A003063 -id:A003063 - cp's OEIS frontend

A005173 Number of rooted trees with 3 nodes of disjoint sets of labels with union {1..n}. If a node has an empty set of labels then it must have at least two children.

0, 1, 12, 61, 240, 841, 2772, 8821, 27480, 84481, 257532, 780781, 2358720, 7108921, 21392292, 64307941, 193185960, 580082161, 1741295052, 5225982301, 15682141200, 47054812201, 141181213812, 423577195861, 1270798696440, 3812530307041, 11437859356572, 34314114940621

Offset: 1

N. J. A. Sloane

			From _Andrew Howroyd_, Mar 28 2025: (Start)
The a(3) = 12 trees up to relabeling have one of the following 3 forms:
     {}         {1}        {1}
    /  \       /   \        |
  {1} {2,3}   {2}  {3}     {2}
                            |
                           {3}
(End)

F. R. McMorris and T. Zaslavsky, The number of cladistic characters, Math. Biosciences, 54 (1981), 3-10.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..500
F. R. McMorris and T. Zaslavsky, The number of cladistic characters, Math. Biosciences, 54 (1981), 3-10. [Annotated scanned copy]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for sequences related to trees
Index entries for linear recurrences with constant coefficients, signature (6, -11, 6).

Column 3 of A094262.

Cf. A003063.

A005173:=-z*(1+6*z)/(z-1)/(3*z-1)/(2*z-1); # conjectured by Simon Plouffe in his 1992 dissertation

CoefficientList[Series[x (1+6 x)/(1-x)/(1-2 x)/(1-3 x),{x,0,30}],x] (* Harvey P. Dale, Jul 03 2023 *)

G.f.: x*(1 + 6*x) / ((1 - x)*(1 - 2*x)*(1 - 3*x)). [corrected by Ray Chandler, Jun 26 2023]
First differences give A003063, 3^(n-1) - 2^n.
From Andrew Howroyd, Mar 28 2025: (Start)
a(n) = (3^(n+1) - 2^(n+3) + 7)/2.
E.g.f.: (3*exp(x)/2 - 1)*(exp(x) - 1)^2. (End)

More terms from Larry Reeves (larryr(AT)acm.org), Feb 06 2001

Name clarified by Andrew Howroyd, Mar 28 2025

A363024 Primes of the form 3^(k-1) - 2^k.

Original entry on oeis.org

11, 179, 601, 1931, 10456158899, 617669101316651, 984770866999239144049, 2153693958571958138940251, 1570042898793851235488822819, 14130386090585813000157964091, 11972515182561981102976512358583456508049, 19088056323407826758511836230558252318494847619

Offset: 1

Sébastien Tao, May 13 2023

nonn

a(23) has 1117 digits. - Michael S. Branicky, May 26 2023

			a(1) = 3^3 - 2^4 = 27 - 16 = 11 (prime).
a(2) = 3^5 - 2^6 = 243 - 64 = 179 (prime).

Michael S. Branicky, Table of n, a(n) for n = 1..22

Prime terms in A003063.

Cf. A162714, A363375, A162715 (subsequence).

Select[Table[3^(k - 1) - 2^k, {k, 1 , 100}], PrimeQ]

A254027 Table T(n,k) = 3^n - 2^k read by antidiagonals.

Original entry on oeis.org

0, 2, -1, 8, 1, -3, 26, 7, -1, -7, 80, 25, 5, -5, -15, 242, 79, 23, 1, -13, -31, 728, 241, 77, 19, -7, -29, -63, 2186, 727, 239, 73, 11, -23, -61, -127, 6560, 2185, 725, 235, 65, -5, -55, -125, -255, 19682, 6559, 2183, 721, 227, 49, -37, -119, -253, -511, 59048, 19681, 6557, 2179, 713, 211, 17, -101, -247, -509, -1023

Offset: 0

K. G. Stier, Jan 22 2015

Table shows differences of a given power of 3 to the powers of 2 (columns), and differences of the powers of 3 to a given power of 2 (rows), respectively.

Note that positive terms (table's upper right area) and negative terms (lower left area) are separated by an imaginary line with slope -log(3)/log(2) = -1.5849625.. (see A020857). This "border zone" of the table is of interest in terms of how close powers of 3 and powers of 2 can get: i.e., those T(n,k) where k/n is a good rational approximation to log(3)/log(2), see A254351 for numerators k and respective A060528 for denominators n.

			Table begins
   0    2   8  26  80..
  -1    1   7  25  79..
  -3   -1   5  23  73..
  -7   -5   1  19  65..
  -15 -13  -7  11  49..
  ..   ..  ..  ..  ..

K. G. Stier, Table of n, a(n) for n = 0..5150

Row 0 (=3^n-1) is A024023.
Row 1 (=3^n-2) is A058481.
Row 2 (=3^n-4) is A168611.
Column 0 (=1-2^n) is (-1)A000225.
Column 1 (=3-2^n) is (-1)A036563.
Column 2 (=9-2^n) is (-1)A185346.
Column 3 (=27-2^n) is (-1)A220087.
0,0-Diagonal (=3^n-2^n) is A001047.
1,0-Diagonal (=3^n-2^(n-1)) for n>0 is A083313 or A064686.
0,1-Diagonal (=3^n-2^(n+1)) is A003063.
0,2-Diagonal (=3^n-2^(n+2)) is A214091.

