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a(n) = -1 + 5*n/6 + n^3/6.
2

%I #42 Sep 08 2022 08:46:18

%S -1,0,2,6,13,24,40,62,91,128,174,230,297,376,468,574,695,832,986,1158,

%T 1349,1560,1792,2046,2323,2624,2950,3302,3681,4088,4524,4990,5487,

%U 6016,6578,7174,7805,8472,9176,9918,10699,11520,12382,13286,14233,15224,16260

%N a(n) = -1 + 5*n/6 + n^3/6.

%C a(n) - a(-n) = 2*a(n) + 2 = A033547(n).

%C (a(n+10) - a(n-10))/10 = 36, 35, 36, 39, 44, 51, ... .

%C a(n) is in the fifth line of

%C 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...

%C 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%C -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, ...

%C 16, 11, 7, 4, 2, 1, 1, 2, 4, 7, 11, 16, 22, 29, ...

%C -26, -15, -8, -4, -2, -1, 0, 2, 6, 13, 24, 40, 62, 91, ...

%C 31, 16, 8, 4, 2, 1, 1, 3, 9, 22, 46, 86, 148, 239, ...

%C etc.

%C First line: A000004. Second line: A000012. Third line: (from 0) A001477 and (backwards) -A001477. Fourth line: (from the second 1 and from the first 1) A000124(n). Fifth line: (from -1) a(n) and -A000125(n). Sixth line: (from 3) A223718, (from the first 1) A000127(n), the second 1 is a separatrix. For the sixth, eighth and tenth lines, Ron Hardin found A223718(n), A223659(n), A225011(n).

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(-n) = - a(n) - 2 = -1, -2, -4, -8, -15, -26, -42, ... = -A000125(n).

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3.

%F a(n) = a(n-1) + A000124(n).

%F G.f.: (2*x^3 -4*x^2 +4*x -1)/(x -1)^4. - _Robert G. Wilson v_, Mar 15 2017

%e G.f. = -1 + 2*x^2 + 6*x^3 + 13*x^4 + 24*x^5 + 40*x^6 + 62*x^7 + ... - _Michael Somos_, Jul 07 2022

%p A283551:=n->-1 + 5*n/6 + n^3/6: seq(A283551(n), n=0..100); # _Wesley Ivan Hurt_, Oct 03 2017

%t Table[-1 + 5 n/6 + n^3/6, {n, 0, 39}] (* _Michael De Vlieger_, Mar 15 2017 *)

%t CoefficientList[ Series[(2x^3 -4x^2 +4x -1)/(x -1)^4, {x, 0, 50}], x] (* or *)

%t LinearRecurrence[{4, -6, 4, -1}, {-1, 0, 2, 6}, 50] (* _Robert G. Wilson v_, Mar 15 2017 *)

%o (Magma) [-1 + 5*n/6 + n^3/6 : n in [0..60]]; // _Wesley Ivan Hurt_, Oct 03 2017

%Y Essentially a duplicate of A003600.

%Y Cf. A000124, A000125, A033547, A059259, A220074.

%K sign,easy

%O 0,3

%A _Paul Curtz_, Mar 10 2017

%E Corrected by _Jeremy Gardiner_, Jan 29 2019