A064686 -id:A064686 - cp's OEIS frontend

A003063 a(n) = 3^(n-1) - 2^n.

-1, -1, 1, 11, 49, 179, 601, 1931, 6049, 18659, 57001, 173051, 523249, 1577939, 4750201, 14283371, 42915649, 128878019, 386896201, 1161212891, 3484687249, 10456158899, 31372671001, 94126401611, 282395982049, 847221500579, 2541731610601, 7625329049531, 22876255584049

Offset: 1

Henrik Johansson (Henrik.Johansson(AT)Nexus.SE)

Binomial transform of A000918: (-1, 0, 2, 6, 14, 30, ...). - Gary W. Adamson, Mar 23 2012

This sequence demonstrates 2^n as a loose lower bound for g(n) in Waring's problem. Since 3^n > 2(2^n) for all n > 2, the number 2^(n + 1) - 1 requires 2^n n-th powers for its representation since 3^n is not available for use in the sum: the gulf between the relevant powers of 2 and 3 widens considerably as n gets progressively larger. - Alonso del Arte, Feb 01 2013

			a(3) = 1 because 3^2 - 2^3 = 9 - 8 = 1.
a(4) = 11 because 3^3 - 2^4 = 27 - 16 = 11.
a(5) = 49 because 3^4 - 2^5 = 81 - 32 = 49.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
D. Knuth, Letter to N. J. A. Sloane, date unknown
Index entries for linear recurrences with constant coefficients, signature (5,-6).

Cf. A000918, A056182 (first differences), A064686, A083313, A214091, A369490.
Cf. A363024 (prime terms).
From the third term onward the first differences of A005173.
Difference between two leftmost columns of A090888.
A diagonal in A254027.
Right edge of irregular triangle A252750.

[3^(n-1) -2^n: n in [1..30]]; // G. C. Greubel, Nov 03 2022

Table[3^(n-1) - 2^n, {n, 25}] (* Alonso del Arte, Feb 01 2013 *)
LinearRecurrence[{5,-6},{-1,-1},30] (* Harvey P. Dale, Feb 02 2015 *)

a(n)=3^(n-1)-2^n \\ Charles R Greathouse IV, Oct 07 2015

[3^(n-1) -2^n for n in range(1,31)] # G. C. Greubel, Nov 03 2022

Let b(n) = 2*(3/2)^n - 1. Then a(n) = -b(1-n)*3^(n-1) for n > 0. A083313(n) = A064686(n) = b(n)*2^(n-1) for n > 0. - Michael Somos, Aug 06 2006
From Colin Barker, May 27 2013: (Start)
a(n) = 5*a(n-1) - 6*a(n-2).
G.f.: -x*(1-4*x) / ((1-2*x)*(1-3*x)). (End)
E.g.f.: (1/3)*(2 - 3*exp(2*x) + exp(3*x)). - G. C. Greubel, Nov 03 2022

A few more terms from Alonso del Arte, Feb 01 2013

A083313 a(0)=1; a(n) = 3^n - 2^(n-1) for n >= 1.

Original entry on oeis.org

1, 2, 7, 23, 73, 227, 697, 2123, 6433, 19427, 58537, 176123, 529393, 1590227, 4774777, 14332523, 43013953, 129074627, 387289417, 1161999323, 3486260113, 10459304627, 31378962457, 94138984523, 282421147873, 847271832227, 2541832273897, 7625530376123

Offset: 0

Paul Barry, Apr 24 2003

Essentially the same as A064686.
Binomial transform of A051049.
Number of skinny Boolean functions f(x_1,...,x_n) that are also Horn functions. - Hugo Pfoertner, Mar 04 2019

Donald E. Knuth, The Art of Computer Programming, Vol. 4, fascicle 1, section 7.1.4, pp. 134, 138, 139, 219, answer to exercise 172, Addison-Wesley, 2009.

Index entries for linear recurrences with constant coefficients, signature (5,-6).

Cf. A083314.

[(2*3^n-2^n+0^n)/2: n in [0..30]]; // Vincenzo Librandi, Feb 01 2015

A083313 := proc(n)
    if n = 0 then
        1;
    else
        3^n-2^(n-1) ;
    end if;
end proc: # R. J. Mathar, Aug 01 2013

CoefficientList[Series[((1 - x) + (1 - 2 x) (1 - 3 x)) / (2 (1 - 2 x) (1 - 3 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 01 2015 *)
LinearRecurrence[{5,-6},{1,2,7},30] (* Harvey P. Dale, Sep 04 2017 *)

