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A319390
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a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), a(0)=1, a(1)=2, a(2)=3, a(3)=6, a(4)=8.
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1
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1, 2, 3, 6, 8, 13, 16, 23, 27, 36, 41, 52, 58, 71, 78, 93, 101, 118, 127, 146, 156, 177, 188, 211, 223, 248, 261, 288, 302, 331, 346, 377, 393, 426, 443, 478, 496, 533, 552, 591, 611, 652, 673, 716, 738, 783, 806, 853, 877, 926, 951, 1002
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OFFSET
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0,2
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COMMENTS
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The bisections A104249(n) = 1, 3, 8, ... and A143689(n+1) = 2, 6, 13, 23, ... are in the following hexagonal spiral:
29--28--28--27--27
/ \
29 17--17--16--16 26
/ / \ \
30 18 9---8---8 15 26
/ / / \ \ \
30 18 9 3---3 7 15 25
/ / / / \ \ \ \
31 19 10 4 1 2 7 14 25
/ / / / / / / /
19 10 4 1---2 6 14 24
\ \ \ / / /
20 11 5---5---6 13 24
\ \ / /
20 11--12--12--13 23
\ /
21--21--22--22--23
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LINKS
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FORMULA
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a(2n) = (3*n^2 + n + 2)/2. a(2n+1) = (3*n^2 + 5*n + 4)/2.
a(-n) = a(n).
G.f.: (1 + x - x^2 + x^3 + x^4) / ((1 - x)^3*(1 + x)^2). - Colin Barker, Jun 05 2019
E.g.f.: ((8 + 7*x + 3*x^2)*cosh(x) + (9 + 5*x + 3*x^2)*sinh(x))/8. - Stefano Spezia, Feb 05 2021
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MATHEMATICA
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LinearRecurrence[{1, 2, -2, -1, 1}, {1, 2, 3, 6, 8}, 100] (* Paolo Xausa, Nov 13 2023 *)
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PROG
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(PARI) Vec((1 + x - x^2 + x^3 + x^4) / ((1 - x)^3*(1 + x)^2) + O(x^50)) \\ Colin Barker, Jun 05 2019
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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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