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 A262997 a(n+3) = a(n) + 24*n + 40, a(0)=0, a(1)=5, a(2)=19. 3
 0, 5, 19, 40, 69, 107, 152, 205, 267, 336, 413, 499, 592, 693, 803, 920, 1045, 1179, 1320, 1469, 1627, 1792, 1965, 2147, 2336, 2533, 2739, 2952, 3173, 3403, 3640, 3885, 4139, 4400, 4669, 4947, 5232, 5525, 5827, 6136, 6453, 6779, 7112, 7453, 7803, 8160, 8525 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The hexasections of A262397(n) are 0, 1, 4, 9, 16, 25, 36, ... = A000290(n) 0, 5, 19, 40, 69, 107, 152, ... = a(n) 0, 1, 5, 11, 18, 28, 40, ... = A240438(n+1) 1, 9, 25, 49, 81, 121, 169, ... = A016754(n) 0, 2, 7, 13, 21, 32, 44, ... = A262523(n) 3, 13, 32, 59, 93, 136, 187, ... = e(n+1). The five-step recurrence in FORMULA is valuable for the six sequences. Consider a(n) extended from right to left with their first two differences: ..., 59, 32, 13, 3, 0, 5, 19, 40, 69, ... ..., -27, -19, -10, -3, 5, 14, 21, 29, 38, ... ..., 8, 9, 7, 8, 9, 7, 8, 9, 7, ... . From 0,the first row is 1) from right to left: e(n) 2) from left to right: a(n). a(n) and e(n) are companions. The third row is of period 3. The last digit of a(n) is of period 15; the same is true of e(n). LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1). FORMULA a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) - a(n-5), n> 4. a(n) = A016742(n) + A042965(n). a(-n) = e(n). a(-n) + a(n) = 8*n^2. a(n+2) - 2*a(n+1) + a(n) = period 3:repeat 9, 7, 8. a(n+3) - a(n-3) = 8*(1 + 6*n). a(n+7) - a(n-7) = 40*(2 + 3*n). a(2n+1) = -a(2n) + 6*n + 3. a(2n+2) = -a(2n+1) + 4*(n+1). 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. G.f.: -x*(x+1)*(3*x^2+4*x+5) / ((x-1)^3*(x^2+x+1)). - Colin Barker, Oct 08 2015 MATHEMATICA 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 *) PROG (PARI) vector(100, n, n--; 4*n^2 + (4*(n+1)-3)\3) \\ Altug Alkan, Oct 07 2015 (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 CROSSREFS Cf. A000290, A008586, A016742, A016754, A016789, A016921, A016945, A022267, A042965, A240438, A262397, A262523. Sequence in context: A129828 A239831 A146600 * A031379 A125202 A024841 Adjacent sequences: A262994 A262995 A262996 * A262998 A262999 A263000 KEYWORD nonn,easy AUTHOR Paul Curtz, Oct 07 2015 STATUS approved

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Last modified February 4 22:05 EST 2023. Contains 360082 sequences. (Running on oeis4.)