A304010 -id:A304010 - cp's OEIS frontend

A304004 Number of n X 2 0..1 arrays with every element unequal to 0, 2, 3 or 5 king-move adjacent elements, with upper left element zero.

1, 4, 7, 13, 26, 49, 99, 194, 387, 773, 1538, 3081, 6147, 12298, 24587, 49157, 98330, 196609, 393251, 786450, 1572883, 3145797, 6291442, 12583033, 25165827, 50331738, 100663451, 201326533, 402653578, 805306225, 1610613155, 3221225698

Offset: 1

R. H. Hardin, May 04 2018

nonn

Column 2 of A304010.

			Some solutions for n=5
..0..0. .0..1. .0..1. .0..1. .0..1. .0..1. .0..1. .0..0. .0..1. .0..1
..1..1. .0..1. .1..0. .0..1. .1..0. .1..0. .1..0. .0..0. .1..0. .0..1
..1..1. .0..1. .0..1. .1..0. .0..1. .1..0. .0..1. .0..0. .1..0. .1..0
..1..1. .0..1. .0..1. .1..0. .0..1. .1..0. .1..0. .0..0. .0..1. .0..1
..0..0. .0..1. .0..1. .1..0. .1..0. .0..1. .0..1. .0..0. .0..1. .0..1

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

Cf. A304010.

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

A304111 Partial sums of f(n) = (-1)^(1-A304109(n)).

Original entry on oeis.org

0, 1, 2, 3, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 4, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 3, 4, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 4, 3, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 4, 3, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 4, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 3, 4, 3, 2, 3, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 2, 3, 2, 1, 2, 1, 0, 1, 2, 3, 2, 3, 2, 3, 2, 3, 4, 3, 2

Offset: 0

Antti Karttunen, May 13 2018

sign

Start from the initial value a(0) = 0, after which, for n > 0, each successive term a(n) is either one more or one less than the previous term a(n-1), depending on whether the binary expansion of n encodes a squarefree (0,1)-polynomial when the factorization is done in polynomial ring GF(2)[X], or whether it encodes a polynomial where at least one of its irreducible divisors occurs more than once.

The first negative term occurs as a(153) = -1. See also comments at A304010.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to polynomials in ring GF(2)[X]

Cf. A304107, A304108, A304109, A304110.

up_to = 128;
A304109(n) = { my(fm=factor(Pol(binary(n))*Mod(1, 2))); for(k=1, #fm~, if(fm[k, 2] > 1, return(0))); (1); };
prepare_v304110(up_to) = { my(v=vector(up_to), c=0); for(n=1, up_to, c += A304109(n); v[n] = c); (v); };
v304110 = prepare_v304110(up_to);
A304110(n) = v304110[n];
A304111(n) = ((2*A304110(n)) - n);
\\ Or just as:
c=0; for(n=0, up_to, if(n>0, c+=((-1)^(1-A304109(n)))); print1(c, ", "));

a(0) = 0, and for n > 0, a(n) = a(n-1) + (-1)^(1-A304109(n)).

For n >= 1, a(n) = (2*A304110(n)) - n.

A304005 Number of nX3 0..1 arrays with every element unequal to 0, 2, 3 or 5 king-move adjacent elements, with upper left element zero.

Original entry on oeis.org

1, 7, 13, 21, 27, 53, 99, 197, 371, 713, 1365, 2645, 5105, 9857, 19079, 36861, 71379, 138165, 267465, 518201, 1003521, 1944649, 3767939, 7302057, 14153291, 27432325, 53179669, 103092661, 199871009, 387518385, 751360183, 1456897269, 2824988675

Offset: 1

R. H. Hardin, May 04 2018

nonn

Column 3 of A304010.

			Some solutions for n=5
..0..1..0. .0..1..0. .0..0..0. .0..1..0. .0..0..0. .0..0..0. .0..0..0
..0..1..1. .1..1..1. .0..0..0. .1..1..1. .1..1..1. .1..1..1. .0..0..0
..1..0..0. .0..0..0. .0..0..0. .1..0..1. .1..1..1. .0..1..0. .0..0..0
..1..1..0. .0..0..0. .0..0..0. .0..0..0. .1..1..1. .0..0..0. .1..1..1
..0..1..0. .1..1..1. .1..1..1. .1..1..1. .0..0..0. .1..0..1. .0..1..0

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

Cf. A304010.

