OFFSET
0,2
COMMENTS
a(n) is the number of partitions of 2*n into powers of 2 less than or equal to 2^4. First differs from A000123 at n=16. - Alois P. Heinz, Apr 02 2012
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..2000
N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, arXiv:math/0312418 [math.CO], 2003.
N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, Discrete Math., 294 (2005), 259-274.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,-2,0,2,0,-2,0,2,-2,2,0,-2,1).
FORMULA
a(n) = (8*floor(n/4)^4 + 8*(m+8)*floor(n/4)^3 - 2*(m^3 - 6*m^2 - 19*m - 86)*floor(n/4)^2 -8*(m^3 - 6*m^2 - 6*m - 22)*floor(n/4) - 7*m^3 + 42*m^2 + 13*m + 54 - (m^3 - 6*m^2 + 5*m + 6)*(-1)^floor(n/4))/48 where m = n mod 4. - Luce ETIENNE, Apr 07 2018
MAPLE
f := proc(n, k) option remember; if k > n then RETURN(0); fi; if k= 0 then if n=0 then RETURN(1) else RETURN(0); fi; fi; if k = 1 then RETURN(1); fi; if n mod 2 = 1 then RETURN(f(n-1, k)); fi; f(n-1, k)+f(n/2, k-1); end; # present sequence is f(2m, 5)
GFF := k->x^(2^(k-2))/((1-x)*mul((1-x^(2^j)), j=0..k-2)); # present g.f. is GFF(5)/x^8
a:= proc(n) local m, r; m := iquo(n, 8, 'r'); r:= r+1; [1, 2, 4, 6, 10, 14, 20, 26][r]+ (((8/3*m +(4*r +28)/3)*m +[0, 4, 9, 14, 20, 26, 33, 40][r] +43/3)*m +[22, 33, 50, 67, 93, 119, 154, 189][r]/3)*m end: seq(a(n), n=0..60); # Alois P. Heinz, Apr 17 2009
MATHEMATICA
CoefficientList[Series[1/((1-x)^2(1-x^2)(1-x^4)(1-x^8)), {x, 0, 60}], x] (* Harvey P. Dale, Apr 22 2011 *)
Table[1 + 1237*n/1536 + 17*n^2/96 + 13*n^3/768 + n^4/1536 + (5/32 + n/32) * Floor[n/4] + (81/256 + 3*n/32 + n^2/128) * Floor[n/2] - Floor[(n+1)/8]/4 - (n+3) * Floor[(n+1)/4]/32 - Floor[(n+2)/8]/4, {n, 0, 100}] (* Vaclav Kotesovec, May 02 2018 *)
Table[Simplify[1023/1024 + 85*n/96 + 341*n^2/1536 + n^3/48 + n^4/1536 + (-1)^n*(113/1024 + n/32 + n^2/512) - (1 + Sqrt[2])*Cos[Pi*n/4]/16 + Cos[Pi*n/2]/64 + (Sqrt[2] - 1) * Cos[3*Pi*n/4]/16 + (1/8 + n/64)*Sin[Pi*n/2]], {n, 0, 100}] (* Vaclav Kotesovec, May 02 2018 *)
PROG
(PARI) Vec(1/((1-x)^2*(1-x^2)*(1-x^4)*(1-x^8))+O(x^99)) \\ Charles R Greathouse IV, Sep 03 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Dec 02 2003
STATUS
approved