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A266883
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Numbers of the form m*(4*m+1)+1, where m = 0,-1,1,-2,2,-3,3,...
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5
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1, 4, 6, 15, 19, 34, 40, 61, 69, 96, 106, 139, 151, 190, 204, 249, 265, 316, 334, 391, 411, 474, 496, 565, 589, 664, 690, 771, 799, 886, 916, 1009, 1041, 1140, 1174, 1279, 1315, 1426, 1464, 1581, 1621, 1744, 1786, 1915, 1959, 2094, 2140, 2281, 2329, 2476, 2526
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OFFSET
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0,2
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COMMENTS
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Also, numbers m such that 16*m-15 is a square. Therefore, the terms 1 and 4 are the only squares in this sequence.
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LINKS
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FORMULA
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O.g.f.: (1 + 3*x + 3*x^3 + x^4)/((1 + x)^2*(1 - x)^3).
E.g.f.: (5 + 8*x + 4*x^2)*exp(x)/4 -(1 - 2*x)*exp(-x)/4.
a(n) = a(-n-1) = n*(n + 1) + 1 - ((2*n + 1)*(-1)^n - 1)/4 = (2*n + 1)*floor((n + 1)/2) + 1.
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MATHEMATICA
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Table[n (n + 1) + 1 - ((2 n + 1) (-1)^n - 1)/4, {n, 0, 50}]
LinearRecurrence[{1, 2, -2, -1, 1}, {1, 4, 6, 15, 19}, 60] (* Vincenzo Librandi, Jan 06 2016 *)
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PROG
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(PARI) vector(50, n, n--; n*(n+1)+1-((2*n+1)*(-1)^n-1)/4)
(PARI) Vec((1+3*x+3*x^3+x^4)/((1+x)^2*(1-x)^3) + O(x^100)) \\ Altug Alkan, Jan 06 2016
(Sage) [n*(n+1)+1-((2*n+1)*(-1)^n-1)/4 for n in range(50)]
(Python) [n*(n+1)+1-((2*n+1)*(-1)**n-1)/4 for n in range(60)]
(Magma) [n*(n+1)+1-((2*n+1)*(-1)^n-1)/4: n in [0..50]];
(Magma) I:=[1, 4, 6, 15, 19]; [n le 5 select I[n] else Self(n-1) + 2*Self(n-2) -2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..60]]; // Vincenzo Librandi, Jan 06 2016
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CROSSREFS
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Cf. A002061: m*(4*m+2)+1 for m = 0,0,-1,1,-2,2,-3,3, ...
Cf. A174114: m*(4*m+3)+1 for m = 0,-1,1,-2,2,-3,3,-4,4, ...
Cf. A054556: m*(4*m+1)+1 for nonpositive m.
Cf. A054567: m*(4*m+1)+1 for nonnegative m.
Cf. A074378: numbers m such that 16*m+1 is a square.
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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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