A288834 -id:A288834 - cp's OEIS frontend

A066373 a(n) = (3n-2)2^(n-3).

2, 7, 20, 52, 128, 304, 704, 1600, 3584, 7936, 17408, 37888, 81920, 176128, 376832, 802816, 1703936, 3604480, 7602176, 15990784, 33554432, 70254592, 146800640, 306184192, 637534208, 1325400064, 2751463424, 5704253440, 11811160064, 24427626496, 50465865728, 104152956928

Offset: 2

N. J. A. Sloane, Jan 04 2002

An elephant sequence, see A175654. For the corner squares 16 A[5] vectors, with decimal values between 59 and 440, lead to this sequence (with a leading 1 added). For the central square these vectors lead to the companion sequence A098156 (without a(1)). - Johannes W. Meijer, Aug 15 2010

a(n) is the total number of 1's in runs of 1's of length >= 2 over all binary words with n bits. - Félix Balado, Jan 15 2024

Harry J. Smith, Table of n, a(n) for n = 2..200
M. Azaola and F. Santos, The number of triangulations of the cyclic polytope C(n,n-4), Discrete Comput. Geom., 27 (2002), 29-48.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Column k=2 of A229079.

Cf. A016789, A098156, A130129, A288834.

seq((3*n-2)*2^(n-3),n=2..30); # Emeric Deutsch, Jul 23 2006

Array[(3 # - 2)*2^(# - 3) &, 28, 2] (* or *)
Drop[CoefficientList[Series[x^2*(2 - x)/(1 - 2 x)^2, {x, 0, 29}], x], 2] (* Michael De Vlieger, Jun 30 2018 *)

a(n) = { (3*n - 2)*2^(n - 3) } /* Harry J. Smith, Feb 11 2010 */

G.f.: x^2*(2-x)/(1-2x)^2. - Emeric Deutsch, Jul 23 2006
a(n) = 2*a(n-1) +3*2^(n-3). - Vincenzo Librandi, Mar 20 2011
a(n+1) - a(n) = A098156(n). - R. J. Mathar, Apr 25 2013
From Paul Curtz, Jun 29 2018: (Start)
a(n) = A130129(n-2) - A130129(n-3) for n >= 2.
Binomial transform of A016789.
Inverse binomial transform of A288834.
Also the main diagonal of the difference table of m -> (-1)^m*(m+2).
    2,  -3,   4,  -5, ...
   -5,   7,  -9,  11, ...
   12, -16,  20, -24, ...
  -28,  36, -44,  52, ... . (End)

A080420 a(n) = (n+1)(n+6)3^n/6.

Original entry on oeis.org

1, 7, 36, 162, 675, 2673, 10206, 37908, 137781, 492075, 1732104, 6022998, 20726199, 70681653, 239148450, 803538792, 2683245609, 8910671247, 29443957164, 96855122250, 317297380491, 1035574967097, 3368233731366, 10920608743932, 35303692060125, 113819103201843

Offset: 0

Paul Barry, Feb 19 2003

a(n-1) is the number of words of length n defined on 5 letters that have exactly one a and no b's or exactly two b's and no a's. For example, for n=3, a(2) = 36 since the words are (number of permutations in parentheses): acc (3), add (3), aee (3), acd (6), ace (6), ade (6), bbc (3), bbd (3), bbe (3). - Enrique Navarrete, Jun 10 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Gregory Gerard Wojnar, Daniel Sz. Wojnar, and Leon Q. Brin, Universal peculiar linear mean relationships in all polynomials, arXiv:1706.08381 [math.GM], 2017. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

T(n,2) in triangle A080419.

Cf. A136158, A288834.

