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1, 3, 11, 17, 33, 43, 67, 81, 113, 131, 171, 193, 241, 267, 323, 353, 417, 451, 523, 561, 641, 683, 771, 817, 913, 963, 1067, 1121, 1233, 1291, 1411, 1473, 1601, 1667, 1803, 1873, 2017, 2091, 2243, 2321, 2481, 2563, 2731, 2817, 2993, 3083, 3267, 3361, 3553, 3651
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history;
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OFFSET
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0,2
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COMMENTS
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Sum of odd square and half of even square. - Vladimir Joseph Stephan Orlovsky, May 20 2011
Numbers m such that 6*m-2 is a square. - Bruno Berselli, Apr 29 2016
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).
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FORMULA
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G.f.: ( -1-2*x-6*x^2-2*x^3-x^4 ) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Feb 28 2011
a(n) = 3*(1+2*n+2*n^2)/4 + (-1)^n*(1+2*n)/4. - R. J. Mathar, Feb 28 2011
a(n+2) = a(n) + A091999(n+2).
Union of A080859 and A126587: a(2*n) = A080859(n) and a(2*n+1) = A126587(n+1).
From Peter Bala, Feb 13 2021: (Start)
Appears to be the sequence of exponents in the following series expansion:
Sum_{n >= 0} (-1)^n * x^n/Product_{k = 1..n} 1 - x^(2*k-1) = 1 - x - x^3 + x^11 + x^17 - x^33 - x^43 + + - - .... Cf. A053253.
More generally, for nonnegative integer N, we appear to have the identity
Product_{j = 1..N} 1/(1 + x^(2*j-1))*( P(N,x) + Sum_{n >= 1} (-1)^n * x^((2*N+1)*n-N)/Product_{k = 1..n} 1 - x^(2*k-1) ) = 1 - x - x^3 + x^11 + x^17 - x^33 - x^43 + + - - ..., where P(N,x) is a polynomial in x of degree N^2 - 1, with the first few values given empirically by
P(0,x) = 0, P(1,x) = 1, P(2,x) = 1 - x^2 + x^3, P(3,x) = 1 - x^2 + x^5 - x^7 + x^8 and P(4,x) = 1 - x^2 - x^4 + x^5 + x^8 - x^9 + x^12 - x^14 + x^15. Cf. A203568. (End)
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MATHEMATICA
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Table[If[OddQ[n], n^2+((n+1)^2)/2, (n^2)/2+(n+1)^2], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, May 20 2011 *)
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PROG
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(Haskell)
a186424 n = a186424_list !! n
a186424_list = filter odd a186423_list
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CROSSREFS
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Cf. A186421, A203568.
Sequence in context: A154501 A045432 A340541 * A018520 A154933 A197225
Adjacent sequences: A186421 A186422 A186423 * A186425 A186426 A186427
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KEYWORD
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nonn,easy,changed
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AUTHOR
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Reinhard Zumkeller, Feb 21 2011
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STATUS
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approved
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