A033113 -id:A033113 - cp's OEIS frontend

A097137 Convolution of 3^n and floor(n/2).

0, 0, 1, 4, 14, 44, 135, 408, 1228, 3688, 11069, 33212, 99642, 298932, 896803, 2690416, 8071256, 24213776, 72641337, 217924020, 653772070, 1961316220, 5883948671, 17651846024, 52955538084, 158866614264, 476599842805

Offset: 0

Paul Barry, Jul 29 2004

nonn

a(n+1) gives partial sums of A033113 and second partial sums of A015518(n+1). Binomial transform of {0,0,1,1,4,4,16,16,...}.

Partial sums of floor(3^n/8) = round(3^n/8). - Mircea Merca, Dec 28 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,3).

Column k=3 of A368296.

Cf. A033113.

a:=[0,0,1,4];; for n in [5..30] do a[n]:=4*a[n-1]-2*a[n-2]-4*a[n-3] +3*a[n-4]; od; a; # G. C. Greubel, Jul 14 2019

[Round((3*3^n-4*n-4)/16): n in [0..30]]; // Vincenzo Librandi, Jun 25 2011

A097137 := proc(n) add( floor(3^i/8),i=0..n) ; end proc:

CoefficientList[Series[x^2/((1-x)^2(1-3x)(1+x)),{x,0,30}],x]  (* Harvey P. Dale, Mar 11 2011 *)

my(x='x+O('x^30)); concat([0,0], Vec(x^2/((1-x)^2*(1-3*x)*(1+x)))) \\ G. C. Greubel, Jul 14 2019

(x^2/((1-x)^2*(1-3*x)*(1+x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jul 14 2019

G.f.: x^2/((1-x)^2*(1-3*x)*(1+x)).
a(n) = 4*a(n-1) - 2*a(n-2) - 4*a(n-3) + 3*a(n-4).
a(n) = Sum_{k=0..n} floor((n-k)/2)*3^k = Sum_{k=0..n} floor(k/2)*3^(n-k).
From Mircea Merca, Dec 26 2010: (Start)
a(n) = round((3*3^n - 4*n - 4)/16) = floor((3*3^n - 4*n - 3)/16) = ceiling((3*3^n - 4*n - 5)/16) = round((3*3^n - 4*n - 3)/16).
a(n) = a(n-2) + (3^(n-1)-1)/2, n > 2. (End)
a(n) = (floor(3^(n+1)/8) - floor((n+1)/2))/2. - Seiichi Manyama, Dec 22 2023

A052929 Expansion of g.f. (2-3x-x^2)/((1-x^2)(1-3*x)).

Original entry on oeis.org

2, 3, 10, 27, 82, 243, 730, 2187, 6562, 19683, 59050, 177147, 531442, 1594323, 4782970, 14348907, 43046722, 129140163, 387420490, 1162261467, 3486784402, 10460353203, 31381059610, 94143178827, 282429536482, 847288609443, 2541865828330, 7625597484987, 22876792454962

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 915.
Index entries for linear recurrences with constant coefficients, signature (3,1,-3).

Cf. A052531: 2^n + (1+(-1)^n)/2.

List([0..30], n-> 3^n + (1+(-1)^n)/2 ); # G. C. Greubel, Oct 17 2019

[&+[(-1)^k+2^k*Binomial(n,k): k in [0..n]]: n in [0..30]]; // Bruno Berselli, Aug 27 2013

spec:= [S, {S=Union(Sequence(Prod(Z,Z)), Sequence(Union(Z,Z,Z)))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
seq(3^n + (1+(-1)^n)/2, n=0..30); # G. C. Greubel, Oct 17 2019