Empirical: a(n) = a(n-1) +5*a(n-2) -a(n-3) -12*a(n-4) -2*a(n-5) +10*a(n-6) +2*a(n-7) +a(n-8) +7*a(n-9) -3*a(n-10) -19*a(n-11) -2*a(n-12) +2*a(n-13) -4*a(n-14) +10*a(n-15) +6*a(n-16) for n>17

A304006 Number of n X 4 0..1 arrays with every element unequal to 0, 2, 3 or 5 king-move adjacent elements, with upper left element zero.

Original entry on oeis.org

1, 13, 21, 26, 64, 115, 211, 439, 870, 1725, 3513, 7141, 14372, 29438, 60518, 123439, 253860, 523477, 1075787, 2216143, 4575316, 9438222, 19480290, 40253428, 83178619, 171897255, 355448881, 735100437, 1520380599, 3145360038, 6508033432

Offset: 1

R. H. Hardin, May 04 2018

nonn

Column 4 of A304010.

			Some solutions for n=5
..0..0..1..1. .0..1..1..1. .0..1..1..0. .0..1..0..0. .0..0..0..0
..0..0..1..1. .1..0..0..0. .0..1..0..1. .1..0..1..1. .0..0..0..0
..0..0..1..1. .0..1..1..0. .0..0..1..0. .1..0..1..0. .1..1..1..1
..0..0..1..1. .0..1..0..1. .0..1..0..0. .0..1..0..0. .1..1..1..1
..0..0..1..1. .1..0..0..1. .0..1..1..1. .1..0..0..1. .1..1..1..1

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

Cf. A304010.

Empirical: a(n) = 5*a(n-1) -6*a(n-2) -9*a(n-4) +8*a(n-5) +36*a(n-6) -29*a(n-7) +34*a(n-8) -88*a(n-9) +8*a(n-10) -70*a(n-11) +205*a(n-12) +66*a(n-13) -377*a(n-14) -29*a(n-15) +420*a(n-16) +897*a(n-17) -1329*a(n-18) -1792*a(n-19) +3267*a(n-20) +1953*a(n-21) -3000*a(n-22) -1986*a(n-23) -1836*a(n-24) +4857*a(n-25) -812*a(n-26) +2460*a(n-27) -5658*a(n-28) -21869*a(n-29) +5879*a(n-30) +37979*a(n-31) +39222*a(n-32) -53345*a(n-33) -60350*a(n-34) +80033*a(n-35) +55646*a(n-36) -80249*a(n-37) -99116*a(n-38) +97246*a(n-39) +113723*a(n-40) -115592*a(n-41) -82690*a(n-42) +63249*a(n-43) +23806*a(n-44) -67251*a(n-45) +52726*a(n-46) +64947*a(n-47) -59223*a(n-48) +4895*a(n-49) +22611*a(n-50) -5950*a(n-51) +1369*a(n-52) +5125*a(n-53) -13309*a(n-54) -19653*a(n-55) +5627*a(n-56) +2405*a(n-57) +1342*a(n-58) +2478*a(n-59) +2088*a(n-60) +435*a(n-61) -1017*a(n-62) -263*a(n-63) -116*a(n-64) +64*a(n-65) -96*a(n-66) +48*a(n-67) for n>70.

A304007 Number of nX5 0..1 arrays with every element unequal to 0, 2, 3 or 5 king-move adjacent elements, with upper left element zero.

Original entry on oeis.org

1, 26, 27, 64, 137, 271, 655, 1412, 3245, 7809, 17290, 40682, 93292, 216048, 499755, 1162587, 2685521, 6238096, 14489376, 33631721, 78120529, 181638185, 422198419, 981730209, 2284138522, 5314049218, 12366970040, 28787274304, 67017999388

Offset: 1

R. H. Hardin, May 04 2018

nonn

Column 5 of A304010.

			Some solutions for n=5
..0..0..0..0..0. .0..0..0..1..1. .0..1..0..1..0. .0..0..1..1..1
..1..1..0..1..1. .1..1..1..0..0. .0..1..1..1..0. .1..1..0..0..0
..0..1..1..1..0. .0..1..0..0..0. .0..0..1..0..0. .1..0..0..0..0
..1..1..0..1..1. .1..1..0..1..1. .0..1..1..1..0. .0..0..0..1..1
..0..0..0..0..0. .0..0..1..1..0. .0..1..0..1..0. .1..0..1..1..0

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

Cf. A304010.

A304008 Number of nX6 0..1 arrays with every element unequal to 0, 2, 3 or 5 king-move adjacent elements, with upper left element zero.

