A210448 -id:A210448 - cp's OEIS frontend

A217764 Array defined by a(n,k) = floor((k+2)/2)3^n - floor((k+1)/2)2^n, read by antidiagonals.

1, 3, 0, 9, 1, 1, 27, 5, 4, 0, 81, 19, 14, 2, 1, 243, 65, 46, 10, 5, 0, 729, 211, 146, 38, 19, 3, 1, 2187, 665, 454, 130, 65, 15, 6, 0, 6561, 2059, 1394, 422, 211, 57, 24, 4, 1, 19683, 6305, 4246, 1330, 665, 195, 84, 20, 7, 0, 59049, 19171, 12866, 4118, 2059, 633, 276, 76, 29, 5, 1

Offset: 0

Ross La Haye, Mar 23 2013

Columns 0,1,2,3 respectively correspond to relations R_3, R_4, R_0, R_1 defined in La Haye paper listed below.

			a(4,4) = 211 because floor((4+2)/2)*3^4 - floor((4+1)/2)*2^4 = 3*3^4 - 2*2^4 = 243 - 32 = 211.

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.

Cf. a(1,k) = A084964(k+2); a(n,0) = A000244(n); a(n,1) = A001047(n); a(n,2) = A027649(n); a(n,3) = A056182(n); a(n,4) = A001047(n+1); a(n,5) = A210448(n); a(n,6) = A166060(n); a(n,7) = A145563(n); a(n,8) = A102485(n).

a(n,k) = floor((k+2)/2)*3^n - floor((k+1)/2)*2^n. a(n,k) = 5*a(n-1,k) - 6*a(n-2,k); a(0,k) = floor((k+2)/2) - floor((k+1)/2), a(1,k) = floor((k+2)/2)*3 - floor((k+1)/2)*2.

A145563 a(0)=0 and a(n+1) = 3*a(n) + 2^(n+2).

Original entry on oeis.org

0, 4, 20, 76, 260, 844, 2660, 8236, 25220, 76684, 232100, 700396, 2109380, 6344524, 19066340, 57264556, 171924740, 516036364, 1548633380, 4646948716, 13942943300, 41833024204, 125507461220, 376539160876, 1129651037060, 3389020220044, 10167194877860

Offset: 0

N. J. A. Sloane, Mar 18 2009

nonn

Suggested by a discussion on the Sequence Fans Mailing List; the formula is due to Andrew V. Sutherland.

First differences of A255459. - Klaus Purath, Apr 25 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..170
Index entries for linear recurrences with constant coefficients, signature (5,-6).

Cf. A001047, A056182, A210448, A217764, A245804.

[ 4*(3^n - 2^n): n in [0..50]]; // Vincenzo Librandi, Apr 24 2011

CoefficientList[Series[4x/((1-2x)(1-3x)),{x,0,40}],x] (* or *) RecurrenceTable[{a[0]==0, a[n]==(3a[n-1]+2^(n+1))},a,{n,40}] (* Harvey P. Dale, Apr 24 2011 *)

a(n) = 4*(3^n - 2^n) \\ Felix Fröhlich, Sep 01 2018

From R. J. Mathar, Mar 18 2009: (Start)
a(n) = 4*(3^n - 2^n) = 4*A001047(n).
G.f.: 4*x/((1-2*x)*(1-3*x)). (End)
a(n) = A056182(n)*2. - Omar E. Pol, Mar 18 2009
a(n) = A217764(n,7). - Ross La Haye, Mar 27 2013
From Klaus Purath, Apr 25 2020: (Start)
a(n) = 5*a(n-1) - 6*a(n-2).
a(n) = 2*A210448(n) - A056182(n).
a(n) = (A056182(n) + A245804(n+1))/2. (End)

A245804 a(n) = 23^n - 32^n.

Original entry on oeis.org

-1, 0, 6, 30, 114, 390, 1266, 3990, 12354, 37830, 115026, 348150, 1050594, 3164070, 9516786, 28599510, 85896834, 257887110, 774054546, 2322950070, 6970423074, 20914414950, 62749536306, 188261191830, 564808741314, 1694476555590, 5083530330066, 15250792316790

Offset: 0

Vincenzo Librandi, Aug 03 2014

Essentially 2 * A210448. - Joerg Arndt, Aug 03 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-6).

Cf. A007283, A008776.

Magma
```
[2*3^n-3*2^n: n in [0..40]];
```

I:=[-1,0]; [n le 2 select I[n] else 5*Self(n-1)-6*Self(n-2): n in [1..30]];

CoefficientList[Series[(-1 + 5 x)/((1 - 2 x) (1 - 3 x)), {x, 0, 30}], x]
Table[(2 3^n - 3 2^n), {n, 0, 30}] (* Vincenzo Librandi, Aug 04 2014 *)

G.f.: (-1 +5*x)/((1-2*x)(1-3*x)).
a(n) = 5*a(n-1) -6*a(n-2) for n>1.
a(n) = A008776(n) - A007283(n). - Michel Marcus, Aug 03 2014

A079268 Triangle read by rows: d(n,k) = number of decreasing labeled trees with n nodes and largest leaf <= k, for 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 7, 15, 15, 1, 15, 57, 105, 105, 1, 31, 195, 561, 945, 945, 1, 63, 633, 2685, 6555, 10395, 10395, 1, 127, 1995, 12105, 40725, 89055, 135135, 135135, 1, 255, 6177, 52605, 237555, 684495, 1381905, 2027025, 2027025, 1, 511, 18915, 223161

Offset: 1

Jeremy Martin (martin(AT)math.umn.edu), Feb 05 2003

			Triangle begins
1,
1, 1,
1, 3, 3,
1, 7, 15, 15,
1, 15, 57, 105, 105,
1, 31, 195, 561, 945, 945,
1, 63, 633, 2685, 6555, 10395, 10395,
...

J. L. Martin, The slopes determined by n points in the plane.
Martin, Jeremy L. The slopes determined by n points in the plane. Duke Math. J. 131 (2006), no. 1, 119-165 (also arXiv math.AG/0302106, but beware errors).

First three columns are A000012, A000225, A210448, rightmost two diagonals are both A001147, difference of second- and third-rightmost diagonals is A000165.

Recurrence: d(n, k) = 1 for n=0 or k=1, d(n, k) = 0 for n>0 and either k<0 or k>n, d(n, k) = d(n-1, k) + d(n, k-1) + Sum_{w=0..k-2, x=0..n-k-1} binomial(k-1, w) * binomial(n-k, x) * d(n-k+w-x, w+1) * d(k-w+x, k-w-1).

cp's OEIS Frontend

A217764 Array defined by a(n,k) = floor((k+2)/2)*3^n - floor((k+1)/2)*2^n, read by antidiagonals.

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A145563 a(0)=0 and a(n+1) = 3*a(n) + 2^(n+2).

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A245804 a(n) = 2*3^n - 3*2^n.

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Magma

Magma

Mathematica

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A079268 Triangle read by rows: d(n,k) = number of decreasing labeled trees with n nodes and largest leaf <= k, for 1 <= k <= n.

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Formula

A217764 Array defined by a(n,k) = floor((k+2)/2)3^n - floor((k+1)/2)2^n, read by antidiagonals.

A245804 a(n) = 23^n - 32^n.