Table[3^n + (1+(-1)^n)/2, {n, 0, 30}] (* Bruno Berselli, Aug 27 2013 *)
LinearRecurrence[{3, 1, -3}, {2, 3, 10}, 40] (* Vincenzo Librandi, Mar 09 2018 *)
Table[3^n + Fibonacci[n+1,0], {n,0,30}] (* G. C. Greubel, Oct 17 2019 *)

x='x+O('x^30); Vec((2-3*x-x^2)/((1-x^2)*(1-3*x))) \\ Altug Alkan, Mar 09 2018

[3^n + (1+(-1)^n)/2 for n in (0..30)] # G. C. Greubel, Oct 17 2019

G.f.: (2-3*x-x^2)/((1-x^2)*(1-3*x)).
a(n) = 3*a(n-1) + a(n-2) - 3*a(n-3), a(0)=2, a(1)=3, a(2)=10.
a(n) = 3^n + Sum_{alpha=RootOf(-1+z^2)} alpha^(-n)/2.
a(n) = 2*A033113(n+1) - 3*A033113(n) - A033113(n-1). - R. J. Mathar, Nov 28 2011
From Bruno Berselli, Aug 27 2013: (Start)
a(n) = 3^n + (1 + (-1)^n)/2.
a(n) = Sum_{k=0..n} (-1)^k + 2^k*binomial(n,k). (End)
E.g.f.: exp(3*x) + cosh(x). - Elmo R. Oliveira, Mar 16 2025

More terms from James Sellers, Jun 05 2000

A089815 a(n) = floor((n+2)^(n+2)/((n+2)^2-1)).

Original entry on oeis.org

1, 3, 17, 130, 1333, 17157, 266305, 4842756, 101010101, 2377597255, 62350352785, 1802828015430, 56984650387477, 1954883439200265, 72340172838076673, 2872362020438669320, 121815504877079063701, 5495610154611982192011, 262801002506265664160401

Offset: 0

Paul Barry, Nov 12 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A000975, A033113-A033119, A059848, A062864, A056830.

[Floor((n+2)^(n+2)/((n+2)^2-1)): n in [0..50]]; // G. C. Greubel, Oct 10 2017

A089815:=n->floor((n+2)^(n+2)/((n+2)^2-1)): seq(A089815(n), n=0..30); # Wesley Ivan Hurt, Apr 11 2017

Table[Floor[(n + 2)^(n + 2)/((n + 2)^2 - 1)], {n, 0, 50}] (* G. C. Greubel, Oct 10 2017 *)

for(n=0,50, print1(floor((n+2)^(n+2)/((n+2)^2-1)), ", ")) \\ G. C. Greubel, Oct 10 2017

A089816 a(n) = floor((n+3)^(n+2)/((n+3)^2-1)).

Original entry on oeis.org

1, 4, 26, 222, 2451, 33288, 538084, 10101010, 216145205, 5195862732, 138679078110, 4070332170534, 130325562613351, 4521260802379792, 168962471790509960, 6767528048726614650, 289242639716420115369

Offset: 0

Paul Barry, Nov 12 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A000975, A033113-A033119, A059848, A062864, A056830.

[Floor((n+3)^(n+2)/((n+3)^2-1)): n in [0..25]]; // G. C. Greubel, Oct 11 2017

Table[Floor[(n + 3)^(n + 2)/((n + 3)^2 - 1)], {n, 0, 50}] (* G. C. Greubel, Oct 11 2017 *)

for(n=0, 25, print1(floor((n+3)^(n+2)/((n+3)^2-1)), ", ")) \\ G. C. Greubel, Oct 11 2017

A340395 a(n) = A340131(A001006(n)).

Original entry on oeis.org

5, 15, 50, 150, 455, 1365, 4100, 12300, 36905, 110715, 332150, 996450, 2989355, 8968065, 26904200, 80712600, 242137805, 726413415, 2179240250, 6537720750, 19613162255, 58839486765, 176518460300, 529555380900, 1588666142705, 4765998428115, 14297995284350

Offset: 2

Gennady Eremin, Jan 06 2021

This sequence is associated with A340131, whose terms are sorted by the length of their ternary code. Elements with the same length of ternary code form a range that has a maximum. The maximal term of the n-range (a set of elements with ternary code length n in A340131) is a(n). Example: numbers 29, 33, 44, 45 and 50 have a ternary length of 4 (see A340131), respectively a(4) = 50.

Ternary code for a(n) is 12..12 for even n and 12..120 for odd n.

			A001006(2) = 2, so a(2) = A340131(2) = 5.
A001006(3) = 4, so a(3) = A340131(4) = 15, etc.

