A181565 -id:A181565 - cp's OEIS frontend

A199108 a(n) = 4*3^n + 1.

5, 13, 37, 109, 325, 973, 2917, 8749, 26245, 78733, 236197, 708589, 2125765, 6377293, 19131877, 57395629, 172186885, 516560653, 1549681957, 4649045869, 13947137605, 41841412813, 125524238437, 376572715309, 1129718145925, 3389154437773, 10167463313317, 30502389939949

Offset: 0

Vincenzo Librandi, Nov 03 2011

An Engel expansion of 3/4 to the base 3 as defined in A181565, with the associated series expansion 3/4 = 3/5 + 3^2/(5*13) + 3^3/(5*13*37) + 3^4/(5*13*37*109) + .... - Peter Bala, Oct 29 2013

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

Cf. A181565.

Magma
```
[4*3^n+1 : n in [0..30]];
```

4*3^Range[0,30]+1 (* or *) LinearRecurrence[{4,-3},{5,13},30] (* or *) NestList[3#-2&,5,30] (* Harvey P. Dale, Mar 01 2012 *)

a(n) = 3*a(n-1) - 2.
a(n) = 4*a(n-1) - 3*a(n-2).
G.f.: (5-7*x)/((1-x)*(1-3*x)). - Bruno Berselli, Nov 03 2011
E.g.f.: exp(x)*(1 + 4*exp(2*x)). - Elmo R. Oliveira, May 07 2025

A245179 Numbers of the form 2^k+3 or 3*2^k+1, k >= 2.

Original entry on oeis.org

7, 11, 13, 19, 25, 35, 49, 67, 97, 131, 193, 259, 385, 515, 769, 1027, 1537, 2051, 3073, 4099, 6145, 8195, 12289, 16387, 24577, 32771, 49153, 65539, 98305, 131075, 196609, 262147, 393217, 524291, 786433, 1048579, 1572865, 2097155, 3145729, 4194307, 6291457

Offset: 1

N. J. A. Sloane, Jul 17 2014

Numbers whose binary expansion is 10..011 or 110..01.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-2).

Essentially the union of A062709 and A181565. Cf. A245178.

&cat [[3*2^i+1,2^(i+2)+3]: i in [1..30]]; // Bruno Berselli, Jul 23 2014

CoefficientList[Series[- (14 x^3 + 8 x^2 - 11 x - 7)/((x - 1) (x + 1) (2 x^2 - 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Jul 23 2014 *)
LinearRecurrence[{0,3,0,-2},{7,11,13,19},50] (* Harvey P. Dale, Mar 05 2015 *)

a(2k) = 2^(k+2)+3, a(2k+1) = 3*2^(k+1)+1. - N. J. A. Sloane, Jul 19 2014

a(n) = 3*a(n-2)-2*a(n-4). G.f.: -x*(14*x^3+8*x^2-11*x-7) / ((x-1)*(x+1)*(2*x^2-1)). - Colin Barker, Jul 19 2014

A275970 a(n) = 3*2^n + n - 1.

Original entry on oeis.org

2, 6, 13, 26, 51, 100, 197, 390, 775, 1544, 3081, 6154, 12299, 24588, 49165, 98318, 196623, 393232, 786449, 1572882, 3145747, 6291476, 12582933, 25165846, 50331671, 100663320, 201326617, 402653210, 805306395, 1610612764, 3221225501, 6442450974, 12884901919, 25769803808, 51539607585, 103079215138, 206158430243, 412316860452, 824633720869

Offset: 0

Miquel Cerda, Aug 15 2016

Carauleanu Marc, Table of n, a(n) for n = 0..400
S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

LinearRecurrence[{4,-5,2},{2,6,13}, 25] (* or *) Table[3*2^n + n - 1, {n,0,25}] (* G. C. Greubel, Aug 18 2016 *)

a(n)=3*2^n+n-1 \\ Charles R Greathouse IV, Aug 27 2016

a(n) = 2*a(n-1) - n + 2.
a(n+1) - a(n) = A181565(n)
a(n) = A007283(n) + n - 1
a(n) = A083706(n) + A000079(n)
a(n) = A145071(n+1) - A000079(n)
a(n) = A079583(n) + A005408(n)
a(n) = A068156(n+1) - A079583(n)
a(n) = (A068156(n+1) + A005408(n)) / 2
a(n) = A000225(n) + A000325(n+1) + A005408(n)
a(n) = A068156(n+1) - A000225(n) - A000325(n+1)
a(n) = A068156(n+1) - A007283(n) + n + 2.
a(n) = A000079(n) + A000225(n) + A000295(n) + A005408(n)
From G. C. Greubel, Aug 18 2016: (Start)
O.g.f.: (2 - 2*x - x^2)/( (1-2*x)*(1-x)^2 ).
E.g.f.: 3*exp(2*x) + (x-1)*exp(x).
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-2). (End)

A324604 Table read by rows: T(n,k) is the number of non-intersecting loops starting at (0,0) on the n X k torus consisting of steps up and to the right, 1 <= k <= n.

