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A104249
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a(n) = (3*n^2 + n + 2)/2.
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16
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1, 3, 8, 16, 27, 41, 58, 78, 101, 127, 156, 188, 223, 261, 302, 346, 393, 443, 496, 552, 611, 673, 738, 806, 877, 951, 1028, 1108, 1191, 1277, 1366, 1458, 1553, 1651, 1752, 1856, 1963, 2073, 2186, 2302, 2421, 2543, 2668, 2796, 2927, 3061, 3198, 3338, 3481
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OFFSET
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0,2
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COMMENTS
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Second differences are all 3.
Related to the sequence of odd numbers A005408 since for these numbers the first differences are all 2.
The number of Hamiltonian nonisomorphic unfoldings in an n-gonal Archimedean antiprism. See sequence A284647. - Rick Mabry, Apr 10 2021
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LINKS
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FORMULA
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G.f.: (1 + 2*x^2)/(1 - x)^3.
Recurrence: (n+3)*u(n+3) + (-5-n)*u(n+2)*(-2+2*n)*u(n+1) + (-2-2*n)*u(n) = 0 for n >= 0 with u(0) = 1, u(1) = 3, and u(2) = 8.
a(0) = 1, a(n) = a(n-1) + 3*n - 1 for n > 0;
a(n) = Sum_{k=0..n} C(n, k)*C(2, k)*J(k+1), where J(n) = A001045(n). (End)
Binomial transform of [1, 2, 3, 0, 0, 0, ...]. - Gary W. Adamson, Apr 23 2008
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EXAMPLE
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The sequence of first differences delta_a(n) = a(n+1) - a(n) is 2, 5, 8, 11, 14, 17, 20, 23, 26, ...
The sequence of second differences delta_delta_a(n) = a(n+2) - 2*a(n+1) + a(n) is: 3, 3, 3, 3, 3, 3, 3, ... E.g., 78 - 2*58 + 41 = 3.
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MAPLE
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a := proc (n) local i, u; option remember; u[0] := 1; u[1] := 3; u[2] := 8; for i from 3 to n do u[i] := -(4*u[i-3]-8*u[i-2]-2*u[i-1]+(-2*u[i-3]+2*u[i-2]-u[i-1])*i)/i end do; [seq(u[i], i = 0 .. n)] end proc;
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {1, 3, 8}, 70] (* Harvey P. Dale, Jul 21 2023 *)
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PROG
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(Haskell)
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CROSSREFS
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KEYWORD
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nonn,easy,changed
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AUTHOR
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STATUS
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approved
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