A237930 -id:A237930 - cp's OEIS frontend

A052909 Expansion of g.f. (1+x-x^2)/((1-x)(1-3x)).

1, 5, 16, 49, 148, 445, 1336, 4009, 12028, 36085, 108256, 324769, 974308, 2922925, 8768776, 26306329, 78918988, 236756965, 710270896, 2130812689, 6392438068, 19177314205, 57531942616, 172595827849, 517787483548, 1553362450645

Offset: 0

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

			Ternary.......................Decimal
1...................................1
12..................................5
121................................16
1211...............................49
12111.............................148
121111............................445
1211111..........................1336
12111111.........................4009
121111111.......................12028
1211111111......................36085, etc. - _Philippe Deléham_, Feb 17 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 889
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A000244, A237930.

Concatenation([1], List([1..30], n-> (11*3^n - 3)/6)); # G. C. Greubel, Oct 15 2019

I:=[1, 5, 16]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 22 2012

spec := [S,{S=Prod(Union(Sequence(Z),Z),Sequence(Union(Z,Z,Z)))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

CoefficientList[Series[(1+x-x^2)/((1-x)*(1-3*x)),{x,0,30}],x] (* Vincenzo Librandi, Jun 22 2012 *)
Join[{1}, (11*3^Range[30] -3)/6] (* G. C. Greubel, Oct 15 2019 *)

vector(30, n, if(n==1, 1, (11*3^(n-1) - 3)/6)) \\ G. C. Greubel, Oct 15 2019

[1]+[(11*3^n -3)/6 for n in (1..30)] # G. C. Greubel, Oct 15 2019

a(n) = 3*a(n-1) + 1, with a(0)=1, a(1)=5, a(2)=16.
a(n) = (11*3^n - 3)/6.
a(n) = 4*a(n-1) - 3*a(n-2). - Vincenzo Librandi, Jun 22 2012
a(n+1) = A237930(n) + 2*A000244(n). - Philippe Deléham, Feb 17 2014
a(n) = Sum_{k=1..3} floor((3^n)/k). - Lechoslaw Ratajczak, Jul 31 2016
E.g.f.: (11*exp(3*x) - 3*exp(x) - 2)/6. - Stefano Spezia, Aug 28 2023

More terms from James Sellers, Jun 08 2000

A199109 a(n) = (7*3^n + 1)/2.

Original entry on oeis.org

4, 11, 32, 95, 284, 851, 2552, 7655, 22964, 68891, 206672, 620015, 1860044, 5580131, 16740392, 50221175, 150663524, 451990571, 1355971712, 4067915135, 12203745404, 36611236211, 109833708632, 329501125895, 988503377684, 2965510133051, 8896530399152, 26689591197455

Offset: 0

Vincenzo Librandi, Nov 03 2011

Also the number of (not necessarily maximal) cliques in the (n+2)-Mycielski graph. - Eric W. Weisstein, Nov 29 2017

			Ternary....................Decimal
11...............................4
102.............................11
1012............................32
10112...........................95
101112.........................284
1011112........................851
10111112......................2552
101111112.....................7655
1011111112...................22964, etc.
- _Philippe Deléham_, Feb 16 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Clique.
Eric Weisstein's World of Mathematics, Mycielski Graph.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A000244, A003462, A005032 (first differences), A199110, A237930.

Magma
```
[(7*3^n+1)/2 : n in [0..30]];
```

Table[(7 3^n + 1)/2, {n, 0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)
(7 3^Range[0, 20] + 1)/2 (* Eric W. Weisstein, Nov 29 2017 *)
LinearRecurrence[{4, -3}, {11, 32}, {0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)
CoefficientList[Series[(4 - 5 x)/(1 - 4 x + 3 x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Nov 29 2017 *)

a(n)=7*3^n\2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 3*a(n-1) - 1.
a(n) = 4*a(n-1) - 3*a(n-2).
G.f.: (4-5*x)/((1-x)*(1-3*x)). - Bruno Berselli, Nov 03 2011
a(n) = A000244(n+1) + A003462(n) + 1 = A237930(n) + 1. - Philippe Deléham, Feb 16 2014
From Elmo R. Oliveira, Apr 02 2025: (Start)
E.g.f.: exp(x)*(7*exp(2*x) + 1)/2.
a(n) = A199110(n)/2. (End)

A027107 a(n) = Sum_{k=0..2n} (k+1) * A027082(n, 2n-k).