Andrew Howroyd, Table of n, a(n) for n = 2..1000
Gennady Eremin, Arithmetization of well-formed parenthesis strings. Motzkin Numbers of the Second Kind, arXiv:2012.12675 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (3,1,-3).

Subsequence of A340131.

Vec(5/(1 - 3*x - x^2 + 3*x^3) + O(x^30)) \\ Andrew Howroyd, Jan 08 2021

a(n) = 5*3^(n-2*k)*(9^k-1)/8 where k = floor(n/2).
a(n+1) = 3*a(n) for even n >= 2; a(n+1) = 3*a(n)+5 for odd n >= 3.
a(n) = 5*A033113(n-1).
a(n) = (5/8)*(3^n - (-1)^(n-1) - 2).
a(n) = 2*a(n-1) + 3*a(n-2) + 5 for n > 3.
From Stefano Spezia, Jan 06 2021: (Start)
G.f.: 5*x^2/(1 - 3*x - x^2 + 3*x^3).
a(n) = 3*a(n-1) + a(n-2) - 3*a(n-3) for n > 4. (End)

A270346 a(n) is the number whose base-11 digits are, in order, the first n terms of the simple periodic sequence: repeat 2,3,5,7.

Original entry on oeis.org

2, 25, 280, 3087, 33959, 373552, 4109077, 45199854, 497198396, 5469182359, 60161005954, 661771065501, 7279481720513, 80074298925646, 880817288182111, 9688990170003228, 106578891870035510, 1172367810570390613, 12896045916274296748, 141856505079017264235

Offset: 1

Harvey P. Dale, Mar 15 2016

The periodic sequence comprises the first four primes, and the selected base is the fifth prime.

			a(8) = 45199854 = 23572357_11.

Index entries for linear recurrences with constant coefficients, signature (11,0,0,1,-11).

Cf. A033113.

a:=[2,25,280,3087,33959];; for n in [6..30] do a[n]:=11*a[n-1]+a[n-4]-11*a[n-5]; od; a; # G. C. Greubel, Jul 14 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( x*(2+3*x+5*x^2+7*x^3)/((1-x^4)*(1-11*x)) )); // G. C. Greubel, Jul 14 2019

Table[FromDigits[PadRight[{},n,{2,3,5,7}],11],{n,30}] (* or *) LinearRecurrence[{11,0,0,1,-11},{2,25,280,3087,33959},31]

a(n) = (-2074+305*(-1)^n+(370+410*I)*(-I)^n+(370-410*I)*I^n+1029*11^n)/4880 \\ Colin Barker, Jul 31 2016

Vec(x*(2+3*x+5*x^2+7*x^3)/((1-x)*(1+x)*(1-11*x)*(1+x^2)) + O(x^30)) \\ Colin Barker, Jul 31 2016

a=(x*(2+3*x+5*x^2+7*x^3)/((1-x^4)*(1-11*x))).series(x, 30).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Jul 14 2019

a(1)=2, a(2)=25, a(3)=280, a(4)=3087, a(5)=33959, a(n) = 11*a(n-1) + a(n-4) - 11*a(n-5). - Harvey P. Dale, Mar 15 2016

G.f.: x*(2+3*x+5*x^2+7*x^3) / ((1-x)*(1+x)*(1-11*x)*(1+x^2)). - Colin Barker, Jul 31 2016

cp's OEIS Frontend

A097137 Convolution of 3^n and floor(n/2).

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A052929 Expansion of g.f. (2-3*x-x^2)/((1-x^2)*(1-3*x)).

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A089815 a(n) = floor((n+2)^(n+2)/((n+2)^2-1)).

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A089816 a(n) = floor((n+3)^(n+2)/((n+3)^2-1)).

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A340395 a(n) = A340131(A001006(n)).

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A270346 a(n) is the number whose base-11 digits are, in order, the first n terms of the simple periodic sequence: repeat 2,3,5,7.

Views

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Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

PARI

Sage

Formula

A052929 Expansion of g.f. (2-3x-x^2)/((1-x^2)(1-3*x)).