Table[3^(n-k) - 2^k, {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 18 2017 *)

for(i=0, 10, {
     for(j=0, i, print1((3^(i-j)-2^j),", "))
});

A369490 a(n) = 3^(n+1) + 2*(-2)^(n+1).

Original entry on oeis.org

-1, 17, 11, 113, 179, 857, 1931, 7073, 18659, 61097, 173051, 539633, 1577939, 4815737, 14283371, 43177793, 128878019, 387944777, 1161212891, 3488881553, 10456158899, 31389448217, 94126401611, 282463090913, 847221500579, 2542000046057

Offset: 0

Philippe Deléham, Jan 24 2024

Index entries for linear recurrences with constant coefficients, signature (1, 6).

Cf. A000079, A000244, A003063, A015441, A085279.

LinearRecurrence[{1,6},{-1,17},26] (* James C. McMahon, Jan 30 2024 *)

def A369490(n): return 3**(n+1)+(1<Chai Wah Wu, Feb 25 2024

a(n) = a(n-1) + 6*a(n-2); a(0) = -1, a(1) = 17.
G.f.: (18*x-1)/((1+2*x)*(1-3*x)).
a(2*n) = A003063(2*n+2).
a(2*n+1) = A085279(2*n+3).
a(n) = 18*A015441(n) - A015441(n+1).

A239926 3^(p-1)-2^(p+1) for primes p > 3.

Original entry on oeis.org

17, 473, 54953, 515057, 42784577, 386371913, 31364282393, 22875718713137, 205886837127353, 150094360419092177, 12157661061010417697, 109418971539326314793, 8862937838177524385273, 6461081871212274789450257, 4710128696093323330314756713

Offset: 1

Vincenzo Librandi, Jun 17 2014

3^(p-1)-2^(p+1) can be written as (3^((p-1)/2)-2^((p+1)/2))*(3^((p-1)/2)+2^((p+1)/2)). Since 3^((p-1)/2)-2^((p+1)/2) > 1 for p > 5, these numbers are all composite after 17 = (3^2-2^3)*(3^2+2^3).

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A000040, A003063, A135171 (numbers of the form 3^p-2^p with p prime), A214091 (supersequence).

[3^(p-1)-2^(p+1): p in PrimesInInterval(4,100)];

Table[3^(Prime[n] - 1) - 2^(Prime[n] + 1), {n, 3, 100}]

A232774 Triangle T(n,k), read by rows, given by T(n,0)=1, T(n,1)=2^(n+1)-n-2, T(n,n)=(-1)^(n-1) for n > 0, T(n,k)=T(n-1,k)-T(n-1,k-1) for 1 < k < n.

Original entry on oeis.org

1, 1, 1, 1, 4, -1, 1, 11, -5, 1, 1, 26, -16, 6, -1, 1, 57, -42, 22, -7, 1, 1, 120, -99, 64, -29, 8, -1, 1, 247, -219, 163, -93, 37, -9, 1, 1, 502, -466, 382, -256, 130, -46, 10, -1, 1, 1013, -968, 848, -638, 386, -176, 56, -11, 1, 2036, -1981, 1816, -1486, 1024

Offset: 0

Philippe Deléham, Nov 30 2013

Row sums are A000079(n) = 2^n.
Diagonal sums are A024493(n+1) = A130781(n).
Sum_{k=0..n} T(n,k)*x^k = -A003063(n+2), A159964(n), A000012(n), A000079(n), A001045(n+2), A056450(n), (-1)^(n+1)*A232015(n+1) for x = -2, -1, 0, 1, 2, 3, 4 respectively.

			Triangle begins:
  1;
  1,    1;
  1,    4,   -1;
  1,   11,   -5,   1;
  1,   26,  -16,   6,   -1;
  1,   57,  -42,  22,   -7,   1;
  1,  120,  -99,  64,  -29,   8,   -1;
  1,  247, -219, 163,  -93,  37,   -9,  1;
  1,  502, -466, 382, -256, 130,  -46, 10,  -1;
  1, 1013, -968, 848, -638, 386, -176, 56, -11, 1;

Columns: A000012, A000295, -A002662, A002663, -A002664, A035038, -A035039, A035040, -A035041, A035042.
Cf. A000079, A055248, A024493, A130781, A000302.
Cf. also A003063, A159964, A000012, A001045, A056450, A232015.

G.f.: Sum_{n>=0, k=0..n} T(n,k)*y^k*x^n=(1+2*(y-1)*x)/((1-2*x)*(1+(y-1)*x)).
|T(2*n,n)| = 4^n = A000302(n).
T(n,k) = (-1)^(k-1) * (Sum_{i=0..n-k} (2^(i+1)-1) * binomial(n-i-1,k-1)) for 0 < k <= n and T(n,0) = 1 for n >= 0. - Werner Schulte, Mar 22 2019

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A005173 Number of rooted trees with 3 nodes of disjoint sets of labels with union {1..n}. If a node has an empty set of labels then it must have at least two children.

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A363024 Primes of the form 3^(k-1) - 2^k.

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A254027 Table T(n,k) = 3^n - 2^k read by antidiagonals.

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A369490 a(n) = 3^(n+1) + 2*(-2)^(n+1).

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A239926 3^(p-1)-2^(p+1) for primes p > 3.

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A232774 Triangle T(n,k), read by rows, given by T(n,0)=1, T(n,1)=2^(n+1)-n-2, T(n,n)=(-1)^(n-1) for n > 0, T(n,k)=T(n-1,k)-T(n-1,k-1) for 1 < k < n.

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