[(n+1)*(n+6)*3^n/6: n in [0..30]]; // Vincenzo Librandi, Aug 05 2013

CoefficientList[Series[(1 - 2 x) / (1 - 3 x)^3, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 05 2013 *)
Table[(n+1)(n+6)3^n/6,{n,0,30}] (* or *) LinearRecurrence[{9,-27,27},{1,7,36},30] (* Harvey P. Dale, Apr 02 2019 *)

[(n+1)*(n+6)*3^n/6 for n in range(31)] # G. C. Greubel, Dec 22 2023

G.f.: (1-2*x)/(1-3*x)^3.
From G. C. Greubel, Dec 22 2023: (Start)
a(n) = (n+6)*A288834(n)/2, for n >= 1.
a(n) = A136158(n+2, 2).
E.g.f.: (1/2)*(2 + 8*x + 3*x^2)*exp(3*x). (End)
From Amiram Eldar, Jan 11 2024: (Start)
Sum_{n>=0} 1/a(n) = 17721/50 - 4356*log(3/2)/5.
Sum_{n>=0} (-1)^n/a(n) = 4392*log(4/3)/5 - 12591/50. (End)

A288836 a(n) = (1/3!)3^(n+1)(n+5)(n+1)(n).

Original entry on oeis.org

18, 189, 1296, 7290, 36450, 168399, 734832, 3070548, 12400290, 48715425, 187067232, 704690766, 2611501074, 9542023155, 34437376800, 122941435176, 434685788658, 1523724783237, 5299912289520, 18305618105250, 62824881337218, 214364018189079, 727538485975056

Offset: 1

Gregory Gerard Wojnar, Jun 17 2017

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (12, -54, 108, -81).

Column k=6 of A287768.

Cf. A288834, A288835, A288838.

LinearRecurrence[{12,-54,108,-81},{18,189,1296,7290},30] (* Harvey P. Dale, May 09 2025 *)

O.g.f.: z*3^2*(2-3*z)/(1-3*z)^4.

a(n) = -A287768(n+5,6).

A288835 a(n) = (1/2!)3^n(n+3)*(n).

Original entry on oeis.org

6, 45, 243, 1134, 4860, 19683, 76545, 288684, 1062882, 3838185, 13640319, 47829690, 165809592, 569173311, 1937102445, 6543101592, 21953827710, 73222472421, 242912646603, 801960412230, 2636009007156, 8629791392475, 28148810469273, 91507169819844

Offset: 1

Gregory Gerard Wojnar, Jun 17 2017

nonn

Index entries for linear recurrences with constant coefficients, signature (9, -27, 27).

Column k=4 of A287768.

Cf. A288834, A288836, A288838.

Table[(1/2!)*3^n*(n + 3) n, {n, 24}] (* Michael De Vlieger, Jun 23 2017 *)
LinearRecurrence[{9,-27,27},{6,45,243},30] (* Harvey P. Dale, Apr 04 2020 *)

O.g.f.: z*3^1*(2-3*z)/(1-3*z)^3.

a(n) = A287768(n+3,4).

A288838 a(n) = (1/4!)3^(n+2)(n+7)(n+2)(n+1)*(n).

Original entry on oeis.org

54, 729, 6075, 40095, 229635, 1194102, 5786802, 26572050, 116917020, 496897335, 2051893701, 8269753401, 32643763425, 126557359740, 482984209620, 1817776934388, 6757388169138, 24843338857125, 90429753439935, 326206114635555, 1167092987918319, 4144371018322194

Offset: 1

Gregory Gerard Wojnar, Jun 17 2017

Index entries for linear recurrences with constant coefficients, signature (15,-90,270,-405,243).

Column k=9 of A287768.

Cf. A288834, A288835, A288836.

Table[1/4! 3^(n+2) (n+7)(n+2)(n+1)n,{n,30}] (* or *) LinearRecurrence[{15,-90,270,-405,243},{54,729,6075,40095,229635},30] (* Harvey P. Dale, May 15 2022 *)

a(n)=3^n*3*n*(n+1)*(n+2)*(n+7)/8 \\ Charles R Greathouse IV, Jun 19 2017

O.g.f.: z*3^3*(2-3*z)/(1-3*z)^5.

a(n) = A287768(n+7,9).