Vec(((1-x)+(1-2*x)*(1-3*x))/(2*(1-2*x)*(1-3*x)) + O(x^30)) \\ Michel Marcus, Jan 31 2015

print1(1,", ",s=2,", " );for(k=2,27,s=2^(k-2)+3*s;print1(s,", ")) \\ Hugo Pfoertner, Mar 04 2019

a(n) = (2*3^n - (2^n - 0^n))/2.
a(0) = 1, a(n) = 3^n - 2^(n-1) for n >= 1.
G.f.: ((1-x) + (1-2*x)*(1-3*x))/(2*(1-2*x)*(1-3*x)).
E.g.f.: (2*exp(3*x) - exp(2*x) + exp(0))/2.
a(n) = A090888(n-1, 4), for n > 0. - Ross La Haye, Sep 21 2004
Let b(n) = 2*(3/2)^n - 1. Then A003063(n) = -b(1-n)*3^(n-1) for n > 0. a(n) = A064686(n) = b(n)*2^(n-1) for n > 0. - Michael Somos, Aug 06 2006
From Alex Ratushnyak, Jul 03 2012: (Start)
a(n) mod 100 = 23 for n = 4*k-1, k >= 1.
a(n) mod 100 = 27 for n = 4*k+1, k >= 1.
(End)

Better name by Alex Ratushnyak, Jul 02 2012

A064544 Biquanimous numbers (or biquams): group the digits into two pieces (not necessarily equal or in order) with the same sum.

Original entry on oeis.org

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 110, 112, 121, 123, 132, 134, 143, 145, 154, 156, 165, 167, 176, 178, 187, 189, 198, 202, 211, 213, 220, 224, 231, 235, 242, 246, 253, 257, 264, 268, 275, 279, 286, 297, 303, 312, 314, 321, 325, 330, 336, 341, 347, 352, 358

Offset: 0

David W. Wilson, Oct 09 2001

This sequence is 10-automatic (decimal expansions form a regular language accepted by a finite automaton).

			143 is in the sequence because its digits {1, 4, 3} may be grouped so that 1+3 = 4.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Index entries for 10-automatic sequences.

Cf. A064671, A064686 (number of n-digit base-3 biquams), A065023, A065024, A065025.

is(n) = { my (d=digits(n), s=[0]); for (k=1, #d, s=setunion(apply(v -> v+d[k], s), apply(v -> v-d[k], s))); setsearch(s, 0)>0 } \\ Rémy Sigrist, Jan 23 2021

A053152 Number of 2-element intersecting families whose union is an n-element set.

Original entry on oeis.org

0, 2, 9, 32, 105, 332, 1029, 3152, 9585, 29012, 87549, 263672, 793065, 2383292, 7158069, 21490592, 64504545, 193579172, 580868589, 1742867912, 5229128025, 15688432652, 47067395109, 141206379632, 423627527505, 1270899359732, 3812731633629, 11438262009752

Offset: 1

Vladeta Jovovic, Goran Kilibarda, Feb 28 2000

G. C. Greubel, Table of n, a(n) for n = 1..1000
V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, (in Russian), Diskretnaya Matematika, 11 (1999), no. 4, 127-138.
V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, (English translation), Discrete Mathematics and Applications, 9, (1999), no. 6.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A036239, A064686 (first differences).

[Floor((3^n-2^n)/2): n in [1..30]]; // Vincenzo Librandi, Mar 17 2015

A053152:=n->floor((3^n-2^n)/2): seq(A053152(n), n=1..30); # Wesley Ivan Hurt, Mar 19 2015

CoefficientList[Series[x (2 - 3 x) / ((1 - x) (1 - 2 x) (1 - 3 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 17 2015 *)
LinearRecurrence[{6,-11,6}, {0,2,9}, 50] (* G. C. Greubel, Oct 06 2017 *)

for(n=1,50, print1((1/2)*(3^n -2^n -1), ", ")) \\ G. C. Greubel, Oct 06 2017

[(3^n - 1)/2-2^(n-1) for n in range(1,27)] # Zerinvary Lajos, Jun 05 2009

a(n) = (1/2!)*(3^n-2^n-1).
From Colin Barker, Jun 26 2012: (Start)
a(n) = 6*a(n-1)-11*a(n-2)+6*a(n-3).
G.f.: x^2*(2-3*x)/((1-x)*(1-2*x)*(1-3*x)). (End)
a(n) = floor((3^n-2^n)/2). - Wesley Ivan Hurt, Mar 16 2015

More terms from James Sellers, Mar 01 2000

a(27)-a(28) from Vincenzo Librandi, Mar 17 2015

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),", "))
});

cp's OEIS Frontend

A003063 a(n) = 3^(n-1) - 2^n.

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A083313 a(0)=1; a(n) = 3^n - 2^(n-1) for n >= 1.

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A064544 Biquanimous numbers (or biquams): group the digits into two pieces (not necessarily equal or in order) with the same sum.

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PARI

A053152 Number of 2-element intersecting families whose union is an n-element set.

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Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

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

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Author

Keywords

Comments

Examples

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Programs

Mathematica

PARI