A107903 -id:A107903 - cp's OEIS frontend

A096053 a(n) = (3*9^n - 1)/2.

1, 13, 121, 1093, 9841, 88573, 797161, 7174453, 64570081, 581130733, 5230176601, 47071589413, 423644304721, 3812798742493, 34315188682441, 308836698141973, 2779530283277761, 25015772549499853, 225141952945498681

Offset: 0

Benoit Cloitre, Jun 18 2004

Generalized NSW numbers. - Paul Barry, May 27 2005
Counts total area under elevated Schroeder paths of length 2n+2, where area under a horizontal step is weighted 3. Case r=4 for family (1+(r-1)x)/(1-2(1+r)x+(1-r)^2*x^2). Case r=2 gives NSW numbers A002315. Fifth binomial transform of (1+8x)/(1-16x^2), A107906. - Paul Barry, May 27 2005
Primes in this sequence include: a(2) = 13, a(4) = 1093, a(7) = 797161. Semiprimes in this sequence include: a(3) = 121 = 11^2, a(5) = 9841 = 13 * 757, a(6) = 88573 = 23 * 3851, a(9) = 64570081 = 1871 * 34511, a(10) = 581130733 = 1597 * 363889, a(12) = 47071589413 = 47 * 1001523179, a(19) = 225141952945498681 = 13097927 * 17189128703.
Sum of divisors of 9^n. - Altug Alkan, Nov 10 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A083420, A096045, A096046, A096047, A096054.

Cf. A107903, A138894 ((5*9^n-1)/4).

[(3*9^n-1)/2: n in [0..20]]; // Vincenzo Librandi, Nov 01 2011

Table[(3*9^n - 1)/2, {n, 0, 18}] (* L. Edson Jeffery, Feb 13 2015 *)

a(n)=(3*9^n-1)/2 \\ Charles R Greathouse IV, Sep 28 2015

vector(30, n, n--; sigma(9^n)) \\ Altug Alkan, Nov 10 2015

From Paul Barry, May 27 2005: (Start)
G.f.: (1+3*x)/(1-10*x+9*x^2);
a(n) = Sum_{k=0..n} binomial(2n+1, 2k)*4^k;
a(n) = ((1+sqrt(4))*(5+2*sqrt(4))^n+(1-sqrt(4))*(5-2*sqrt(4))^n)/2. (End)
a(n-1) = (-9^n/3)*B(2n,1/3)/B(2n) where B(n,x) is the n-th Bernoulli polynomial and B(k)=B(k,0) is the k-th Bernoulli number.
a(n) = 10*a(n-1) - 9*a(n-2).
a(n) = 9*a(n-1) + 4. - Vincenzo Librandi, Nov 01 2011
a(n) = A000203(A001019(n)). - Altug Alkan, Nov 10 2015
a(n) = A320030(3^n-1). - Nathan M Epstein, Jan 02 2019

Edited by N. J. A. Sloane, at the suggestion of Andrew S. Plewe, Jun 15 2007

A102591 a(n) = Sum_{k=0..n} binomial(2n+1, 2k)*3^(n-k).

Original entry on oeis.org

1, 6, 44, 328, 2448, 18272, 136384, 1017984, 7598336, 56714752, 423324672, 3159738368, 23584608256, 176037912576, 1313964867584, 9807567290368, 73204678852608, 546407161659392, 4078438577864704, 30441879976280064

Offset: 0

Paul Barry, Jan 22 2005

In general, Sum_{k=0..n} binomial(2n+1,2k)*r^(n-k) has g.f. (1-(r-1)x)/(1-2(r+1)+(r-1)^2x^2) and a(n) = ((sqrt(r)-1)^(2n+1) + (sqrt(r)+1)^(2n+1))/(2*sqrt(r)).

Index entries for linear recurrences with constant coefficients, signature (8,-4).

Bisection of A080879 or A002605.

Cf. A066443, A079935, A099156, A102592, A107903.