A354314 Expansion of e.g.f. 1/(1 - x/3 * (exp(3 * x) - 1)).

Original entry on oeis.org

1, 0, 2, 9, 60, 495, 4986, 58401, 780984, 11749779, 196446870, 3612882933, 72484364052, 1575418827879, 36875093680530, 924769734574185, 24737895033896304, 703105981990977915, 21159355356941587470, 672148402091190649629, 22475238194908656800460

Offset: 0

Seiichi Manyama, May 23 2022

nonn

Cf. A052848, A354313.

Cf. A288834, A328182, A353999, A354312.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x/3*(exp(3*x)-1))))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j*3^(j-2)*binomial(i, j)*v[i-j+1])); v;

a(n) = n!*sum(k=0, n\2, 3^(n-2*k)*k!*stirling(n-k, k, 2)/(n-k)!);

a(0) = 1; a(n) = Sum_{k=2..n} k * 3^(k-2) * binomial(n,k) * a(n-k).

a(n) = n! * Sum_{k=0..floor(n/2)} 3^(n-2*k) * k! * Stirling2(n-k,k)/(n-k)!.

A288842 Triangle (sans apex) of coefficients of terms of the form (eM_1)^j*(eM_2)^k re construction of triangle A287768.

Original entry on oeis.org

1, 2, 3, 9, 6, 9, 36, 45, 18, 27, 135, 243, 189, 54, 81, 486, 1134, 1296, 729, 162, 243, 1701, 4860, 7290, 6075, 2673, 486, 729, 5832, 19683, 36450, 40095, 26244, 9477, 1458, 2187, 19683, 76545, 168399, 229635, 199017, 107163, 32805, 4374, 6561, 65610, 288684, 734832, 1194102, 1285956, 918540, 419904, 111537, 13122

Offset: 1

Gregory Gerard Wojnar, Jun 17 2017

			Triangle begins:
  1,  2;
  3,  9,  6;
  9, 36, 45, 18;

G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, Table GW.n=3, p.22, arXiv:1706.08381 [math.GM], 2017.

Columns are: A000244, A288834, A288835, A288836, A288838, etc.

Cf. A287768.

T[1, 1] = 1; T[1, 2] = 2;
T[n_, k_] /; 1 <= k <= n+1 := T[n, k] = 3 T[n-1, k-1] + 3 T[n-1, k];
T[, ] = 0;
Table[T[n, k], {n, 1, 9}, {k, 1, n+1}] // Flatten (* Jean-François Alcover, Nov 16 2018 *)

T(n+1,k+1) = 3*T(n,k) + 3*T(n,k+1).

cp's OEIS Frontend

A066373 a(n) = (3*n-2)*2^(n-3).

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A080420 a(n) = (n+1)*(n+6)*3^n/6.

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A288836 a(n) = (1/3!)*3^(n+1)*(n+5)*(n+1)*(n).

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A288835 a(n) = (1/2!)*3^n*(n+3)*(n).

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A288838 a(n) = (1/4!)*3^(n+2)*(n+7)*(n+2)*(n+1)*(n).

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A354314 Expansion of e.g.f. 1/(1 - x/3 * (exp(3 * x) - 1)).

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PARI

PARI

PARI

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A288842 Triangle (sans apex) of coefficients of terms of the form (eM_1)^j*(eM_2)^k re construction of triangle A287768.

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A066373 a(n) = (3n-2)2^(n-3).

A080420 a(n) = (n+1)(n+6)3^n/6.

A288836 a(n) = (1/3!)3^(n+1)(n+5)(n+1)(n).

A288835 a(n) = (1/2!)3^n(n+3)*(n).

A288838 a(n) = (1/4!)3^(n+2)(n+7)(n+2)(n+1)*(n).