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Number of ways to reciprocally link elements of an n X n array either to themselves or to exactly two horizontal and antidiagonal neighbors, without consecutive collinear links
+10
1
1, 1, 3, 13, 107, 1617, 44976, 2282408, 216083336, 37166744488, 11934907610044, 6974153590170208, 7595477025017870022, 15083748097547577433271, 55729708477784939752304144, 376141278066033748724918163244
EXAMPLE
All solutions for n=3 0=self 3=ne 4=w 6=e 7=sw (reciprocal directions total 10)
.00.00.00...00.67.47...00.00.00
.00.00.00...36.34.00...00.67.47
.00.00.00...00.00.00...36.34.00
Number of ways to reciprocally link elements of an n X 5 array either to themselves or to exactly two horizontal and antidiagonal neighbors, without consecutive collinear links.
+10
1
1, 5, 12, 37, 107, 317, 932, 2749, 8101, 23881, 70392, 207497, 611639, 1802937, 5314536, 15665721, 46178025, 136119501, 401241028, 1182742829, 3486384739, 10276856693, 30293209548, 89295644789, 263217806797, 775890178961
FORMULA
Empirical: a(n) = 2*a(n-1) + 3*a(n-2) - a(n-4) - 2*a(n-5) - a(n-6).
Empirical g.f.: x*(1 + 3*x - x^2 - 2*x^3 - 2*x^4 - x^5) / ((1 - x)*(1 + x)*(1 - 2*x - 2*x^2 - 2*x^3 - x^4)). - Colin Barker, Aug 02 2018
EXAMPLE
Some solutions for n=3 0=self 3=ne 4=w 6=e 7=sw (reciprocal directions total 10):
.00.00.67.47.00...00.00.00.00.00...00.00.00.00.00...00.67.47.00.00
.00.36.34.67.47...00.67.47.67.47...00.00.00.00.00...36.34.00.00.00
.00.00.36.34.00...36.34.36.34.00...00.00.00.00.00...00.00.00.00.00
Number of ways to reciprocally link elements of an n X 6 array either to themselves or to exactly two horizontal and antidiagonal neighbors, without consecutive collinear links.
+10
1
1, 8, 24, 105, 405, 1617, 6412, 25449, 101029, 400986, 1591697, 6317904, 25077948, 99542634, 395117919, 1568354512, 6225321052, 24710371599, 98083689723, 389326813396, 1545367714146, 6134078853549, 24348200775585, 96646113488531
FORMULA
Empirical: a(n) = 3*a(n-1) + 5*a(n-2) - 3*a(n-3) - 6*a(n-4) - a(n-5) - a(n-6) + 4*a(n-7) + a(n-8) - a(n-9) - a(n-10).
Empirical g.f.: x*(1 + x)*(1 + 4*x - 9*x^2 + 5*x^3 - 5*x^4 + 3*x^5 + x^6 - x^8) / (1 - 3*x - 5*x^2 + 3*x^3 + 6*x^4 + x^5 + x^6 - 4*x^7 - x^8 + x^9 + x^10). - Colin Barker, Aug 02 2018
EXAMPLE
Some solutions for n=3 0=self 3=ne 4=w 6=e 7=sw (reciprocal directions total 10):
.00.00.67.47.67.47...00.00.00.00.67.47...00.00.00.00.00.00...00.00.00.00.00.00
.00.36.34.36.34.00...00.67.47.36.34.00...00.00.67.47.67.47...00.00.00.00.00.00
.00.00.00.00.00.00...36.34.00.00.00.00...00.36.34.36.34.00...00.00.00.00.00.00
Number of ways to reciprocally link elements of an n X 7 array either to themselves or to exactly two horizontal and antidiagonal neighbors, without consecutive collinear links.
