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A281026
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a(n) = floor(3*n*(n+1)/4).
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6
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0, 1, 4, 9, 15, 22, 31, 42, 54, 67, 82, 99, 117, 136, 157, 180, 204, 229, 256, 285, 315, 346, 379, 414, 450, 487, 526, 567, 609, 652, 697, 744, 792, 841, 892, 945, 999, 1054, 1111, 1170, 1230, 1291, 1354, 1419, 1485, 1552, 1621, 1692, 1764, 1837, 1912, 1989, 2067, 2146
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OFFSET
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0,3
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LINKS
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FORMULA
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O.g.f.: x*(1 + x + x^2)/((1 + x^2)*(1 - x)^3).
E.g.f.: -(1 - 6*x - 3*x^2)*exp(x)/4 - (1 + i)*(i - exp(2*i*x))*exp(-i*x)/8, where i=sqrt(-1).
a(n) = a(-n-1) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5) = a(n-4) + 6*n - 9.
a(n) = 3*n*(n+1)/4 + (i^(n*(n+1)) - 1)/4. Therefore:
a(4*k+r) = 12*k^2 + 3*(2*r+1)*k + r^2, where 0 <= r <= 3.
a(n) = n^2 - floor((n-1)*(n-2)/4).
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MAPLE
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MATHEMATICA
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Table[Floor[3 n (n + 1)/4], {n, 0, 60}]
LinearRecurrence[{3, -4, 4, -3, 1}, {0, 1, 4, 9, 15}, 60] (* Harvey P. Dale, Jun 04 2023 *)
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PROG
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(PARI) vector(60, n, n--; floor(3*n*(n+1)/4))
(Python) [int(3*n*(n+1)/4) for n in range(60)]
(Sage) [floor(3*n*(n+1)/4) for n in range(60)]
(Maxima) makelist(floor(3*n*(n+1)/4), n, 0, 60);
(Magma) [3*n*(n+1) div 4: n in [0..60]];
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CROSSREFS
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Cf. sequences with formula floor(k*n*(n+1)/4): A011848 (k=1), A000217 (k=2), this sequence (k=3), A002378 (k=4).
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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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