A153972 -id:A153972 - cp's OEIS frontend

A250742 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with nondecreasing x(i,j)-x(i,j-1) in the i direction and nonincreasing x(i,j)-x(i-1,j) in the j direction.

6, 10, 10, 18, 14, 18, 34, 22, 22, 34, 66, 38, 30, 38, 66, 130, 70, 46, 46, 70, 130, 258, 134, 78, 62, 78, 134, 258, 514, 262, 142, 94, 94, 142, 262, 514, 1026, 518, 270, 158, 126, 158, 270, 518, 1026, 2050, 1030, 526, 286, 190, 190, 286, 526, 1030, 2050, 4098, 2054, 1038

Offset: 1

R. H. Hardin, Nov 27 2014

Table starts
....6...10...18...34...66..130..258..514.1026.2050.4098..8194.16386.32770.65538
...10...14...22...38...70..134..262..518.1030.2054.4102..8198.16390.32774.65542
...18...22...30...46...78..142..270..526.1038.2062.4110..8206.16398.32782.65550
...34...38...46...62...94..158..286..542.1054.2078.4126..8222.16414.32798.65566
...66...70...78...94..126..190..318..574.1086.2110.4158..8254.16446.32830.65598
..130..134..142..158..190..254..382..638.1150.2174.4222..8318.16510.32894.65662
..258..262..270..286..318..382..510..766.1278.2302.4350..8446.16638.33022.65790
..514..518..526..542..574..638..766.1022.1534.2558.4606..8702.16894.33278.66046
.1026.1030.1038.1054.1086.1150.1278.1534.2046.3070.5118..9214.17406.33790.66558
.2050.2054.2062.2078.2110.2174.2302.2558.3070.4094.6142.10238.18430.34814.67582

			Some solutions for n=4 k=4
..0..0..0..0..0....1..0..1..0..1....0..1..0..0..1....1..1..1..1..1
..0..0..0..0..0....1..0..1..0..1....0..1..0..0..1....1..1..1..1..1
..0..0..0..0..0....1..0..1..0..1....0..1..0..0..1....0..0..0..0..0
..1..1..1..1..1....1..0..1..0..1....0..1..0..0..1....0..0..0..0..0
..1..1..1..1..1....1..0..1..0..1....0..1..0..0..1....1..1..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..1450

Column 1 is A052548(n+1)
Column 2 is A153972(n+1)
Diagonal is A000918(n+2)

The constraints apparently result in horizontally or vertically banded arrays, hence:
Empirical: T(n,k) = 2^(k+1)+2^(n+1)-2
Empirical for column k:
k=1: a(n) = 3*a(n-1) -2*a(n-2); a(n) = 2^(n+1) +2
k=2: a(n) = 3*a(n-1) -2*a(n-2); a(n) = 2^(n+1) +6
k=3: a(n) = 3*a(n-1) -2*a(n-2); a(n) = 2^(n+1) +14
k=4: a(n) = 3*a(n-1) -2*a(n-2); a(n) = 2^(n+1) +30
k=5: a(n) = 3*a(n-1) -2*a(n-2); a(n) = 2^(n+1) +62
k=6: a(n) = 3*a(n-1) -2*a(n-2); a(n) = 2^(n+1) +126
k=7: a(n) = 3*a(n-1) -2*a(n-2); a(n) = 2^(n+1) +254

A153973 a(n) = 3a(n-1) - 2a(n-2), with a(1) = 9, a(2) = 12.

Original entry on oeis.org

9, 12, 18, 30, 54, 102, 198, 390, 774, 1542, 3078, 6150, 12294, 24582, 49158, 98310, 196614, 393222, 786438, 1572870, 3145734, 6291462, 12582918, 25165830, 50331654, 100663302, 201326598, 402653190, 805306374, 1610612742, 3221225478

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A146529, A153972

I:=[9,12]; [n le 2 select I[n] else 3*Self(n-1)-2*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Sep 01 2016

a=9;lst={a};Do[a=(a-2)*2-2;AppendTo[lst,a],{n,6!}];lst
NestList[2#-6&,9,30] (* or *) LinearRecurrence[{3,-2},{9,12},31]
Table[ (3/2)*(4 + 2^n), {n, 1, 25}] (* G. C. Greubel, Sep 01 2016 *)

a(n) = 3*a(n-1) - 2*a(n-2), with a(1) = 9, a(2) = 12. - Harvey P. Dale, May 09 2012
From G. C. Greubel, Sep 01 2016: (Start)
a(n) = (3/2)*(4 + 2^n).
G.f.: 3*x*(3 - 5*x)/((1 - x)*(1 - 2*x)).
E.g.f.: (3/2)*(-5 + 4*exp(x) + exp(2*x)). (End)

Definition adapted to offset by Georg Fischer, Jun 18 2021

A245728 Numbers k that divide 2^k + 6.