+10
1
1, 13, 48, 298, 1520, 8338, 44976, 244029, 1322551, 7171769, 38885648, 210854845, 1143330649, 6199601829, 33616711696, 182283373322, 988413912488, 5359579124490, 29061800830768, 157584813722329, 854488462716575
FORMULA
Empirical: a(n) = 5*a(n-1) +6*a(n-2) -17*a(n-3) -19*a(n-4) +10*a(n-5) +16*a(n-6) -46*a(n-7) +7*a(n-8) +15*a(n-9) -14*a(n-10) +21*a(n-11) +3*a(n-12) +10*a(n-13) +8*a(n-14) +2*a(n-15) +5*a(n-16) +a(n-17) +2*a(n-18) -a(n-19) -a(n-20).
EXAMPLE
Some solutions for n=3 0=self 3=ne 4=w 6=e 7=sw (reciprocal directions total 10)
.00.00.00.00.00.00.00...00.00.00.00.00.00.00...00.67.47.00.00.00.00
.00.00.67.47.00.00.00...00.00.67.47.67.47.00...36.34.00.00.00.67.47
.00.36.34.00.00.00.00...00.36.34.36.34.00.00...00.00.00.00.36.34.00
Number of ways to reciprocally link elements of a 4 X n array either to themselves or to exactly two horizontal and antidiagonal neighbors, without consecutive collinear links.
+10
1
1, 1, 5, 13, 37, 105, 298, 846, 2404, 6826, 19394, 55081, 156473, 444445, 1262497, 3586113, 10186570, 28935186, 82191652, 233468038, 663174914, 1883771569, 5350922525, 15199487245, 43174696525, 122639283705, 348361331050
FORMULA
Empirical: a(n) = a(n-1) + 6*a(n-2) - 6*a(n-4) - a(n-5) + a(n-6) for n>7.
Empirical g.f.: x*(1 - 2*x^2 + 2*x^3 - 3*x^5 + x^6) / ((1 - x)*(1 + x)*(1 - x - 5*x^2 - x^3 + x^4)). - Colin Barker, Aug 02 2018
EXAMPLE
All solutions for n=3 0=self 3=ne 4=w 6=e 7=sw (reciprocal directions total 10):
.00.00.00...00.00.00...00.67.47...00.67.47...00.00.00
.00.67.47...00.00.00...36.34.00...36.34.00...00.00.00
.36.34.00...00.00.00...00.00.00...00.67.47...00.67.47
.00.00.00...00.00.00...00.00.00...36.34.00...36.34.00
Number of ways to reciprocally link elements of an 5 X n array either to themselves or to exactly two horizontal and antidiagonal neighbors, without consecutive collinear links.
+10
1
1, 1, 8, 28, 107, 405, 1520, 5706, 21418, 80390, 301736, 1132543, 4250895, 15955408, 59887275, 224782132, 843701335, 3166765105, 11886195940, 44613876758, 167454575870, 628527215306, 2359126067724, 8854788952291, 33235734276043
FORMULA
Empirical: a(n) = 3*a(n-1) + 7*a(n-2) - 14*a(n-3) - 11*a(n-4) + 17*a(n-5) + 5*a(n-6) - 6*a(n-7) for n>11.
