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A104249
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a(n) = (3*n^2 + n + 2)/2.
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15
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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.
Column 2 of A114202. - Paul Barry, Nov 17 2005
Equals third row of A167560 divided by 2. - Johannes W. Meijer, Nov 12 2009
A242357(a(n)) = n + 1. - Reinhard Zumkeller, May 11 2014
Also, this sequence is related to A011379, for n>0, by A011379(n) = n*a(n) - Sum_{i=0..n-1} a(i). - Bruno Berselli, Jul 08 2015
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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Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Rick Mabry, Fibonacci Numbers, Integer Compositions, and Nets of Antiprisms, The American Mathematical Monthly, Vol. 126 (2019), no. 9, pp. 786-801.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
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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.
From Paul Barry, Nov 17 2005: (Start)
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
E.g.f.: exp(x)*(2 + 4*x + 3*x^2)/2. - Stefano Spezia, Apr 10 2021
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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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A104249[n_] := (3*n^2 + n + 2)/2; Table[A104249[n], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)
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PROG
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(Magma) [(3*n^2+n+2)/2: n in [0..50]]; // Vincenzo Librandi, May 09 2011
(Haskell)
a104249 n = n*(3*n+1) `div` 2 + 1 -- Reinhard Zumkeller, May 11 2014
(PARI) a(n)=n*(3*n+1)/2+1 \\ Charles R Greathouse IV, Oct 07 2015
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CROSSREFS
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Cf. A001399, A002597, A005408, A011379, A016777, A143689.
Counts special cases of A284647.
Sequence in context: A115006 A211480 A122796 * A225253 A254875 A025202
Adjacent sequences: A104246 A104247 A104248 * A104250 A104251 A104252
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KEYWORD
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nonn,easy
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AUTHOR
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Thomas Wieder, Feb 26 2005
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STATUS
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approved
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