A143208 a(1)=2; for n>1, a(n) = (4-9*n+3*n^2)/2.

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

Author

Roger L. Bagula and Gary W. Adamson, Oct 20 2008

Keywords

• sign
• easy

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.

Examples

			G.f. = 2*x - x^2 + 2*x^3 + 8*x^4 + 17*x^5 + 29*x^6 + 44*x^7 + 62*x^8 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A095794.

Programs

Magma

[n eq 1 select 2 else (4-9*n+3*n^2)/2: n in [1..60]]; // G. C. Greubel, Jul 19 2024

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 *)

PARI

Vec(x*(3*x^3-11*x^2+7*x-2)/(x-1)^3 + O(x^100)) \\ Colin Barker, Apr 14 2014

SageMath

[(4-9*n+3*n^2)/2 + 3*int(n==1) for n in range(1,61)] # G. C. Greubel, Jul 19 2024

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

Extensions

Better name and edits by Colin Barker and Joerg Arndt, Apr 14 2014