A088932 G.f.: 1/((1-x)^2*(1-x^2)*(1-x^4)*(1-x^8)).

1, 2, 4, 6, 10, 14, 20, 26, 36, 46, 60, 74, 94, 114, 140, 166, 201, 236, 280, 324, 380, 436, 504, 572, 656, 740, 840, 940, 1060, 1180, 1320, 1460, 1625, 1790, 1980, 2170, 2390, 2610, 2860, 3110, 3396, 3682, 4004, 4326, 4690, 5054, 5460, 5866, 6321, 6776, 7280, 7784

Offset: 0

Author

N. J. A. Sloane, Dec 02 2003

Keywords

• nonn
• easy

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).

Crossrefs

See A000027, A002620, A008804, A088954, A000123 for similar sequences.

Column k=4 of A181322.

Cf. A010873.

Programs

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 *)

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

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