Original entry on oeis.org

1, 2, 10, 1030, 10009593662, 13957196317, 55299492770, 3764656723270

Offset: 1

Derek Orr, Jul 30 2014

No other terms below 10^15. Some larger terms: 2962089521722084981, 1376243703434217460265762. - Max Alekseyev, Sep 23 2016

			2^10 + 6 = 1030 is divisible by 10. Thus 10 is a term of this sequence.

OEIS Wiki, 2^n mod n

Cf. A015910, A128122, A153972.

select(n -> 2 &^ n + 6 mod n = 0, [$1..10^6]); # Robert Israel, Jul 30 2014

Select[Range[10^5], Divisible[2^# + 6, #] &] (* Robert Price, Oct 12 2018 *)

for(n=1,10^9,if(Mod(2,n)^n==Mod(-6,n),print1(n,", ")))

a(5) from Jason G. Wurtzel, Sep 25 2014

a(6)-a(8) from Max Alekseyev, Sep 23 2016

A209724 1/4 the number of (n+1) X 6 0..2 arrays with every 2 X 2 subblock having distinct clockwise edge differences.

Original entry on oeis.org

8, 9, 10, 12, 14, 18, 22, 30, 38, 54, 70, 102, 134, 198, 262, 390, 518, 774, 1030, 1542, 2054, 3078, 4102, 6150, 8198, 12294, 16390, 24582, 32774, 49158, 65542, 98310, 131078, 196614, 262150, 393222, 524294, 786438, 1048582, 1572870, 2097158

Offset: 1

R. H. Hardin, Mar 12 2012

nonn

Column 5 of A209727.

Conjecture: a(1) = 8; for n > 1, a(n) is the smallest integer m such that m = ((2x * a(n-1)) /(x+1)) - x , with x a positive nontrivial divisor of m. (This is true at least for a(1) to a(100).) - Enric Reverter i Bigas, Oct 11 2020

			Some solutions for n=4:
..2..1..2..1..2..1....2..0..2..0..1..0....2..1..2..1..2..1....0..1..0..1..0..2
..0..2..0..2..0..2....1..2..1..2..0..2....0..2..0..2..0..2....2..0..2..0..2..1
..1..0..1..0..1..0....2..0..2..0..1..0....2..1..2..1..2..1....0..1..0..1..0..2
..0..2..0..2..0..2....1..2..1..2..0..2....0..2..0..2..0..2....2..0..2..0..2..1
..1..0..1..0..1..0....2..0..2..0..1..0....1..0..1..0..1..0....0..1..0..1..0..2

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A153972, A209727.

Empirical: a(n) = a(n-1) +2*a(n-2) -2*a(n-3).
Conjectures from Colin Barker, Mar 07 2018: (Start)
G.f.: x*(8 + x - 15*x^2) / ((1 - x)*(1 - 2*x^2)).
a(n) = 3*2^(n/2-1) + 6 for n even.
a(n) = 2^((n+1)/2) + 6 for n odd.
(End)

A242475 a(n) = 2^n + 8.

Original entry on oeis.org

9, 10, 12, 16, 24, 40, 72, 136, 264, 520, 1032, 2056, 4104, 8200, 16392, 32776, 65544, 131080, 262152, 524296, 1048584, 2097160, 4194312, 8388616, 16777224, 33554440, 67108872, 134217736, 268435464, 536870920, 1073741832

Offset: 0

Vincenzo Librandi, May 20 2014

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

Cf. A000051, A052548, A062709, A140504, A168614, A153972, A168415, A188165.

Magma
```
[2^n+8: n in [0..40]];
```

Table[2^n + 8, {n, 0, 40}] (* or *) CoefficientList[Series[(9 - 17 x)/((1 - x) (1 - 2 x)),{x, 0, 30}], x]
LinearRecurrence[{3,-2},{9,10},40] (* Harvey P. Dale, May 21 2025 *)

G.f.: (9 - 17*x)/((1 - x)*(1 - 2*x)).
a(n) = 2*a(n-1) - 8 = 3*a(n-1) - 2*a(n-2).
a(n) = A052548(n)+6 = A140504(n)+4 = A153972(n)+2.
E.g.f.: exp(2*x) + 8*exp(x). - Elmo R. Oliveira, Nov 11 2023

A246139 a(n) = 2^n + 10.

