n-1)*(n+10)-4: n in [0..30]]; // Vincenzo Librandi, Sep 28 2011

Vec((1+2*x-7*x^2)/((1-x)*(1-2*x)^2) + O(x^40)) \\ Colin Barker, Feb 18 2016

From Benoit Cloitre, Jun 17 2003: (Start)
Main diagonal of the array defined by T(0, j) = j+1 with j>=0, T(i, 0) = i+1 with i>=0, T(i,j) = T(i-1,j-1)+T(i-1,j)+ 4. Therefore, for i = j = n:
a(n) = 2^(n-1)*(n+10)-4. (End)
a(0)=1, a(1)=7, a(2)=20; for n>2, a(n) = 5*a(n-1)-8*a(n-2)+4*a(n-3). - Vincenzo Librandi, Sep 28 2011
a(n) = 2^(n-1)*(n+10)-4. G.f.: (1+2*x-7*x^2) / ((1-x)*(1-2*x)^2). - Colin Barker, Feb 18 2016

A048467 a(n) = T(6,n), array T given by A047858.

Original entry on oeis.org

1, 8, 23, 55, 123, 267, 571, 1211, 2555, 5371, 11259, 23547, 49147, 102395, 212987, 442363, 917499, 1900539, 3932155, 8126459, 16777211, 34603003, 71303163, 146800635, 301989883, 620756987, 1275068411, 2617245691

Offset: 0

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (5, -8, 4).

n-th difference of a(n), a(n-1), ..., a(0) is (7, 8, 9, ...).

[2^(n-1)*(n+12)-5: n in [0..30]]; // Vincenzo Librandi, Sep 28 2011

LinearRecurrence[{5,-8,4},{1,8,23},30] (* or *) CoefficientList[ Series[ (-9x^2+3x+1)/((1-x)(1-2x)^2),{x,0,30}],x] (* Harvey P. Dale, Jul 07 2011 *)

Vec((-9*x^2+3*x+1)/((1-x)*(1-2*x)^2) + O(x^40)) \\ Andrew Howroyd, Feb 15 2018

G.f.: (-9*x^2 + 3*x + 1)/((1-x)*(1-2*x)^2).
a(n) = 5*a(n-1) - 8*a(n-2) + 4*a(n-3). - Harvey P. Dale, Jul 07 2011
a(n) = 2^(n-1)*(n+12) - 5. - Vincenzo Librandi, Sep 28 2011

A048468 a(n) = T(7,n), array T given by A047858.

Original entry on oeis.org

1, 9, 26, 62, 138, 298, 634, 1338, 2810, 5882, 12282, 25594, 53242, 110586, 229370, 475130, 983034, 2031610, 4194298, 8650746, 17825786, 36700154, 75497466, 155189242, 318767098, 654311418, 1342177274, 2751463418, 5637144570, 11542724602, 23622320122

Offset: 0

Clark Kimberling

n-th difference of a(n), a(n-1), ..., a(0) is (8, 9, 10, ...).

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (5, -8, 4).

[2^(n-1)*(n+14)-6: n in [0..30]]; // Vincenzo Librandi, Sep 28 2011

LinearRecurrence[{5,-8,4},{1,9,26},30] (* Harvey P. Dale, Apr 19 2012 *)

Vec((1+4*x-11*x^2)/((1-x)*(1-2*x)^2) + O(x^40)) \\ Colin Barker, Feb 18 2016

a(n) = 2^(n-1)*(n+14)-6. a(0)=1, a(1)=9, a(2)=26, a(n) = 5*a(n-1)-8*a(n-2)+4*a(n-3). - Vincenzo Librandi, Sep 28 2011

a(n) = 2^(n-1)*(n+14)-6. G.f.: (1+4*x-11*x^2) / ((1-x)*(1-2*x)^2). - Colin Barker, Feb 18 2016

A048469 a(n) = T(8,n), array T given by A047858.