Original entry on oeis.org

1, 6, 20, 62, 188, 566, 1700, 5102, 15308, 45926, 137780, 413342, 1240028, 3720086, 11160260, 33480782, 100442348, 301327046, 903981140, 2711943422, 8135830268, 24407490806, 73222472420, 219667417262, 659002251788

Offset: 0

Clark Kimberling

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

[1] cat [7*3^n-1: n in [0..30]]; // Vincenzo Librandi, Mar 24 2013

Join[{1}, Table[7 3^(n-1) - 1, {n, 30}]] (* Vincenzo Librandi, Mar 24 2013 *)

For n>0, a(n) = 7*3^(n-1) - 1.
G.f.: (1+2*x-x^2)/(1-4*x+3*x^2). [Bruno Berselli, Mar 25 2013]
a(n) = 2*A237930(n-1), n>0. - R. J. Mathar, Jun 24 2020

A329774 a(n) = n+1 for n <= 2; otherwise a(n) = 3*a(n-3)+1.

Original entry on oeis.org

1, 2, 3, 4, 7, 10, 13, 22, 31, 40, 67, 94, 121, 202, 283, 364, 607, 850, 1093, 1822, 2551, 3280, 5467, 7654, 9841, 16402, 22963, 29524, 49207, 68890, 88573, 147622, 206671, 265720, 442867, 620014, 797161, 1328602, 1860043, 2391484, 3985807

Offset: 0

N. J. A. Sloane, Nov 27 2019

Robert Fathauer observed that if the "warp and woof" construction used by Jim Conant in his recursive dissection of a square (see A328078) is applied to a triangle, one obtains the Sierpinski gasket.

The present sequence gives the number of regions after the n-th generation of this dissection of a triangle.

Robert Fathauer, Email to N. J. A. Sloane, Oct 14 2019.

Colin Barker, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,0,3,-3).

A mixture of A003462, A060816, and A237930. Cf. A328078.

f:=proc(n) option remember;
if n<=2 then n+1 else 3*f(n-3)+1; fi; end;
[seq(f(n),n=0..50)];

Vec((1 + x + x^2 - 2*x^3) / ((1 - x)*(1 - 3*x^3)) + O(x^40)) \\ Colin Barker, Nov 27 2019

From Colin Barker, Nov 27 2019: (Start)
G.f.: (1 + x + x^2 - 2*x^3) / ((1 - x)*(1 - 3*x^3)).
a(n) = a(n-1) + 3*a(n-3) - 3*a(n-4) for n>3.
(End)

A330246 a(n) = 4^(n+1) + (4^n-1)/3.

Original entry on oeis.org

4, 17, 69, 277, 1109, 4437, 17749, 70997, 283989, 1135957, 4543829, 18175317, 72701269, 290805077, 1163220309, 4652881237, 18611524949, 74446099797, 297784399189, 1191137596757, 4764550387029, 19058201548117, 76232806192469, 304931224769877, 1219724899079509

Offset: 0

Vincenzo Librandi, Jan 09 2020

After 4, these numbers are the third column of the rectangular array in A238475.

Similar to A272743.
Cf. A000302, A002450, A052909, A178415, A237930, A238475, A347834.
Together with 1: first bisection of A136326.

Magma
```
[4^(n+1)+(4^n-1)/3: n in [0..30]];
```

Table[(4^(n + 1) + (4^n - 1) / 3), {n, 0, 30}]

G.f.: (4 - 3*x) / ((1 - x)*(1 - 4*x)).
a(n) = 5*a(n-1) - 4*a(n-2) for n > 1.
a(n) = 4*a(n-1) + 1 for n > 0.
a(n) = (13*4^n -1)/3, for n >= 0. - Wolfdieter Lang, Sep 16 2021
a(n) = A178415(5, n) = A347834(7, n-1), arrays, for n >= 1. - Wolfdieter Lang, Nov 29 2021

A238055 a(n) = (13*3^n-1)/2.

Original entry on oeis.org

6, 19, 58, 175, 526, 1579, 4738, 14215, 42646, 127939, 383818, 1151455, 3454366, 10363099, 31089298, 93267895, 279803686, 839411059, 2518233178, 7554699535, 22664098606, 67992295819, 203976887458, 611930662375, 1835791987126, 5507375961379, 16522127884138

Offset: 0

Philippe Deléham, Feb 17 2014

			Ternary....................Decimal
20...............................6
201.............................19
2011............................58
20111..........................175
201111.........................526
2011111.......................1579
20111111......................4738
201111111....................14215, etc.

Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A000244, A003462, A052909, A060816, A237930.

a(n) = 3*a(n-1) + 1, a(0)=6.
a(n) = 4*a(n-1) - 3*a(n-2), a(0)=6, a(1)=19.
a(n) = 2*A237930(n) - A003462(n).
a(n) = A052909(n+1) + A000244(n).
a(n) = A237930(n) + A000244(n+1).
a(n) = 13*A003462(n) + 6.
G.f.: (6-5*x)/((1-x)*(1-3*x)).
E.g.f.: exp(x)*(13*exp(2*x) - 1)/2. - Stefano Spezia, Aug 28 2023

A238206 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) is A007494(k) and T(n,k) = 3*T(n-1,k) + 1 for n>0.

Original entry on oeis.org

0, 2, 1, 3, 7, 4, 5, 10, 22, 13, 6, 16, 31, 67, 40, 8, 19, 49, 94, 202, 121, 9, 25, 58, 148, 283, 607, 364, 11, 28, 76, 175, 445, 850, 1822, 1093, 12, 34, 85, 229, 526, 1336, 2551, 5467, 3280, 14, 37, 103, 256, 688, 1579, 4009, 7654, 16402, 9841, 15, 43, 112, 310

Offset: 0

Philippe Deléham, Feb 20 2014

Permutation of nonnegative integers.

			Square array begins:
0, 2, 3, 5, 6, 8, 9, ...
1, 7, 10, 16, 19, 25, 28, ...
4, 22, 31, 49, 58, 76, 85, ...
13, 67, 94, 148, 175, 229, 256, ...
40, 202, 283, 445, 523, 688, 769, ...
121, 607, 850, 1336, 1579, 2065, 2308, ...
364, 1822, 2551, 4009, 4738, 6196, 6925, ...
1093, 5467, 7654, 12028, 14215, 18589, 20776, ...
3280, 16402, 22963, 36085, 42646, 55768, 62329, ...
9841, 49207, 68890, 108256, 127939, 167305, 186988, ...
...

Cf. A003462, A007494, A052909, A060816, A237930, A238055

T(n,k) = T(0,k)*3^n + T(n,0) where T(0,k) = (6*k + 1 -(-1)^k)/4 = A007494(k) and T(n,0) = (3^n - 1)/2 = A003462(n).

A370481 a(0) = 33. a(n) = 3a(n-1) + 2n + 1 for n >= 1.

Original entry on oeis.org

33, 102, 311, 940, 2829, 8498, 25507, 76536, 229625, 688894, 2066703, 6200132, 18600421, 55801290, 167403899, 502211728, 1506635217, 4519905686, 13559717095, 40679151324, 122037454013, 366112362082, 1098337086291, 3295011258920, 9885033776809, 29655101330478

Offset: 0

Paul Curtz, Mar 31 2024

Last digit is of period 10: repeat [3, 2, 1, 0, 9, 8, 7, 6, 5, 4].

			a(1) = 3*33 + 3 = 102, a(2) = 3*102 + 5 = 311.

Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

Cf. A144396, A237930.

LinearRecurrence[{5, -7, 3}, {33, 102, 311}, 26] (* Amiram Eldar, Apr 01 2024 *)

a(n) = 4*a(n-1) - 3*a(n-2) + 2 with a(0) = 33 and a(1) = 102 for n >= 2.
a(n) = 10*A237930(n) + 3 - n.
a(n) = 35*3^n - n - 2. - Hugo Pfoertner, Mar 31 2024

cp's OEIS Frontend

A052909 Expansion of g.f. (1+x-x^2)/((1-x)*(1-3*x)).

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A199109 a(n) = (7*3^n + 1)/2.

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A027107 a(n) = Sum_{k=0..2n} (k+1) * A027082(n, 2n-k).

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A329774 a(n) = n+1 for n <= 2; otherwise a(n) = 3*a(n-3)+1.

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A330246 a(n) = 4^(n+1) + (4^n-1)/3.

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Magma

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A238055 a(n) = (13*3^n-1)/2.

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A238206 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) is A007494(k) and T(n,k) = 3*T(n-1,k) + 1 for n>0.

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A370481 a(0) = 33. a(n) = 3*a(n-1) + 2*n + 1 for n >= 1.

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A052909 Expansion of g.f. (1+x-x^2)/((1-x)(1-3x)).

A370481 a(0) = 33. a(n) = 3a(n-1) + 2n + 1 for n >= 1.