A048473 -id:A048473 - cp's OEIS frontend

A195732 Numbers k such that 2*(3^k-k)-1 is prime.

1, 2, 3, 11, 30, 62, 534, 620, 803, 2436, 2669, 3975, 4530, 4827, 5294, 15987, 17589, 51960

Offset: 1

Juri-Stepan Gerasimov, Sep 23 2011

According to Maple version 14, the next entries are 534, 620 and 803, but these involve primes of 256 and more digits and may not be certified primes. - R. J. Mathar, Sep 23 2011

a(19) > 10^5, if it exists. - Michael S. Branicky, Aug 26 2024

			1 is in the sequence because 2*3^1-2*1-1 = 3 is a prime.
2 is in the sequence because 2*3^2-2*2-1 = 13 is a prime.
3 is in the sequence because 2*3^3-2*3-1=47 ts a prime.
4 is not in the sequence because 2*3^4-2*4-1 = 153 = 3^2*17 is not prime.

Cf. A048473.

is(n)=ispseudoprime(2*(3^n-n)-1) \\ Charles R Greathouse IV, Jun 13 2017

a(10)-a(18) from Michael S. Branicky, Jul 14 2023

A198686 4*7^n-1.

Original entry on oeis.org

3, 27, 195, 1371, 9603, 67227, 470595, 3294171, 23059203, 161414427, 1129900995, 7909306971, 55365148803, 387556041627, 2712892291395, 18990246039771, 132931722278403, 930522055948827, 6513654391641795, 45595580741492571

Offset: 0

Vincenzo Librandi, Oct 29 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. A048473, A081655, A198480, A198647.

Magma
```
[4*7^n-1: n in [0..30]]
```

4*7^Range[0,20]-1 (* or *) LinearRecurrence[{8,-7},{3,27},20] (* Harvey P. Dale, Dec 27 2011 *)

a(n) = 7*a(n-1)+6. a(n) = 8*a(n-1)-7*a(n-2), n>1.

G.f. ( 3+3*x ) / ( (7*x-1)*(x-1) ). - R. J. Mathar, Oct 30 2011

A198687 5*7^n-1.

Original entry on oeis.org

4, 34, 244, 1714, 12004, 84034, 588244, 4117714, 28824004, 201768034, 1412376244, 9886633714, 69206436004, 484445052034, 3391115364244, 23737807549714, 166164652848004, 1163152569936034, 8142067989552244, 56994475926865714

Offset: 0

Vincenzo Librandi, Oct 29 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. A048473, A081655, A198480, A198647.

Magma
```
[5*7^n-1: n in [0..30]]
```

CoefficientList[Series[(4+2*x)/((1-x)*(1-7*x)),{x,0,40}],x] (* Vincenzo Librandi, Jul 06 2012 *)
LinearRecurrence[{8,-7},{4,34},20] (* Harvey P. Dale, Jul 23 2024 *)

a(n) = 7*a(n-1)+6 = 8*a(n-1)-7*a(n-2), n>1.

G.f.:(4+2*x)/((1-x)*(1-7*x)). - Vincenzo Librandi, Jul 06 2012

A198689 8*7^n-1.

Original entry on oeis.org

7, 55, 391, 2743, 19207, 134455, 941191, 6588343, 46118407, 322828855, 2259801991, 15818613943, 110730297607, 775112083255, 5425784582791, 37980492079543, 265863444556807, 1861044111897655, 13027308783283591, 91191161482985143

Offset: 0

Vincenzo Librandi, Oct 29 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. A024075, A048473, A081655, A198480, A198647.

Magma
```
[8*7^n-1: n in [0..30]]
```

a(n) = 7*a(n-1)+6. a(n) = 8*a(n-1)-7*a(n-2), n>1.

G.f. ( 7-x ) / ( (7*x-1)*(x-1) ). - R. J. Mathar, Oct 30 2011

A198690 9*7^n-1.

Original entry on oeis.org

8, 62, 440, 3086, 21608, 151262, 1058840, 7411886, 51883208, 363182462, 2542277240, 17795940686, 124571584808, 872001093662, 6104007655640, 42728053589486, 299096375126408, 2093674625884862, 14655722381194040, 102590056668358286

Offset: 0

Vincenzo Librandi, Oct 29 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. A024075, A048473, A081655, A198480, A198647.

