A228754 -id:A228754 - cp's OEIS frontend

A230699 Sequence of pairs k,g such that k2^n-1, k2^n-1+g, k2^n-1+2g, and k2^n+3g are four consecutive primes in arithmetic progression for the smallest odd k.

135, -6, 63, 6, 415, -6, 987, 6, 55, -6, 273, 6, 1195, -6, 299, 18, 1371, 6, 5, -6, 189, 6, 1077, 6, 7111, 6, 15, -6, 2821, -18, 15465, 24, 1081, 6, 11475, -6, 17155, -6, 3393, 12, 9751, 6, 16523, -24, 165, -6, 7395, -6, 8695, -6, 20325, -6, 7153, 18, 2235, -6

Offset: 1

Pierre CAMI, Oct 30 2013

sign

The number g may be negative.

g is always 0 mod 6 so a multiple of 6.

			135*2^1-1=269, 135*2^1-1-6=263, 135*2^1-1-2*6=257, 135*2^1-1-3*6=251
269, 263, 257, 251 are four consecutive primes in arithmetic progression so a(1)=135, a(2)=-6.
63*2^2-1=251, 63*2^2-1+6=257, 63*2^2-1+2*6=263, 63*2^2-1-3*6=269
251, 257, 263, 269 are four consecutive primes in arithmetic progression so a(3)=63 a(4)=6.

Pierre CAMI, Table of n, a(n) for n = 1..390

Cf. A227888, A228452, A228754.

A228749 Number of n X n binary arrays with top left element equal to 1 and no two ones adjacent horizontally or antidiagonally.

Original entry on oeis.org

1, 3, 39, 780, 43862, 4823826, 1316714732, 762713002760, 1040109467257908, 3118624187066762728, 21501868781585259257676, 331091404725670052219822756, 11603262095526618070600084170304, 914315533886530112431287075923892544

Offset: 1

R. H. Hardin Sep 02 2013

nonn

Diagonal of A228754

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

R. H. Hardin, Table of n, a(n) for n = 1..30

A228750 Number of n X 4 binary arrays with top left element equal to 1 and no two ones adjacent horizontally or antidiagonally.

Original entry on oeis.org

3, 20, 126, 780, 4808, 29608, 182288, 1122240, 6908896, 42533440, 261849728, 1612032128, 9924194048, 61096565760, 376130326016, 2315580595200, 14255467112448, 87761291061248, 540287045521408, 3326182739886080

Offset: 1

R. H. Hardin, Sep 02 2013

nonn

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

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

Column 4 of A228754.

Empirical: a(n) = 8*a(n-1) - 12*a(n-2) + 4*a(n-3).

Empirical g.f.: x*(3 - 4*x + 2*x^2) / (1 - 8*x + 12*x^2 - 4*x^3). - Colin Barker, Sep 12 2018

A228751 Number of n X 5 binary arrays with top left element equal to 1 and no two ones adjacent horizontally or antidiagonally.

Original entry on oeis.org

5, 50, 482, 4599, 43862, 418370, 3990739, 38067290, 363121586, 3463797759, 33040991486, 315176359538, 3006451507579, 28678390419458, 273561730649282, 2609491655262279, 24891810279661862, 237441732282894050

Offset: 1

R. H. Hardin, Sep 02 2013

nonn

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

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

Column 5 of A228754.

Empirical: a(n) = 13*a(n-1) - 36*a(n-2) + 29*a(n-3) - 5*a(n-4) for n>5.

Empirical g.f.: x*(5 - 15*x + 12*x^2 - 12*x^3 + 2*x^4) / ((1 - x)*(1 - 12*x + 24*x^2 - 5*x^3)). - Colin Barker, Sep 12 2018

A228752 Number of n X 6 binary arrays with top left element equal to 1 and no two ones adjacent horizontally or antidiagonally.

Original entry on oeis.org

8, 119, 1712, 24246, 342207, 4823826, 67970044, 957616341, 13491214832, 190066959598, 2677695197199, 37723794440794, 531458704862804, 7487272853619205, 105481862639606840, 1486044856081515654

Offset: 1

R. H. Hardin, Sep 02 2013

nonn

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

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

Column 6 of A228754.

Empirical: a(n) = 21*a(n-1) - 112*a(n-2) + 217*a(n-3) - 157*a(n-4) + 36*a(n-5) for n>6.

Empirical g.f.: x*(8 - 49*x + 109*x^2 - 114*x^3 + 218*x^4 - 78*x^5) / (1 - 21*x + 112*x^2 - 217*x^3 + 157*x^4 - 36*x^5). - Colin Barker, Sep 12 2018

A228753 Number of nX7 binary arrays with top left element equal to 1 and no two ones adjacent horizontally or antidiagonally.

Original entry on oeis.org

13, 289, 6277, 134440, 2876170, 61534448, 1316714732, 28177227352, 602998827928, 12904414518912, 276160275058160, 5909958226524512, 126475877308730464, 2706643227463513344, 57923438257867988416, 1239588830689830585216

Offset: 1

R. H. Hardin Sep 02 2013

nonn

Column 7 of A228754

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

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

Empirical: a(n) = 34*a(n-1) -324*a(n-2) +1264*a(n-3) -2236*a(n-4) +1816*a(n-5) -624*a(n-6) +64*a(n-7) for n>9

A228755 Number of 3 X n binary arrays with top left element equal to 1 and no two ones adjacent horizontally or antidiagonally.

