OFFSET
0,3
COMMENTS
a(n) - a(-n) = 2*a(n) + 2 = A033547(n).
(a(n+10) - a(n-10))/10 = 36, 35, 36, 39, 44, 51, ... .
a(n) is in the fifth line of
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, ...
16, 11, 7, 4, 2, 1, 1, 2, 4, 7, 11, 16, 22, 29, ...
-26, -15, -8, -4, -2, -1, 0, 2, 6, 13, 24, 40, 62, 91, ...
31, 16, 8, 4, 2, 1, 1, 3, 9, 22, 46, 86, 148, 239, ...
etc.
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).
LINKS
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
a(-n) = - a(n) - 2 = -1, -2, -4, -8, -15, -26, -42, ... = -A000125(n).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3.
a(n) = a(n-1) + A000124(n).
G.f.: (2*x^3 -4*x^2 +4*x -1)/(x -1)^4. - Robert G. Wilson v, Mar 15 2017
EXAMPLE
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
MAPLE
MATHEMATICA
Table[-1 + 5 n/6 + n^3/6, {n, 0, 39}] (* Michael De Vlieger, Mar 15 2017 *)
CoefficientList[ Series[(2x^3 -4x^2 +4x -1)/(x -1)^4, {x, 0, 50}], x] (* or *)
LinearRecurrence[{4, -6, 4, -1}, {-1, 0, 2, 6}, 50] (* Robert G. Wilson v, Mar 15 2017 *)
PROG
(Magma) [-1 + 5*n/6 + n^3/6 : n in [0..60]]; // Wesley Ivan Hurt, Oct 03 2017
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Paul Curtz, Mar 10 2017
EXTENSIONS
Corrected by Jeremy Gardiner, Jan 29 2019
STATUS
approved