Original entry on oeis.org

2, 2, 4, 2, 7, 22, 2, 13, 66, 258, 2, 25, 189, 898, 3528, 2, 49, 537, 3118, 14545, 87830, 2, 97, 1544, 11462, 75446, 746589, 8295536, 2, 193, 4508, 44990, 447667

Offset: 1

Peter Kagey, Mar 09 2019

Conjecture: Second column is A181565.

			The T(3,2) = 7 walks on the 3 X 2 torus are:
  (0,0),(0,1),(0,0)
  (0,0),(1,0),(2,0),(0,0)
  (0,0),(0,1),(1,1),(1,0),(2,0),(0,0)
  (0,0),(0,1),(1,1),(2,1),(2,0),(0,0)
  (0,0),(1,0),(1,1),(2,1),(0,1),(0,0)
  (0,0),(1,0),(1,1),(2,1),(2,0),(0,0)
  (0,0),(1,0),(2,0),(2,1),(0,1),(0,0)
Table begins:
  2
  2,  4,
  2,  7,   22,
  2, 13,   66,   258,
  2, 25,  189,   898,  3528,
  2, 49,  537,  3118, 14545,  87830,
  2, 97, 1544, 11462, 75446, 746589, 8295536

Peter Kagey, Cycles on the torus, Programming Puzzles & Code Golf Stack Exchange.

Cf. A181565, A324603.

A248416 Rectangular array by antidiagonals: for n >= 0, row n gives the positions in the Thue-Morse sequence A010059 at which the first 2^n terms occur.

Original entry on oeis.org

1, 4, 1, 6, 4, 1, 7, 7, 7, 1, 10, 11, 13, 13, 1, 11, 13, 21, 25, 25, 1, 13, 16, 25, 41, 49, 49, 1, 16, 19, 31, 49, 81, 97, 97, 1, 18, 21, 37, 61, 97, 161, 193, 193, 1, 19, 25, 41, 73, 121, 193, 321, 385, 385, 1, 21, 28, 49, 81, 145, 241, 385, 641, 769, 769, 1, 24, 31, 55, 97, 161, 289, 481, 769, 1281, 1537, 1537, 1

Offset: 1

Clark Kimberling, Oct 06 2014

Each row contains contains its successor as a proper subsequence.
Note that this supposes that the Thue-Morse sequence A010059 has offset 1, whereas the true offset is 0. So really the entries should all be reduced by 1. - N. J. A. Sloane, Jul 01 2016
Apparently T(n,3) = A004119(n+1) for n>0. Apparently T(n,4) = A083575(n) for n>0. - R. J. Mathar, Nov 06 2018

			Northwest corner, n>=0, k>=1:
   1    4    6    7   10   11   13   16   18   19
   1    4    7   11   13   16   19   21   25   28
   1    7   13   21   25   31   37   41   49   55
   1   13   25   41   49   61   73   81   97  109
   1   25   49   81   97  121  145  161  193  217
   1   49   97  161  193  241  289  321  385  433
   1   97  193  321  385  481  577  641  769  865
The Thue-Morse sequence A010059 begins with 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, from which we see that the first 4 terms (=1,0,0,1) occur at positions 1, 7, 13, ..., as indicated for row n=2.

Cf. A010059 (Thue-Morse), A026147 (row 0), A091855 (row 1?), A157971 (row 2?),

Column 1 is essentially A004119 (or A181565).