Empirical g.f.: x*(1 - 2*x - 2*x^2 + 11*x^3 - 8*x^4 - 6*x^5 + 14*x^6 - 18*x^7 - 3*x^8 + 18*x^9 - 8*x^10) / ((1 - x)*(1 + x)*(1 - 3*x - 6*x^2 + 11*x^3 + 5*x^4 - 6*x^5)). - Colin Barker, Aug 02 2018
EXAMPLE
Some solutions for n=3 0=self 3=ne 4=w 6=e 7=sw (reciprocal directions total 10):
.00.00.00...00.00.00...00.67.47...00.67.47...00.00.00...00.00.00...00.00.00
.00.67.47...00.00.00...36.34.00...36.34.00...00.00.00...00.67.47...00.00.00
.36.34.00...00.00.00...00.00.00...00.00.00...00.67.47...36.34.00...00.00.00
.00.00.00...00.67.47...00.00.00...00.67.47...36.34.00...00.67.47...00.00.00
.00.00.00...36.34.00...00.00.00...36.34.00...00.00.00...36.34.00...00.00.00
Number of ways to reciprocally link elements of an 6Xn array either to themselves or to exactly two horizontal and antidiagonal neighbors, without consecutive collinear links
+10
1
1, 1, 13, 60, 317, 1617, 8338, 42873, 221082, 1139020, 5870960, 30259204, 155962396, 803874071, 4143371101, 21356146495, 110075467413, 567360680620, 2924337273221, 15072870834560, 77689863937436, 400435729023800
FORMULA
Empirical: a(n) = a(n-1) +33*a(n-2) -a(n-3) -351*a(n-4) -75*a(n-5) +1760*a(n-6) +436*a(n-7) -4708*a(n-8) -997*a(n-9) +6997*a(n-10) +1085*a(n-11) -5748*a(n-12) -564*a(n-13) +2512*a(n-14) +131*a(n-15) -543*a(n-16) -7*a(n-17) +49*a(n-18) -a(n-19) -a(n-20) for n>24
EXAMPLE
Some solutions for n=3 0=self 3=ne 4=w 6=e 7=sw (reciprocal directions total 10)
.00.67.47...00.00.00...00.00.00...00.00.00...00.67.47...00.00.00...00.00.00
.36.34.00...00.67.47...00.00.00...00.00.00...36.34.00...00.67.47...00.00.00
.00.00.00...36.34.00...00.00.00...00.00.00...00.67.47...36.34.00...00.67.47
.00.67.47...00.67.47...00.00.00...00.00.00...36.34.00...00.00.00...36.34.00
.36.34.00...36.34.00...00.00.00...00.67.47...00.00.00...00.00.00...00.00.00
.00.00.00...00.00.00...00.00.00...36.34.00...00.00.00...00.00.00...00.00.00
Number of ways to reciprocally link elements of an 7Xn array either to themselves or to exactly two horizontal and antidiagonal neighbors, without consecutive collinear links
+10
1
1, 1, 21, 129, 932, 6412, 44976, 311193, 2168111, 15034974, 104478616, 725286308, 5037269200, 34977393156, 242898931716, 1686719471828, 11713075485508, 81338093590180, 564832581201972, 3922330554994484, 27237632538545092
FORMULA
Empirical: a(n) = a(n-1) +76*a(n-2) -4*a(n-3) -2156*a(n-4) -520*a(n-5) +32432*a(n-6) +9800*a(n-7) -296352*a(n-8) -80128*a(n-9) +1755472*a(n-10) +365536*a(n-11) -6970672*a(n-12) -1003248*a(n-13) +18839680*a(n-14) +1679040*a(n-15) -34716928*a(n-16) -1650048*a(n-17) +43135744*a(n-18) +820736*a(n-19) -35163136*a(n-20) -61440*a(n-21) +17813504*a(n-22) -116736*a(n-23) -5017600*a(n-24) +36864*a(n-25) +589824*a(n-26) for n>32
EXAMPLE
Some solutions for n=3 0=self 3=ne 4=w 6=e 7=sw (reciprocal directions total 10)
.00.67.47...00.00.00...00.00.00...00.67.47...00.67.47...00.67.47...00.00.00
.36.34.00...00.00.00...00.67.47...36.34.00...36.34.00...36.34.00...00.00.00
.00.00.00...00.67.47...36.34.00...00.67.47...00.67.47...00.67.47...00.67.47
.00.00.00...36.34.00...00.00.00...36.34.00...36.34.00...36.34.00...36.34.00
.00.00.00...00.00.00...00.67.47...00.00.00...00.00.00...00.67.47...00.00.00
.00.00.00...00.00.00...36.34.00...00.67.47...00.00.00...36.34.00...00.67.47
.00.00.00...00.00.00...00.00.00...36.34.00...00.00.00...00.00.00...36.34.00
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