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A143208
a(1)=2; for n>1, a(n) = (4-9*n+3*n^2)/2.
3
2, -1, 2, 8, 17, 29, 44, 62, 83, 107, 134, 164, 197, 233, 272, 314, 359, 407, 458, 512, 569, 629, 692, 758, 827, 899, 974, 1052, 1133, 1217, 1304, 1394, 1487, 1583, 1682, 1784, 1889, 1997, 2108, 2222, 2339, 2459, 2582, 2708, 2837, 2969, 3104, 3242, 3383, 3527, 3674
OFFSET
1,1
COMMENTS
Old Name was: A sequence based on odd numbers of the type 3*n + 2: a(n) = a(n - 1) + n - 1; A000096; f(n) = 3*a(n)+2.
FORMULA
From Colin Barker, Apr 14 2014: (Start)
a(n) = (4 - 9*n + 3*n^2)/2 for n>1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>4.
G.f.: x*(2 - 7*x + 11*x^2 - 3*x^3)/ (1-x)^3. (End).
a(n) = (n-2)*A095794(n) - (n-1)*A095794(n-1) for n>1. [Bruno Berselli, May 19 2015]
E.g.f.: (1/2)*(4 - 6*x + 3*x^2)*exp(x) - 2 + 3*x. - G. C. Greubel, Jul 19 2024
EXAMPLE
G.f. = 2*x - x^2 + 2*x^3 + 8*x^4 + 17*x^5 + 29*x^6 + 44*x^7 + 62*x^8 + ...
MATHEMATICA
a[0] = 0; a[1] = -1; a[n_] := a[n] = a[n - 1] + n - 1; a1 = Table[a[n], {n, 0, 30}]; f[n_] := 3*a[n] + 2; Table[f[n], {n, 0, 50}]
LinearRecurrence[{3, -3, 1}, {2, -1, 2, 8}, 60] (* Harvey P. Dale, Mar 22 2018 *)
PROG
(PARI) Vec(x*(3*x^3-11*x^2+7*x-2)/(x-1)^3 + O(x^100)) \\ Colin Barker, Apr 14 2014
(Magma) [n eq 1 select 2 else (4-9*n+3*n^2)/2: n in [1..60]]; // G. C. Greubel, Jul 19 2024
(SageMath) [(4-9*n+3*n^2)/2 + 3*int(n==1) for n in range(1, 61)] # G. C. Greubel, Jul 19 2024
CROSSREFS
Cf. A095794.
Sequence in context: A137305 A282885 A242841 * A353581 A188664 A326572
KEYWORD
sign,easy
AUTHOR
EXTENSIONS
Better name and edits by Colin Barker and Joerg Arndt, Apr 14 2014
STATUS
approved