A010060 := proc(n)
    local i;
    add(i, i=convert(n, base, 2)) mod 2 ;
end proc:
A010059 := proc(n)
    1-A010060(n) ;
end proc:
A248416Off0 := proc(n,k)
    option remember ;
    local strtN,binpat,src,thue ;
    if k = 1 then
        strtN := 0 ;
    else
        strtN := 1+procname(n,k-1) ;
    end if;
    binpat := [seq(A010059(i),i=0..n-1)] ;
    for src from strtN do
        thue := [seq(A010059(i),i=src..src+nops(binpat)-1)] ;
        if binpat=thue then
            return src ;
        end if;
    end do:
end proc:
A248416 := proc(n,k)
    1+A248416Off0(2^n,k) ;
end proc:
for d from 1 to 11 do
    for k from d to 1 by -1 do
        printf("%d,",A248416(d-k,k)) ;
end do: # R. J. Mathar, Nov 06 2018

z = 3000; u = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 20]; Length[u]
t[p_, q_] := t[p, q] = Table[u[[k]], {k, p, q}];
r[n_] := Select[Range[z], t[#, # + 2^(n - 1)] == t[1, 1 + 2^(n - 1)] &]
TableForm[Table[r[n], {n, 0, 10}]]

Definitions and examples clarified. - R. J. Mathar, Nov 06 2018

A288870 Triangle T from array A(k,n) = (2k+1)2^n + 1, k >=0, n >= 0 read by downwards antidiagonals.

Original entry on oeis.org

2, 3, 4, 5, 7, 6, 9, 13, 11, 8, 17, 25, 21, 15, 10, 33, 49, 41, 29, 19, 12, 65, 97, 81, 57, 37, 23, 14, 129, 193, 161, 113, 73, 45, 27, 16, 257, 385, 321, 225, 145, 89, 53, 31, 18, 513, 769, 641, 449, 289, 177, 105, 61, 35, 20, 1025, 1537, 1281, 897, 577, 353, 209, 121, 69, 39, 22

Offset: 0

Wolfdieter Lang, Jun 21 2017

This entry was motivated by a class work of Ferran D.

			The array A begins:
k\n  0  1  2   3   4   5    6    7    8    9    10 ...
0:   2  3  5   9  17  33   65  129  257  513  1025
1:   4  7 13  25  49  97  193  385  769 1537  3073
2:   6 11 21  41  81 161  321  641 1281 2561  5121
3:   8 15 29  57 113 225  449  897 1793 3585  7169
4:  10 19 37  73 145 289  577 1153 2305 4609  9217
5:  12 23 45  89 177 353  705 1409 2817 5633 11265
6:  14 27 53 105 209 417  833 1665 3329 6657 13313
7:  16 31 61 121 241 481  961 1921 3841 7681 15361
8:  18 35 69 137 273 545 1089 2177 4353 8705 17409
9:  20 39 77 153 305 609 1217 2433 4865 9729 19457
...
The triangle T begins:
m\k    0    1    2   3   4   5   6   7  8  9 10 ...
0:     2
1:     3    4
2:     5    7    6
3:     9   13   11   8
4:    17   25   21  15  10
5:    33   49   41  29  19  12
6:    65   97   81  57  37  23  14
7:   129  193  161 113  73  45  27 16
8:   257  385  321 225 145  89  53 31 18
9:   513  769  641 449 289 177 105 61 35 20
10: 1025 1537 1281 897 577 353 209 121 69 39 22
...

Cf. A288871. Columns of T (no 0's, or rows of A): A000051, A181565, A083575, A083686, A083705, A083683, A168596.

Row sums give A077802(n+1) or A095151(n+1).

Table[(2 k + 1)*2^(m - k) + 1, {m, 0, 10}, {k, 0, m}] // Flatten (* Michael De Vlieger, Jun 25 2017 *)

A(n, k) = (2*n + 1)*2^k + 1;
for(n=0, 10, for(k=0, n, print1(A(k, n - k),", "))) \\ Indranil Ghosh, Jun 22 2017

Array A(k, n) = (2*k+1)*2^n + 1 for k >= 0 and n >= 0.
Triangle T(m, k) = A(k, m-k) = (2*k+1)*2^(m-k) + 1, k >= m >= 0, otherwise T(m, k) = 0.
O.g.f. for column k of T: x^k*(2*(k+1) - (2*k+3)*x)/((1-2*x)*(1-x)), k >= 0.
E.g.f. for column k of T (without leading 0's): (2*k+1)*exp(2*x) + exp(x), k>=0.
E.g.f. for column k of T: 2^(-k)*(2*k+1)*exp(2*x) + exp(x) - S(k,x), with S(k, x) = 2^(-k)* Sum_{m=1..k} A288871(k,m)*x^(m-1)/(m-1)! if k >=1 and S(0,x) = 0.

