Search: a270700 -id:a270700
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A019557
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Coordination sequence for G_2 lattice.
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+10
25
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1, 12, 30, 48, 66, 84, 102, 120, 138, 156, 174, 192, 210, 228, 246, 264, 282, 300, 318, 336, 354, 372, 390, 408, 426, 444, 462, 480, 498, 516, 534, 552, 570, 588, 606, 624, 642, 660, 678, 696, 714, 732, 750, 768, 786, 804, 822, 840, 858, 876, 894, 912, 930, 948, 966, 984, 1002, 1020, 1038, 1056
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OFFSET
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0,2
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COMMENTS
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Also, coordination sequence of Dual(3.12.12) tiling with respect to a 12-valent node. - N. J. A. Sloane, Jan 22 2018
For n > 1, also the number of minimum vertex colorings of the n-Andrásfai graph. - Eric W. Weisstein, Mar 03 2024
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LINKS
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FORMULA
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a(n) = 18*n-6, n >= 1.
G.f.: (1 + 10*x + 7*x^2)/(1-x)^2.
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EXAMPLE
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Illustration of initial terms:
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o o
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o o o o o o o o o o o o
o o o o o o o o o o o o
o o o o o o o o o o o o
o o o o o o o
o o o o o o o o o o o o
o o o o o o o o o o o o
o o o o o o o o o o o o
o o o
o o
o
1 12 30 48
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MATHEMATICA
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CoefficientList[Series[(1 + 10 x + 7 x^2)/(1 - x)^2, {x, 0, 59}], x] (* Michael De Vlieger, Mar 21 2016 *)
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PROG
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(PARI) x='x+O('x^100); Vec((1+10*x+7*x^2)/(1-x)^2) \\ Altug Alkan, Mar 20 2016
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CROSSREFS
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List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.
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KEYWORD
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nonn,easy
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AUTHOR
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Michael Baake (mbaake(AT)sunelc3.tphys.physik.uni-tuebingen.de)
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STATUS
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approved
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A271114
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Expansion of (1+x)*(2+x)/(1-x)^2.
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+10
2
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2, 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 97, 103, 109, 115, 121, 127, 133, 139, 145, 151, 157, 163, 169, 175, 181, 187, 193, 199, 205, 211, 217, 223, 229, 235, 241, 247, 253, 259, 265, 271, 277, 283, 289, 295, 301, 307, 313, 319, 325
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listen;
history;
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internal format)
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OFFSET
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0,1
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LINKS
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FORMULA
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G.f.: (1+x)*(2+x)/(1-x)^2.
a(n) = 2*a(n-1)-a(n-2) for n>2.
a(5*m+1) = 30*m + 7 = A132231(m+1).
a(5*m+2) = 30*m + 13 = A082369(m+1).
a(5*m+4) = 30*m + 25 = 5*A016969(m).
a(5*m+5) = 30*m + 31 = A128470(m+1). (End)
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MATHEMATICA
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Join[{2}, LinearRecurrence[{2, -1}, {7, 13}, 100]] (* G. C. Greubel, Mar 31 2016 *)
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PROG
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(PARI) Vec((1+x)*(2+x)/(1-x)^2 + O(x^70))
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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