[go: up one dir, main page]
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
Search: a288834 -id:a288834
Displaying 1-7 of 7 results found. page 1
     Sort: relevance | references | number | modified | created      Format: long | short | data
A066373 a(n) = (3*n-2)*2^(n-3). +10
7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,1
COMMENTS
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
LINKS
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).
FORMULA
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)
MAPLE
seq((3*n-2)*2^(n-3), n=2..30); # Emeric Deutsch, Jul 23 2006
MATHEMATICA
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 *)
PROG
(PARI) { for (n=2, 200, write("b066373.txt", n, " ", (3*n - 2)*2^(n - 3)) ) } /* Harry J. Smith, Feb 11 2010 */
CROSSREFS
Column k=2 of A229079.
Cf. A016789, A098156, A130129, A288834.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jan 04 2002
STATUS
approved
A080420 a(n) = (n+1)*(n+6)*3^n/6. +10
7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
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).
FORMULA
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)
MATHEMATICA
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 *)
PROG
(Magma) [(n+1)*(n+6)*3^n/6: n in [0..30]]; // Vincenzo Librandi, Aug 05 2013
(SageMath) [(n+1)*(n+6)*3^n/6 for n in range(31)] # G. C. Greubel, Dec 22 2023
CROSSREFS
T(n,2) in triangle A080419.
Cf. A136158, A288834.
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Feb 19 2003
STATUS
approved
A288836 a(n) = (1/3!)*3^(n+1)*(n+5)*(n+1)*(n). +10
4
18, 189, 1296, 7290, 36450, 168399, 734832, 3070548, 12400290, 48715425, 187067232, 704690766, 2611501074, 9542023155, 34437376800, 122941435176, 434685788658, 1523724783237, 5299912289520, 18305618105250, 62824881337218, 214364018189079, 727538485975056 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n = 1..23
Index entries for linear recurrences with constant coefficients, signature (12, -54, 108, -81).
FORMULA
O.g.f.: z*3^2*(2-3*z)/(1-3*z)^4.
a(n) = -A287768(n+5,6).
CROSSREFS
Column k=6 of A287768.
Cf. A288834, A288835, A288838.
KEYWORD
nonn
AUTHOR
Gregory Gerard Wojnar, Jun 17 2017
STATUS
approved
A288835 a(n) = (1/2!)*3^n*(n+3)*(n). +10
3
6, 45, 243, 1134, 4860, 19683, 76545, 288684, 1062882, 3838185, 13640319, 47829690, 165809592, 569173311, 1937102445, 6543101592, 21953827710, 73222472421, 242912646603, 801960412230, 2636009007156, 8629791392475, 28148810469273, 91507169819844 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..24.
Index entries for linear recurrences with constant coefficients, signature (9, -27, 27).
FORMULA
O.g.f.: z*3^1*(2-3*z)/(1-3*z)^3.
a(n) = A287768(n+3,4).
MATHEMATICA
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 *)
CROSSREFS
Column k=4 of A287768.
Cf. A288834, A288836, A288838.
KEYWORD
nonn
AUTHOR
Gregory Gerard Wojnar, Jun 17 2017
STATUS
approved
A288838 a(n) = (1/4!)*3^(n+2)*(n+7)*(n+2)*(n+1)*(n). +10
3
54, 729, 6075, 40095, 229635, 1194102, 5786802, 26572050, 116917020, 496897335, 2051893701, 8269753401, 32643763425, 126557359740, 482984209620, 1817776934388, 6757388169138, 24843338857125, 90429753439935, 326206114635555, 1167092987918319, 4144371018322194 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..22.
Index entries for linear recurrences with constant coefficients, signature (15,-90,270,-405,243).
FORMULA
O.g.f.: z*3^3*(2-3*z)/(1-3*z)^5.
a(n) = A287768(n+7,9).
MATHEMATICA
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 *)
PROG
(PARI) a(n)=3^n*3*n*(n+1)*(n+2)*(n+7)/8 \\ Charles R Greathouse IV, Jun 19 2017
CROSSREFS
Column k=9 of A287768.
Cf. A288834, A288835, A288836.
KEYWORD
nonn,easy
AUTHOR
Gregory Gerard Wojnar, Jun 17 2017
STATUS
approved
A354314 Expansion of e.g.f. 1/(1 - x/3 * (exp(3 * x) - 1)). +10
2
1, 0, 2, 9, 60, 495, 4986, 58401, 780984, 11749779, 196446870, 3612882933, 72484364052, 1575418827879, 36875093680530, 924769734574185, 24737895033896304, 703105981990977915, 21159355356941587470, 672148402091190649629, 22475238194908656800460 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Table of n, a(n) for n=0..20.
FORMULA
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)!.
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x/3*(exp(3*x)-1))))
(PARI) 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;
(PARI) a(n) = n!*sum(k=0, n\2, 3^(n-2*k)*k!*stirling(n-k, k, 2)/(n-k)!);
CROSSREFS
Cf. A052848, A354313.
Cf. A288834, A328182, A353999, A354312.
KEYWORD
nonn
AUTHOR
Seiichi Manyama, May 23 2022
STATUS
approved
A288842 Triangle (sans apex) of coefficients of terms of the form (eM_1)^j*(eM_2)^k re construction of triangle A287768. +10
0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Table of n, a(n) for n=1..54.
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.
FORMULA
T(n+1,k+1) = 3*T(n,k) + 3*T(n,k+1).
EXAMPLE
Triangle begins:
1, 2;
3, 9, 6;
9, 36, 45, 18;
MATHEMATICA
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 *)
CROSSREFS
Columns are: A000244, A288834, A288835, A288836, A288838, etc.
Cf. A287768.
KEYWORD
nonn,tabf
AUTHOR
Gregory Gerard Wojnar, Jun 17 2017
STATUS
approved
page 1

Search completed in 0.008 seconds

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 30 00:57 EDT 2024. Contains 375520 sequences. (Running on oeis4.)