A334164 a(n) is the number of ON-cells in the completed n-th level of a triangular wedge in the hexagonal grid of A151723 (i.e., after 2^k >= n generations of the automaton in A151723 have been computed).

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 4, 8, 2, 10, 6, 10, 4, 14, 8, 16, 2, 18, 10, 16, 8, 20, 12, 22, 6, 26, 14, 22, 8, 30, 16, 32, 2, 34, 18, 28, 16, 34, 18, 32, 14, 40, 22, 34, 16, 42, 24, 44, 10, 50, 26, 40, 20, 48, 28, 50, 14, 58, 30, 46, 16, 62, 32, 64

Offset: 1

Hartmut F. W. Hoft, Apr 17 2020

nonn

Conjecture 1: Except for a(2^n + 1) = 2, n >= 1, for odd-numbered completed levels a lower bound of the ratio of ON-cells to the length of the level is (2^n + 2)/(3*2^(n+1) + 1) with limit 1/6, determined by the subsequence of levels starting with: 13, 25, 49, 97, 193, 385, 769, 1537, 3073, ..., and the associated ON-cell counts: 4, 6, 10, 18, 34, 66, 130, 258, 514, ..., as listed in the second column of each of the two triangles below.
The ON-cell counts for the indices in each row define a line of slope 1/2.  The formula for the indices of levels in row k >= 2 is  L(k, i) = 1 + Sum_{j = 0, ..., i} 2^(k-j), 0 <= i <= k - 2, and the formula for the associated numbers of ON-cells is C(k, i) = 2 + Sum_{j = 1..i} 2^(k-1-j), 0 <= i <= k - 2:
Index of the level: L(k, i) number of ON-cells: C(k, i)
k\i  0   1   2   3   4   5   6    k/i  0   1   2   3   4   5   6
2:   5                             2:  2
3:   9  13                         3:  2   4
4:  17  25  29                     4:  2   6   8
5:  33  49  57  61                 5:  2  10  14  16
6:  65  97 113 121 125             6:  2  18  26  30  32
7: 129 193 225 241 249 253         7:  2  34  50  58  62  64
8: 257 385 449 481 497 505 509     8:  2  66  98 114 122 126 128
...
The pairs ( L(k, i), C(k, i) ), for 0 <= i <= k-2, define a line of slope 1/2 for each k >= 3.
For triangle L(k, i): column 0 is A000051(n), n >= 2; column 1 is A181565(n), n >= 3; column 2 is A083686(n), n >= 2; columns 3 is A195744(n), n >= 1; column 4 is A206371(n), n >= 2; column 5 is A196657(n), n >= 1; the bounding diagonal is A036563(n), n >= 3.
For triangle C(k, i): column 1 is A052548(n), n >= 1; column 2 is A164094(n), n >= 1.
Conjecture 2: In an even-numbered completed level 2*n the fraction of ON-cells is bounded below by (23 * 2^n - 24)/(2^(n+5) - 36) with limit 23/32, determined by the subsequence of levels starting with: 28, 92, 220, 476, 988, 2012, 4060, ... .
There are 16 numbers less than 1000 that do not occur as the number of ON-cells in a completed level through level 16384: 136, 164, 330, 334, 402, 444, 526, 570, 598, 604, 614, 714, 740, 822, 832, 878.
Sequence A334169 of even-numbered completed levels in which all cells are ON-cells is a subsequence of this sequence.

Hartmut F. W. Hoft, Diagram of ON-cell counts

Cf. A000051, A036563, A052548, A083686, A151723, A164094, A181565, A195744, A196657, A206371, A334169.

(* a169781[] and support functions are defined in A169781 and create the list nTriangle *)
a334164[n_] := Module[{k, levels={}}, a169781[n]; For[k=1, k<=n, k++, AppendTo[levels, Count[nTriangle[[k]], 1] - 2]]; levels]/;(n>=3 && IntegerQ[Log[2,n]])
a334164[64] (* sequence data *)

A172097 Table T(n,k) read by rows which contains in row n the 2^n terms of A171968 starting at the (n+1)st 3.

Original entry on oeis.org

3, 3, 7, 3, 7, 11, 15, 3, 7, 11, 15, 19, 23, 27, 31, 3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59, 63, 3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59, 63, 67, 71, 75, 79, 83, 87, 91, 95, 99, 103, 107, 111, 115, 119, 123, 127, 3

Offset: 0

Paul Curtz, Nov 19 2010

The positions of the 3's in A171968 are 3, 6, 12, ... = A007283(n), or A181565(n) if A171968 is interpreted with offset 1.

