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A262997 a(n+3) = a(n) + 24*n + 40, a(0)=0, a(1)=5, a(2)=19.

%I A262997 #27 Dec 16 2024 19:19:30
%S A262997 0,5,19,40,69,107,152,205,267,336,413,499,592,693,803,920,1045,1179,
%T A262997 1320,1469,1627,1792,1965,2147,2336,2533,2739,2952,3173,3403,3640,
%U A262997 3885,4139,4400,4669,4947,5232,5525,5827,6136,6453,6779,7112,7453,7803,8160,8525
%N A262997 a(n+3) = a(n) + 24*n + 40, a(0)=0, a(1)=5, a(2)=19.
%C A262997 The hexasections of A262397(n) are
%C A262997 0,  1,  4,  9, 16,  25,  36, ...  = A000290(n)
%C A262997 0,  5, 19, 40, 69, 107, 152, ...  = a(n)
%C A262997 0,  1,  5, 11, 18,  28,  40, ...  = A240438(n+1)
%C A262997 1,  9, 25, 49, 81, 121, 169, ...  = A016754(n)
%C A262997 0,  2,  7, 13, 21,  32,  44, ...  = A262523(n)
%C A262997 3, 13, 32, 59, 93, 136, 187, ...  = e(n+1).
%C A262997 The five-step recurrence in FORMULA is valuable for the six sequences.
%C A262997 Consider a(n) extended from right to left with their first two differences:
%C A262997 ...,  59,  32,  13,  3, 0,  5, 19, 40, 69, ...
%C A262997 ..., -27, -19, -10, -3, 5, 14, 21, 29, 38, ...
%C A262997 ...,   8,   9,   7,  8, 9,  7,  8,  9,  7, ... .
%C A262997 From 0,the first row is
%C A262997 1) from right to left: e(n)
%C A262997 2) from left to right: a(n).
%C A262997 a(n) and e(n) are companions.
%C A262997 The third row is of period 3.
%C A262997 The last digit of a(n) is of period 15; the same is true of e(n).
%H A262997 Colin Barker, <a href="/A262997/b262997.txt">Table of n, a(n) for n = 0..1000</a>
%H A262997 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).
%F A262997 a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) - a(n-5), n> 4.
%F A262997 a(n) = A016742(n) + A042965(n).
%F A262997 a(-n) = e(n).
%F A262997 a(-n) + a(n) = 8*n^2.
%F A262997 a(n+2) - 2*a(n+1) + a(n) = period 3:repeat 9, 7, 8.
%F A262997 a(n+3) - a(n-3) =  8*(1 + 6*n).
%F A262997 a(n+7) - a(n-7) = 40*(2 + 3*n).
%F A262997 a(2n+1) = -a(2n) + 6*n + 3.
%F A262997 a(2n+2) = -a(2n+1) + 4*(n+1).
%F A262997 a(3n) = 4*n*(9*n+1) = 8*A022267(n), a(3n+1) = 36*n^2 +28*n +5, a(3n+2) = 36*n^2 +52*n +19.
%F A262997 G.f.: -x*(x+1)*(3*x^2+4*x+5) / ((x-1)^3*(x^2+x+1)). - _Colin Barker_, Oct 08 2015
%t A262997 a[0] = 0; a[1] = 5; a[2] = 19; a[n_] := a[n] = a[n - 3] + 24 (n - 3) + 40; Table[a@ n, {n, 0, 46}] (* _Michael De Vlieger_, Oct 09 2015 *)
%t A262997 LinearRecurrence[{2,-1,1,-2,1},{0,5,19,40,69},60] (* _Harvey P. Dale_, Dec 16 2024 *)
%o A262997 (PARI) vector(100, n, n--; 4*n^2 + (4*(n+1)-3)\3) \\ _Altug Alkan_, Oct 07 2015
%o A262997 (PARI) concat(0, Vec(-x*(x+1)*(3*x^2+4*x+5)/((x-1)^3*(x^2+x+1)) + O(x^100))) \\ _Colin Barker_, Oct 08 2015
%Y A262997 Cf. A000290, A008586, A016742, A016754, A016789, A016921, A016945, A022267, A042965, A240438, A262397, A262523.
%K A262997 nonn,easy
%O A262997 0,2
%A A262997 _Paul Curtz_, Oct 07 2015