Original entry on oeis.org

4, 8, 39, 126, 482, 1712, 6277, 22700, 82580, 299648, 1088499, 3952186, 14352786, 52119040, 189266297, 687294648, 2495834292, 9063317432, 32912374319, 119517358582, 434013128786, 1576067091632, 5723300581661, 20783486354532

Offset: 1

R. H. Hardin, Sep 02 2013

nonn

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

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

Row 3 of A228754.

Empirical: a(n) = 2*a(n-1) + 6*a(n-2) - a(n-4).

Empirical g.f.: x*(2 - x)*(2 + x) / (1 - 2*x - 6*x^2 + x^4). - Colin Barker, Sep 12 2018

A228756 Number of 4 X n binary arrays with top left element equal to 1 and no two ones adjacent horizontally or antidiagonally.

Original entry on oeis.org

8, 21, 168, 780, 4599, 24246, 134440, 728537, 3988862, 21739002, 118720559, 647758554, 3535719686, 19295840383, 105313615878, 574764393110, 3136909949941, 17120293581344, 93437607513222, 509954720089427

Offset: 1

R. H. Hardin, Sep 02 2013

nonn

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

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

Row 4 of A228754.

Empirical: a(n) = a(n-1) + 20*a(n-2) + 27*a(n-3) - 14*a(n-4) - 25*a(n-5) + 4*a(n-6) + 5*a(n-7) - a(n-8).

Empirical g.f.: x*(8 + 13*x - 13*x^2 - 24*x^3 + 4*x^4 + 5*x^5 - x^6) / ((1 + x)*(1 + 2*x - x^2)*(1 - 4*x - 9*x^2 + 5*x^3 + 4*x^4 - x^5)). - Colin Barker, Sep 13 2018

A228757 Number of 5Xn binary arrays with top left element equal to 1 and no two ones adjacent horizontally or antidiagonally.

Original entry on oeis.org

16, 55, 723, 4808, 43862, 342207, 2876170, 23326164, 192379841, 1574217212, 12928573892, 106000461919, 869759215471, 7134075563384, 58525595955046, 480089753747591, 3938341082031018, 32307082900690396, 265023929648410933

Offset: 1

R. H. Hardin Sep 02 2013

nonn

Row 5 of A228754

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

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

Empirical: a(n) = a(n-1) +49*a(n-2) +107*a(n-3) -154*a(n-4) -404*a(n-5) +250*a(n-6) +472*a(n-7) -278*a(n-8) -168*a(n-9) +131*a(n-10) -9*a(n-11) -7*a(n-12) +a(n-13)

A228758 Number of 6Xn binary arrays with top left element equal to 1 and no two ones adjacent horizontally or antidiagonally.

Original entry on oeis.org

32, 144, 3111, 29608, 418370, 4823826, 61534448, 746135864, 9275632316, 113914525214, 1407229306722, 17335288371612, 213833281859269, 2636011878793640, 32504748521697000, 400762354405626468

Offset: 1

R. H. Hardin Sep 02 2013

nonn

Row 6 of A228754

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

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

Empirical: a(n) = a(n-1) +119*a(n-2) +397*a(n-3) -1395*a(n-4) -5213*a(n-5) +8797*a(n-6) +24443*a(n-7) -36756*a(n-8) -45888*a(n-9) +84092*a(n-10) +19388*a(n-11) -83412*a(n-12) +24912*a(n-13) +22741*a(n-14) -12257*a(n-15) -1459*a(n-16) +1607*a(n-17) -49*a(n-18) -71*a(n-19) +3*a(n-20) +a(n-21)

cp's OEIS Frontend

A230699 Sequence of pairs k,g such that k*2^n-1, k*2^n-1+g, k*2^n-1+2*g, and k*2^n+3*g are four consecutive primes in arithmetic progression for the smallest odd k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A228749 Number of n X n binary arrays with top left element equal to 1 and no two ones adjacent horizontally or antidiagonally.

Views

Author

Keywords

Comments

Examples

Links

A228750 Number of n X 4 binary arrays with top left element equal to 1 and no two ones adjacent horizontally or antidiagonally.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A228751 Number of n X 5 binary arrays with top left element equal to 1 and no two ones adjacent horizontally or antidiagonally.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A228752 Number of n X 6 binary arrays with top left element equal to 1 and no two ones adjacent horizontally or antidiagonally.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A228753 Number of nX7 binary arrays with top left element equal to 1 and no two ones adjacent horizontally or antidiagonally.

Views

Author

Keywords

Comments

Examples

Links

Formula

A228755 Number of 3 X n binary arrays with top left element equal to 1 and no two ones adjacent horizontally or antidiagonally.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A228756 Number of 4 X n binary arrays with top left element equal to 1 and no two ones adjacent horizontally or antidiagonally.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A228757 Number of 5Xn binary arrays with top left element equal to 1 and no two ones adjacent horizontally or antidiagonally.

Views

Author

Keywords

Comments

Examples

Links

Formula

A228758 Number of 6Xn binary arrays with top left element equal to 1 and no two ones adjacent horizontally or antidiagonally.

Views

Author

Keywords

Comments

Examples

Links

Formula

A230699 Sequence of pairs k,g such that k2^n-1, k2^n-1+g, k2^n-1+2g, and k2^n+3g are four consecutive primes in arithmetic progression for the smallest odd k.