Original entry on oeis.org

11, 12, 14, 18, 26, 42, 74, 138, 266, 522, 1034, 2058, 4106, 8202, 16394, 32778, 65546, 131082, 262154, 524298, 1048586, 2097162, 4194314, 8388618, 16777226, 33554442, 67108874, 134217738, 268435466, 536870922, 1073741834, 2147483658, 4294967306

Offset: 0

Vincenzo Librandi, Aug 18 2014

First trisection of A085688. [Bruno Berselli, Aug 19 2014]

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

Cf. Sequences of the form 2^n + k: A000079 (k=0), A000051 (k=1), A052548 (k=2), A062709 (k=3), A140504 (k=4), A168614 (k=5), A153972 (k=6), A168415 (k=7), A242475 (k=8), A188165 (k=9), this sequence (k=10).

Cf. A085688.

Magma
```
[2^n+10: n in [0..40]];
```
Mathematica
```
Table[2^n + 10, {n, 0, 40}]
```

vector(50, n, 2^(n-1)+10) \\ Derek Orr, Aug 18 2014

G.f.: (11 - 21*x)/(1 - 3*x + 2*x^2).
a(n) = A000079(n) + 10.
a(n) = 3*a(n-1) - 2*a(n-2) for n > 1.
E.g.f.: exp(2*x) + 10*exp(x). - Elmo R. Oliveira, Nov 11 2023

A267615 a(n) = 2^n + 11.

Original entry on oeis.org

12, 13, 15, 19, 27, 43, 75, 139, 267, 523, 1035, 2059, 4107, 8203, 16395, 32779, 65547, 131083, 262155, 524299, 1048587, 2097163, 4194315, 8388619, 16777227, 33554443, 67108875, 134217739, 268435467, 536870923, 1073741835, 2147483659, 4294967307, 8589934603, 17179869195, 34359738379

Offset: 0

Ilya Gutkovskiy, Jan 18 2016

Recurrence relation b(n) = 3*b(n - 1) - 2*b(n - 2) for n>1, b(0) = k, b(1) = k + 1, gives the closed form b(n) = 2^n + k - 1.

Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. sequences with closed form 2^n + k - 1: A168616 (k=-4), A028399 (k=-3), A036563 (k=-2), A000918 (k=-1), A000225 (k=0), A000079 (k=1), A000051 (k=2), A052548 (k=3), A062709 (k=4), A140504 (k=5), A168614 (k=6), A153972 (k=7), A168415 (k=8), A242475 (k=9), A188165 (k=10), A246139 (k=11), this sequence (k=12).

Cf. A156940.

[2^n+11: n in [0..30]]; // Vincenzo Librandi, Jan 19 2016

Table[2^n + 11, {n, 0, 35}]
LinearRecurrence[{3, -2}, {12, 13}, 40] (* Vincenzo Librandi, Jan 19 2016 *)

a(n) = 2^n + 11; \\ Altug Alkan, Jan 18 2016

G.f.: (12 - 23*x)/(1 - 3*x + 2*x^2).
a(n) = 3*a(n - 1) - 2*a(n - 2) for n>1, a(0)=12, a(1)=13.
a(n) = A000079(n) + A010850(n).
Sum_{n>=0} 1/a(n) = 0.367971714327125...
Lim_{n->oo} a(n + 1)/a(n) = 2.
E.g.f.: exp(2*x) + 11*exp(x). - Elmo R. Oliveira, Nov 08 2023

cp's OEIS Frontend

A250742 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with nondecreasing x(i,j)-x(i,j-1) in the i direction and nonincreasing x(i,j)-x(i-1,j) in the j direction.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A153973 a(n) = 3*a(n-1) - 2*a(n-2), with a(1) = 9, a(2) = 12.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A245728 Numbers k that divide 2^k + 6.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A209724 1/4 the number of (n+1) X 6 0..2 arrays with every 2 X 2 subblock having distinct clockwise edge differences.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A242475 a(n) = 2^n + 8.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A246139 a(n) = 2^n + 10.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A267615 a(n) = 2^n + 11.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A153973 a(n) = 3a(n-1) - 2a(n-2), with a(1) = 9, a(2) = 12.