LinearRecurrence[{8,-4},{1,6},20] (* Harvey P. Dale, Sep 28 2021 *)

G.f.: (1-2x)/(1-8x+4x^2);
a(n) = 8*a(n-1) - 4*a(n-2);
a(n) = sqrt(3)*(sqrt(3)-1)^(2n+1)/6 + sqrt(3)*(sqrt(3)+1)^(2n+1)/6.
a(n) = 2^n*A079935(n). - R. J. Mathar, Sep 20 2012
a(n) = 2^(2*n+1)*Sum_{k >= n} binomial(2*k,2*n)*(1/3)^(k+1). Cf. A099156. - Peter Bala, Nov 29 2021
3*a(n)^2 = A107903(n)^2 + 2^(2*n+1). - Philippe Deléham, Mar 21 2023

A180028 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1 + 3x)/(1 - 6x - 3*x^2).

Original entry on oeis.org

1, 9, 57, 369, 2385, 15417, 99657, 644193, 4164129, 26917353, 173996505, 1124731089, 7270376049, 46996449561, 303789825513, 1963728301761, 12693739287105, 82053620627913, 530402941628793, 3428578511656497

Offset: 0

Johannes W. Meijer, Aug 09 2010; edited Jun 21 2013

The a(n) represent the number of n-move routes of a fairy chess piece starting in the center square (m = 5) on a 3 X 3 chessboard. This fairy chess piece behaves like a white queen on the eight side and corner squares but on the central square the queen explodes with fury and turns into a red queen.
On a 3 X 3 chessboard there are 2^9 = 512 ways to explode with fury on the center square (off the center square the piece behaves like a normal queen). The red queen is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program and A180140. For the center square the 512 red queens lead to 17 red queen sequences, see the overview of red queen sequences and the crossreferences.
The sequence above corresponds to just one red queen vector, i.e., A[5] = [111 111 111] vector. The other squares lead for this vector to A090018.
This sequence belongs to a family of sequences with g.f. (1+k*x)/(1 - 6*x - k*x^2). The members of this family that are red queen sequences are A180028 (k=3; this sequence), A180029 (k=2), A015451 (k=1), A000400 (k=0), A001653 (k=-1), A180034 (k=-2), A084120 (k=-3), A154626 (k=-4) and A000012 (k=-5). Other members of this family are A123362 (k=5), 6*A030192(k=-6).
Inverse binomial transform of A107903.

Gary Chartrand, Introductory Graph Theory, pp. 217-221, 1984.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Johannes W. Meijer, The red queen sequences.
Wikipedia, Alice in Wonderland (2010 film).
Index entries for linear recurrences with constant coefficients, signature (6, 3).

Cf. A180140 (berserker sequences)
Cf. A180032 (Corner and side squares).
Cf. Red queen sequences center square [decimal value A[5]]: A180028 [511], A180029 [255], A180031 [495], A015451 [127], A152240 [239], A000400 [63], A057088 [47], A001653 [31], A122690 [15], A180034 [23], A180036 [7], A084120 [19], A180038 [3], A154626 [17], A015449 [1], A000012 [16], A000007 [0].

I:=[1,9]; [n le 2 select I[n] else 6*Self(n-1)+3*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 15 2011

nmax:=19; m:=5; A[1]:=[0,1,1,1,1,0,1,0,1]: A[2]:=[1,0,1,1,1,1,0,1,0]: A[3]:=[1,1,0,0,1,1,1,0,1]: A[4]:=[1,1,0,0,1,1,1,1,0]: A[5]:=[1,1,1,1,1,1,1,1,1]: A[6]:=[0,1,1,1,1,0,0,1,1]: A[7]:=[1,0,1,1,1,0,0,1,1]: A[8]:=[0,1,0,1,1,1,1,0,1]: A[9]:=[1,0,1,0,1,1,1,1,0]: A:=Matrix([A[1], A[2], A[3], A[4], A[5], A[6], A[7], A[8], A[9]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

