A023054 Simon Plouffe's conjectured extension of sequence A008368.
1, 1, 3, 4, 7, 8, 13, 14, 20, 22, 29, 31, 40, 42, 52, 55, 66, 69, 82, 85, 99, 103, 118, 122, 139, 143, 161, 166, 185, 190, 211, 216, 238, 244, 267, 273, 298, 304, 330, 337, 364, 371, 400, 407, 437, 445, 476, 484, 517, 525, 559, 568, 603, 612, 649, 658, 696, 706, 745, 755
Offset: 0
Examples
G.f. = 1 + x + 3*x^2 + 4*x^3 + 7*x^4 + 8*x^5 + 13*x^6 + 14*x^7 + 20*x^8 + ...
Links
- J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
- Index entries for linear recurrences with constant coefficients, signature (0,2,1,-1,-2,0,1).
Programs
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Mathematica
CoefficientList[Series[(1-x^5)/((1-x)*(1-x^2)^2*(1-x^3)), {x, 0, 59}], x] (* Georg Fischer, Oct 13 2020 *) -
PARI
{a(n) = if( n%2, (n + 1) * (5*n + 7) + 8 * (n%6 == 3), (n + 2) * (5*n + 8) + 8 * (n%6 == 0) ) / 24}; /* Michael Somos, May 22 2014 */ -
PARI
{a(n) = if( n<0, n = -3 - n); polcoeff( (1 - x^5) / ((1 - x) * (1 - x^2)^2 * (1 - x^3)) + x * O(x^n), n)}; /* Michael Somos, May 22 2014 */
Formula
G.f.: (1-x^5)/((1-x)*(1-x^2)^2*(1-x^3)).
Euler transform of length 5 sequence [ 1, 2, 1, 0, -1]. - Michael Somos, May 22 2014
a(-3 - n) = a(n). - Michael Somos, May 22 2014
a(2*n + 2) - a(2*n) = A032793(n + 2). a(2*n + 3) - a(2*n + 1) = A042706(n + 2). - Michael Somos, May 22 2014