Magma
```
[9*7^n-1: n in [0..30]]
```

9*7^Range[0,30]-1 (* or *) LinearRecurrence[{8,-7},{8,62},30] (* Harvey P. Dale, Apr 22 2019 *)

a(n) = 7*a(n-1)+6. a(n) = 8*a(n-1)-7*a(n-2), n>1.

G.f. ( 8-2*x ) / ( (7*x-1)*(x-1) ). - R. J. Mathar, Oct 30 2011

A198691 10*7^n-1.

Original entry on oeis.org

9, 69, 489, 3429, 24009, 168069, 1176489, 8235429, 57648009, 403536069, 2824752489, 19773267429, 138412872009, 968890104069, 6782230728489, 47475615099429, 332329305696009, 2326305139872069, 16284135979104489, 113988951853731429

Offset: 0

Vincenzo Librandi, Oct 29 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. A024075, A048473, A081655, A198480, A198647.

Magma
```
[10*7^n-1: n in [0..30]]
```

10*7^Range[0,20]-1 (* or *) LinearRecurrence[{8,-7},{9,69},20] (* Harvey P. Dale, Mar 30 2016 *)

a(n) = 7*a(n-1)+6. a(n) = 8*a(n-1)-7*a(n-2), n>1.

G.f. ( 9-3*x ) / ( (7*x-1)*(x-1) ). - R. J. Mathar, Oct 30 2011

A198692 a(n) = 11*7^n-1.

Original entry on oeis.org

10, 76, 538, 3772, 26410, 184876, 1294138, 9058972, 63412810, 443889676, 3107227738, 21750594172, 152254159210, 1065779114476, 7460453801338, 52223176609372, 365562236265610, 2558935653859276, 17912549577014938

Offset: 0

Vincenzo Librandi, Oct 29 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. A024075, A048473, A081655, A198480, A198647.

Magma
```
[11*7^n-1: n in [0..30]]
```

11 (7^Range[0, 30]) - 1 (* Wesley Ivan Hurt, Jan 21 2017 *)

a(n) = 7*a(n-1)+6. a(n) = 8*a(n-1)-7*a(n-2), n>1.

G.f.: ( 10-4*x ) / ( (7*x-1)*(x-1) ). - R. J. Mathar, Oct 30 2011

A213666 Irregular triangle read by rows: T(n,k) is the number of dominating subsets with k vertices of the graph G(n) obtained by taking n copies of the path P_3 and identifying one of their endpoints (a star with n branches of length 2).

Original entry on oeis.org

1, 3, 1, 0, 3, 8, 5, 1, 0, 0, 7, 20, 18, 7, 1, 0, 0, 0, 15, 48, 56, 32, 9, 1, 0, 0, 0, 0, 31, 112, 160, 120, 50, 11, 1, 0, 0, 0, 0, 0, 63, 256, 432, 400, 220, 72, 13, 1, 0, 0, 0, 0, 0, 0, 127, 576, 1120, 1232, 840, 364, 98, 15, 1

Offset: 1

Emeric Deutsch, Jul 01 2012

Rows also give the coefficients of the domination polynomial of the n-helm graph (divided by x, i.e., with initial 0 dropped from rows). - Eric W. Weisstein, May 28 2017
Row n contains 2n + 1 entries (first n-1 of which are 0).
Sum of entries in row n = 2*3^{n-1} - 1 = A048473(n).
Sum of entries in column k = A213667(k).

			Row 2 is 0,3,8,5,1 because G(2) is the path P_5 abcde; no domination subset of size 1, three of size 2 (ad, bd, be), all subsets of size 3 with the exception of abc and cde are dominating (binomial(5,3)-2=8), all binomial(5,4)=5 subsets of size 4 are dominating, and abcde is dominating.
Triangle starts:
  1, 3, 1;
  0, 3, 8,  5,  1;
  0, 0, 7, 20, 18,  7,  1;
  0, 0, 0, 15, 48, 56, 32, 9, 1;

S. Alikhani and Y. H. Peng, Introduction to domination polynomial of a graph, arXiv:0905.2251 [math.CO], 2009.
É. Czabarka, L. Székely, and S. Wagner, The inverse problem for certain tree parameters, Discrete Appl. Math., 157, 2009, 3314-3319.
T. Kotek, J. Preen, F. Simon, P. Tittmann, and M. Trinks, Recurrence relations and splitting formulas for the domination polynomial, arXiv:1206.5926 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Domination Polynomial.
Eric Weisstein's World of Mathematics, Helm Graph.