LinearRecurrence[{6,3},{1,9},50] (* Vincenzo Librandi, Nov 15 2011 *)

G.f.: (1+3*x)/(1 - 6*x - 3*x^2).
a(n) = 6*a(n-1) + 3*a(n-2) with a(0) = 1 and a(1) = 9.
a(n) = ((1-A)*A^(-n-1) + (1-B)*B^(-n-1))/4 with A=(-1+2*sqrt(3)/3) and B=(-1-2*sqrt(3)/3).
Lim_{k->infinity} a(n+k)/a(k) = (-1)^(n-1)*A108411(n+1)/(A041017(n-1)*sqrt(12) - A041016(n-1)) for n >= 1.

A270444 Expansion of 2(1+2x) / (1-8x+4x^2).

Original entry on oeis.org

2, 20, 152, 1136, 8480, 63296, 472448, 3526400, 26321408, 196465664, 1466439680, 10945654784, 81699479552, 609813217280, 4551707820032, 33974409691136, 253588446248960, 1892809931227136, 14128125664821248, 105453765593661440

Offset: 1

Altug Alkan, Mar 17 2016

nonn

If p is an odd prime, a((p+1)/2) == 2 mod p. In other words, a((p+1)/2) - 2^p is divisible by p where p is an odd prime.

			a(2) = 20 because (1 + sqrt(3))^3 + (1 - sqrt(3))^3 = 20.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (8, -4).

Cf. A077444, A080040, A080041, A107903.

CoefficientList[Series[2(1+2x)/(1-8x+4x^2),{x,0,30}],x] (* or *) LinearRecurrence[{8,-4},{2,20},30] (* Harvey P. Dale, Jun 09 2020 *)

PARI
```
Vec(2*(1+2*x)/(1-8*x+4*x^2) + O(x^100))
```

G.f.: 2*(1+2*x)/(1-8*x+4*x^2).
a(n) = (1+sqrt(3))^(2*n-1) + (1-sqrt(3))^(2*n-1).
a(n) = 2 * A107903(n-1).

A107904 Expansion of (1+6x)/(1-12x^2).

Original entry on oeis.org

1, 6, 12, 72, 144, 864, 1728, 10368, 20736, 124416, 248832, 1492992, 2985984, 17915904, 35831808, 214990848, 429981696, 2579890176, 5159780352, 30958682112, 61917364224, 371504185344, 743008370688, 4458050224128, 8916100448256, 53496602689536, 106993205379072

Offset: 0

Paul Barry, May 27 2005

Fourth binomial transform is A107903.

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

Cf. A094015, A107903, A108411.

LinearRecurrence[{0,12},{1,6},30] (* Harvey P. Dale, Sep 22 2014 *)

a(n) = ((1+sqrt(3))*(2*sqrt(3))^n + (1-sqrt(3))*(-2*sqrt(3))^n)/2.
a(2n) = 12^n, a(2n+1) = 6*12^n.
a(n) = 2^n*A108411(n+1). - R. J. Mathar, Aug 15 2023
From Amiram Eldar, Dec 06 2024: (Start)
Sum_{n>=0} 1/a(n) = 14/11.
Sum_{n>=0} (-1)^n/a(n) = 10/11. (End)

cp's OEIS Frontend

A096053 a(n) = (3*9^n - 1)/2.

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A102591 a(n) = Sum_{k=0..n} binomial(2n+1, 2k)*3^(n-k).

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A180028 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1 + 3*x)/(1 - 6*x - 3*x^2).

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A270444 Expansion of 2*(1+2*x) / (1-8*x+4*x^2).

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Programs

Mathematica

PARI

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A107904 Expansion of (1+6x)/(1-12x^2).

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Mathematica

Formula

A180028 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1 + 3x)/(1 - 6x - 3*x^2).

A270444 Expansion of 2(1+2x) / (1-8x+4x^2).