Original entry on oeis.org

1, 10, 29, 69, 153, 329, 697, 1465, 3065, 6393, 13305, 27641, 57337, 118777, 245753, 507897, 1048569, 2162681, 4456441, 9175033, 18874361, 38797305, 79691769, 163577849, 335544313, 687865849, 1409286137, 2885681145, 5905580025, 12079595513, 24696061945

Offset: 0

Clark Kimberling

n-th difference of a(n), a(n-1), ..., a(0) is (9, 10, 11, ...).

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

[2^(n-1)*(n+16)-7: n in [0..30]]; // Vincenzo Librandi, Sep 28 2011

LinearRecurrence[{5,-8,4},{1,10,29},40] (* Harvey P. Dale, Aug 15 2020 *)

Main diagonal of the array defined by T(0, j) = j+1 for j>=0, T(i, 0) = i+1 for i>=0, T(i, j) = T(i-1, j-1) + T(i-1, j) +  7. - Benoit Cloitre, Jun 17 2003
a(n) = 2^(n-1)*(n+16)-7. a(0)=1, a(1)=10, a(2)=29, a(n) = 5*a(n-1)-8*a(n-2)+4*a(n-3). - Vincenzo Librandi, Sep 28 2011
G.f.: (1+5*x-13*x^2) / ((1-x)*(1-2*x)^2). - Colin Barker, Feb 18 2016

A195857 a(n) = T(9, n), array T given by A047858.

Original entry on oeis.org

1, 11, 32, 76, 168, 360, 760, 1592, 3320, 6904, 14328, 29688, 61432, 126968, 262136, 540664, 1114104, 2293752, 4718584, 9699320, 19922936, 40894456, 83886072, 171966456, 352321528, 721420280, 1476395000, 3019898872, 6174015480, 12616466424, 25769803768

Offset: 0

Vincenzo Librandi, Sep 28 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

Magma
```
[2^(n-1)*(n+18)-8: n in [0..30]]
```

Vec((1+6*x-15*x^2)/((1-x)*(1-2*x)^2) + O(x^40)) \\ Colin Barker, Aug 24 2016

a(n) = 2^(n-1)*(n+18)-8.
a(0)=1, a(1)=11, a(2)=32, a(n) = 5*a(n-1)-8*a(n-2)+4*a(n-3).
G.f.: (1+6*x-15*x^2) / ((1-x)*(1-2*x)^2). - Colin Barker, Aug 24 2016

A195858 a(n) = T(10, n), array T given by A047858.

Original entry on oeis.org

1, 12, 35, 83, 183, 391, 823, 1719, 3575, 7415, 15351, 31735, 65527, 135159, 278519, 573431, 1179639, 2424823, 4980727, 10223607, 20971511, 42991607, 88080375, 180355063, 369098743, 754974711, 1543503863, 3154116599, 6442450935, 13153337335, 26843545591

Offset: 0

Vincenzo Librandi, Sep 28 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

Cf. A047858.

[2^(n-1)*(n+20)-9: n in [0..30]]; // Vincenzo Librandi, Sep 28 2011

LinearRecurrence[{5,-8,4},{1,12,35},40] (* Harvey P. Dale, Jul 24 2019 *)

Vec((1+7*x-17*x^2)/((1-x)*(1-2*x)^2) + O(x^40)) \\ Colin Barker, Aug 24 2016

a(n) = 2^(n-1)*(n+20)-9.
a(0)=1, a(1)=12, a(2)=35, a(n) = 5*a(n-1)-8*a(n-2)+4*a(n-3).
G.f.: (1+7*x-17*x^2) / ((1-x)*(1-2*x)^2). - Colin Barker, Aug 24 2016

cp's OEIS Frontend

A047859 a(n) = T(2, n), where T is the array given by A047858.

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A047860 a(n) = T(3,n), array T given by A047858.

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A036542 a(n) = T(n, n), array T given by A047858.

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A047861 a(n) = T(4,n), array T given by A047858.

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A047862 a(n) = T(5,n), array T given by A047858.

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A048467 a(n) = T(6,n), array T given by A047858.

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A048468 a(n) = T(7,n), array T given by A047858.

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A048469 a(n) = T(8,n), array T given by A047858.

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