Cf. A048473, A213667.

T := proc (n, k) if k = n then 2^n-1 else 2^(2*n-k)*(2*binomial(n, k-n-1) + binomial(n, k-n)) end if end proc: for n to 10 do seq(T(n, k), k = 1 .. 2*n+1) end d; # yields sequence in triangular form

T[n_, n_] := 2^n - 1;
T[n_, k_] := 2^(2*n - k)*(2*Binomial[n, k - n - 1] + Binomial[n, k - n]);
Table[T[n, k], {n, 1, 10}, {k, 1, 2*n + 1}] // Flatten (* Jean-François Alcover, Dec 02 2017 *)

T(n,k) = 2^(2*n-k)*(2*binomial(n,k-n-1)+binomial(n,k-n)) if k > n; T(n,n)=2^n - 1.
The generating polynomial of row n is g[n] = g[n,x] = (1+x)(x*(2+x))^n - x^n (= domination polynomial of the graph G(n)).
Bivariate g.f.: G(x,z) = x*z*(1+x)*(2+x)/(1-2*x*z-x^2*z)-x*z/(1-xz).

A230445 Triangle read by rows: T(n,m) = 3^m*2^(n-m)-1, the number of neighbors in an n-dimensional cubic array.

Original entry on oeis.org

0, 1, 2, 3, 5, 8, 7, 11, 17, 26, 15, 23, 35, 53, 80, 31, 47, 71, 107, 161, 242, 63, 95, 143, 215, 323, 485, 728, 127, 191, 287, 431, 647, 971, 1457, 2186, 255, 383, 575, 863, 1295, 1943, 2915, 4373, 6560, 511, 767, 1151, 1727, 2591, 3887, 5831, 8747, 13121

Offset: 0

Ron R. King, Oct 18 2013

Let n be the dimension of the cubic array.
Let m be the "placement depth" of the cell within the array. m = (number of horizontal or vertical neighbors)-n. 0 <= m <= n.
Let T(n,m) represent the number of neighbors (horizontally, vertically, or diagonally) a cell has in an n-dimensional cube that has at least 3^n cells.
The sequence forms a triangle structure similar to Pascal’s triangle: T(0,0) in row one, T(1,0), T(1,1) in row two, etc.
The triangle in A094615 is a subtriangle. - Philippe Deléham, Oct 31 2013
In a finite n-dimensional hypercube lattice, the sequence gives the number of nodes situated at a Chebyshev distance of 1 for a node, situated on an m-cube bound, which is not on an (m-1)-cube bound. The number of m-cube bounds for n-cube is given by A013609. In cellular automata theory, the cell surrounding with Chebyshev distance 1 is called Moore's neighborhood. For von Neumann neighborhood (with Manhattan distance 1), an analogous sequence is represented by A051162. - Dmitry Zaitsev, Oct 22 2015

			Triangle starts:
n \ m  0    1    2    3    4    5     6     7     8     9    10 ...
0:     0
1:     1    2
2:     3    5    8
3:     7   11   17   26
4:    15   23   35   53   80
5:    31   47   71  107  161  242
6:    63   95  143  215  323  485   728
7:   127  191  287  431  647  971  1457  2186
8:   255  383  575  863 1295 1943  2915  4373  6560
9:   511  767 1151 1727 2591 3887  5831  8747 13121 19682
10: 1023 1535 2303 3455 5183 7775 11663 17495 26243 39365 59048
... (reformatted (and extended) by _Wolfdieter Lang_, May 04 2022)
For a 3-d cube, at a corner, the number of horizontal and vertical neighbors is 3, hence m = 3-3 = 0.
Along the edge, the number of horizontal and vertical neighbors is 4, hence m = 4-3 = 1.
In a face, the number of horizontal and vertical neighbors is 5, hence m = 5-3 = 2.
In the interior, the number of horizontal and vertical neighbors is 6, hence m = 6-3 = 3.
T(3,2) = 17 because a cell on the face of a 3-d cube has 17 neighbors.

