A098293 -id:A098293

Search: a098293 -id:a098293

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A258439

Powers of 3 alternating with powers of 2.

+10
1

1, 1, 3, 2, 9, 4, 27, 8, 81, 16, 243, 32, 729, 64, 2187, 128, 6561, 256, 19683, 512, 59049, 1024, 177147, 2048, 531441, 4096, 1594323, 8192, 4782969, 16384, 14348907, 32768, 43046721, 65536, 129140163, 131072, 387420489, 262144, 1162261467

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

a(n)*A098293(n) = A000400(floor(n/2)).

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,5,0,-6).

FORMULA

a(n) = ((5+(-1)^n)/2)^((2*n-1+(-1)^n)/4).

a(n) = 5*a(n-2)-6*a(n-4). - Colin Barker, May 30 2015

G.f.: -(3*x^3+2*x^2-x-1) / ((2*x^2-1)*(3*x^2-1)). - Colin Barker, May 30 2015

MAPLE

seq(op([3^n, 2^n]), n=0..20); # Muniru A Asiru, Jul 16 2018

MATHEMATICA

Flatten[Table[{3^n, 2^n}, {n, 0, 25}]] (* Vincenzo Librandi, Jul 17 2018 *)

PROG

(PARI) Vec(-(3*x^3+2*x^2-x-1)/((2*x^2-1)*(3*x^2-1)) + O(x^100)) \\ Colin Barker, May 30 2015

(GAP) Flat(List([0..20], n->[3^n, 2^n])); # Muniru A Asiru, Jul 16 2018

(Magma) &cat[[3^n, 2^n]: n in [0..35]]; // Vincenzo Librandi, Jul 17 2018

CROSSREFS

Cf. A098293, A006899.

KEYWORD

nonn,easy

AUTHOR

Luce ETIENNE, May 30 2015

STATUS

approved

A375966

Powers of 3 alternating with powers of 4.

+10
1

1, 1, 3, 4, 9, 16, 27, 64, 81, 256, 243, 1024, 729, 4096, 2187, 16384, 6561, 65536, 19683, 262144, 59049, 1048576, 177147, 4194304, 531441, 16777216, 1594323, 67108864, 4782969, 268435456, 14348907, 1073741824, 43046721, 4294967296, 129140163, 17179869184

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..35.

Index entries for linear recurrences with constant coefficients, signature (0,7,0,-12).

FORMULA

a(n) = 7*a(n-2) - 12*a(n-4) for n >= 4.

From Stefano Spezia, Sep 06 2024: (Start)

G.f.: (1 + x - 4*x^2 - 3*x^3)/((1 - 2*x)*(1 + 2*x)*(1 - 3*x^2)).

a(n) = (4*3^(n/2)*A059841(n) - (-2)^n + 2^n)/4.

E.g.f.: cosh(sqrt(3)*x) + cosh(x)*sinh(x). (End)

MATHEMATICA

seq[len_] := Module[{m = Ceiling[len/2] - 1}, Riffle @@ Map[#^Range[0, m] &, {3, 4}]]; seq[36] (* Amiram Eldar, Sep 05 2024 *)

PROG

(Python)

def A375966(n): return 1<<(n^1) if n&1 else 3**(n>>1) # Chai Wah Wu, Sep 24 2024

CROSSREFS

Cf. A000244 and A000302 interleaved.

Cf. A000035, A059841, A072345, A098293.

KEYWORD

nonn,easy

AUTHOR

Paul Curtz, Sep 04 2024

STATUS

approved

A097294

Contains exactly once every triple i,j,k such that i>j>k>0.

+10
0

3, 2, 1, 4, 2, 1, 5, 2, 1, 4, 3, 1, 6, 2, 1, 5, 3, 1, 4, 3, 2, 7, 2, 1, 6, 3, 1, 5, 4, 1, 5, 3, 2, 8, 2, 1, 7, 3, 1, 6, 4, 1, 6, 3, 2, 5, 4, 2, 9, 2, 1, 8, 3, 1, 7, 4, 1, 6, 5, 1, 7, 3, 2, 6, 4, 2, 5, 4, 3, 10, 2, 1, 9, 3, 1, 8, 4, 1, 7, 5, 1, 8, 3, 2, 7, 4, 2, 6, 5, 2, 6, 4, 3, 11, 2, 1, 10, 3, 1, 9, 4, 1, 8, 5

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..104.

FORMULA

Obtained by reversing triples in A097293.

CROSSREFS

Cf. A098293.

KEYWORD

nonn

AUTHOR

Clark Kimberling, Aug 05 2004

STATUS

approved

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