Original entry on oeis.org

1, 49, 53, 115, 271, 538, 1458, 2973, 7826, 19113, 47074, 115148, 283156, 698411, 1720946, 4258788, 10518418, 26019796, 64391560, 159345205, 394533973, 976872015, 2419079852, 5991435758, 14839400772, 36760610050, 91064615465

Offset: 1

R. H. Hardin, May 04 2018

nonn

Column 6 of A304010.

			Some solutions for n=5
..0..1..0..0..0..1. .0..1..1..1..0..1. .0..1..0..1..0..1. .0..0..0..0..1..0
..1..1..1..0..1..0. .1..0..0..1..0..0. .0..1..1..1..0..0. .1..1..0..1..1..0
..0..1..1..1..1..0. .1..1..0..0..1..1. .0..0..1..1..1..0. .1..0..1..0..0..1
..0..0..1..1..0..0. .0..0..1..0..0..1. .0..1..1..0..1..0. .0..0..1..0..1..1
..1..0..1..1..0..1. .1..0..1..1..1..0. .1..0..0..0..0..1. .1..0..1..1..0..0

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

Cf. A304010.

A304009 Number of nX7 0..1 arrays with every element unequal to 0, 2, 3 or 5 king-move adjacent elements, with upper left element zero.

Original entry on oeis.org

1, 99, 99, 211, 655, 1458, 4224, 10633, 28365, 75175, 201031, 544217, 1452266, 3908208, 10505646, 28266755, 76097248, 204983456, 552420015, 1488384837, 4011805443, 10816167549, 29162780211, 78639733211, 212076547630

Offset: 1

R. H. Hardin, May 04 2018

nonn

Column 7 of A304010.

			Some solutions for n=5
..0..0..0..0..1..1..1. .0..0..0..0..0..0..1. .0..1..1..0..1..1..0
..1..1..1..0..1..1..1. .1..1..0..1..1..1..0. .0..1..0..0..0..1..0
..0..1..0..1..1..1..1. .0..1..1..1..0..0..1. .1..0..0..0..0..0..1
..0..0..0..1..1..0..0. .1..1..1..0..1..0..1. .0..1..0..0..0..1..0
..1..0..1..1..0..0..1. .0..1..0..0..1..1..0. .1..0..1..1..1..0..1

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

Cf. A304010.

A304003 Number of n X n 0..1 arrays with every element unequal to 0, 2, 3 or 5 king-move adjacent elements, with upper left element zero.

Original entry on oeis.org

1, 4, 13, 26, 137, 538, 4224, 30468, 264104, 2993696, 40315467

Offset: 1

R. H. Hardin, May 04 2018

nonn

Diagonal of A304010.

			Some solutions for n=5
..0..0..1..1..0. .0..1..1..1..1. .0..0..0..1..1. .0..0..1..1..1
..1..1..0..0..1. .0..1..1..1..1. .0..0..0..1..1. .0..0..1..1..1
..1..1..1..0..1. .0..1..1..1..1. .0..0..0..0..0. .0..0..0..0..0
..0..0..1..0..1. .0..1..1..1..1. .1..1..1..0..0. .1..1..0..0..0
..1..0..0..1..0. .0..1..1..1..1. .1..1..1..0..0. .1..1..0..0..0

Cf. A304010.

cp's OEIS Frontend

A304004 Number of n X 2 0..1 arrays with every element unequal to 0, 2, 3 or 5 king-move adjacent elements, with upper left element zero.

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A304111 Partial sums of f(n) = (-1)^(1-A304109(n)).

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A304005 Number of nX3 0..1 arrays with every element unequal to 0, 2, 3 or 5 king-move adjacent elements, with upper left element zero.

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A304006 Number of n X 4 0..1 arrays with every element unequal to 0, 2, 3 or 5 king-move adjacent elements, with upper left element zero.

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A304007 Number of nX5 0..1 arrays with every element unequal to 0, 2, 3 or 5 king-move adjacent elements, with upper left element zero.

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A304008 Number of nX6 0..1 arrays with every element unequal to 0, 2, 3 or 5 king-move adjacent elements, with upper left element zero.

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A304009 Number of nX7 0..1 arrays with every element unequal to 0, 2, 3 or 5 king-move adjacent elements, with upper left element zero.

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A304003 Number of n X n 0..1 arrays with every element unequal to 0, 2, 3 or 5 king-move adjacent elements, with upper left element zero.

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