			The table starts in row n as:
3;
3,7;
3,7,11,15;
3,7,11,15,19,23,27,31;

Cf. A006257.

A213657 Irregular triangle read by rows: T(n,k) is the number of dominating subsets with k vertices of the graph G(n) consisting of an edge ab, n vertices c_1, c_2, ..., c_n, and 2n edges ac_i, bc_i (i=1..n). (n triangles with a common edge).

Original entry on oeis.org

3, 3, 1, 2, 6, 4, 1, 2, 7, 10, 5, 1, 2, 9, 16, 15, 6, 1, 2, 11, 25, 30, 21, 7, 1, 2, 13, 36, 55, 50, 28, 8, 1, 2, 15, 49, 91, 105, 77, 36, 9, 1, 2, 17, 64, 140, 196, 182, 112, 45, 10, 1, 2, 19, 81, 204, 336, 378, 294, 156, 55, 11, 1

Offset: 1

Emeric Deutsch, Jun 29 2012

Row n contain n + 2 entries.

Sum of entries in row n = 1 + 3*2^n = A181565(n).

			Row 1 is 3,3,1 because the graph G(1) is the triangle abc; there are 3 dominating subsets of size 1 ({a}, {b}, {c}), 3 dominating subsets of size 2 ({a,b}, {a,c}, {b,c}), and 1 dominating subset of size 3 ({a,b,c}).
T(n,1)=2 for n >= 2 because {a} and {b} are the only dominating subsets of size k=1.
Triangle starts:
  3, 3,  1;
  2, 6,  4,  1;
  2, 7, 10,  5, 1;
  2, 9, 16, 15, 6, 1;

S. Alikhani and E. Deutsch, Graphs with domination roots in the right half-plane, arXiv preprint arXiv:1305.3734, 2013
S. Alikhani and Y. H. Peng, Introduction to domination polynomial of a graph, arXiv:0905.2251.
T. Kotek, J. Preen, F. Simon, P. Tittmann, and M. Trinks, Recurrence relations and splitting formulas for the domination polynomial, arXiv:1206.5926.

Cf. A181565

T := proc (n, k) if k = n then (1/2)*(n+1)*(n+2) else 2*binomial(n, k-1)+binomial(n, k-2) end if end proc: for n to 12 do seq(T(n, k), k = 1 .. n+2) end do; # yields sequence in triangular form

T[n_, k_] := If[k==n, (n+1)*(n+2)/2, 2*Binomial[n, k-1]+Binomial[n, k-2]];
Table[T[n, k], {n, 1, 10}, {k, 1, n+2}] // Flatten (* Jean-François Alcover, Dec 09 2017 *)

Generating polynomial of row n is x^n + x*(2+x)*(1+x)^n; this is the domination polynomial of the graph G(n).

T(n,n) = (n+1)*(n+3)/2; T(n,k) = 2*binomial(n, k-1) + binomial(n, k-2) if k != n.

cp's OEIS Frontend

A199108 a(n) = 4*3^n + 1.

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A245179 Numbers of the form 2^k+3 or 3*2^k+1, k >= 2.

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A275970 a(n) = 3*2^n + n - 1.

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A324604 Table read by rows: T(n,k) is the number of non-intersecting loops starting at (0,0) on the n X k torus consisting of steps up and to the right, 1 <= k <= n.

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A248416 Rectangular array by antidiagonals: for n >= 0, row n gives the positions in the Thue-Morse sequence A010059 at which the first 2^n terms occur.

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A288870 Triangle T from array A(k,n) = (2*k+1)*2^n + 1, k >=0, n >= 0 read by downwards antidiagonals.

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A334164 a(n) is the number of ON-cells in the completed n-th level of a triangular wedge in the hexagonal grid of A151723 (i.e., after 2^k >= n generations of the automaton in A151723 have been computed).

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A172097 Table T(n,k) read by rows which contains in row n the 2^n terms of A171968 starting at the (n+1)st 3.

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A213657 Irregular triangle read by rows: T(n,k) is the number of dominating subsets with k vertices of the graph G(n) consisting of an edge ab, n vertices c_1, c_2, ..., c_n, and 2n edges ac_i, bc_i (i=1..n). (n triangles with a common edge).

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A288870 Triangle T from array A(k,n) = (2k+1)2^n + 1, k >=0, n >= 0 read by downwards antidiagonals.