Dmitry A. Zaitsev, A generalized neighborhood for cellular automata, Theoretical Computer Science, 2016, Volume 666, 1 March 2017, Pages 21-35.

Cf. A000225, A024023, A051162, A013609.
Sequence numbers are 1 less than A036561.
Cf. A048473, A094615.

void a10(){int p3[10], p2[10], n, m, a; p3[0]=1; p2[0]=1;
for(n=1;n<10;n++){ p2[n]=p2[n-1]*2; p3[n]=p3[n-1]*3;
  for(m=0;m<=d;m++){ a=p3[m]*p2[n-m]-1; printf("%d ",a); }
  printf("\n"); } } /* Dmitry Zaitsev, Oct 23 2015 */

Table[3^m 2^(n - m) - 1, {n, 0, 9}, {m, 0, n}] // Flatten (* Michael De Vlieger, Oct 23 2015 *)

T(n,m) = 3^m*2^(n-m)-1, 0 <= m <= n.
T(n,0) = 2^n-1. (A000225)
T(n,n) = 3^n-1. (A024023)
T(n,m) = (3*T(n,m-1)+1)/2, first part of the Collatz sequence for the number 2^n-1, for n >= 1.
T(n,m) = (T(n-1,m) + T(n,m+1))/2, 0 <= m <= n-1.
T(n,m) = 1 + T(n-1,m-1) + T(n,m-1), 1 <= m <= n.
m = T2(n,k)-n, where T2(n,k) is A051162.
From Wolfdieter Lang, May 04 2022: (Start)
G.f. for column m: G(m, x) = x^m*(3^m - 1 - (3^m - 2)*x)/((1 - 2*x)*(1 - x)).
G.f. for row polynomials R(n, x) = Sum_{m=1..n} T(n, m)*x^m, for n >= 0: G(z, x) = z*(1 + (2 - 5*z)*x)/((1 - 2*z)*(1 - z)*(1 - 3*x*z)*(1 - x*z)).
(End)

A277106 a(n) = 8*3^n - 12.

Original entry on oeis.org

12, 60, 204, 636, 1932, 5820, 17484, 52476, 157452, 472380, 1417164, 4251516, 12754572, 38263740, 114791244, 344373756, 1033121292, 3099363900, 9298091724, 27894275196, 83682825612, 251048476860, 753145430604, 2259436291836, 6778308875532

Offset: 1

Emeric Deutsch, Nov 05 2016

a(n) is the first Zagreb index of the Sierpiński [Sierpinski] gasket graph S[n].
The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternately, it is the sum of the degree sums d(i)+d(j) over all edges ij of the graph.
The M-polynomial of the Sierpinski gasket graph S[n] is  M(S[n],x,y) = 6*x^2*y^4 + (3^n - 6)*x^4*y^4.

E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
I. Gutman and K. C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50, 2004, 83-92.
Eric Weisstein's World of Mathematics, Sierpiński Gasket Graph.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A277107.

Maple
```
seq(8*3^n-12, n = 1..30);
```

Array[8*3^# - 12 &, 25] (* Robert G. Wilson v, Nov 05 2016 *)
LinearRecurrence[{4,-3},{12,60},40] (* Harvey P. Dale, Oct 25 2020 *)

G.f.: 12*x*(1 + x)/((1 - x)*(1 - 3*x)).
a(n) = 4*a(n-1) - 3*a(n-2).
a(n)=12*A048473(n-1). - R. J. Mathar, Apr 07 2022

cp's OEIS Frontend

A195732 Numbers k such that 2*(3^k-k)-1 is prime.

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A198686 4*7^n-1.

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Magma

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A198687 5*7^n-1.

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Magma

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A198689 8*7^n-1.

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Magma

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A198690 9*7^n-1.

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Magma

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A198691 10*7^n-1.

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Magma

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A198692 a(n) = 11*7^n-1.

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Magma

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A213666 Irregular triangle read by rows: T(n,k) is the number of dominating subsets with k vertices of the graph G(n) obtained by taking n copies of the path P_3 and identifying one of their endpoints (a star with n branches of length 2).

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Maple