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A008642 Quarter-squares repeated. 7
1, 1, 2, 2, 4, 4, 6, 6, 9, 9, 12, 12, 16, 16, 20, 20, 25, 25, 30, 30, 36, 36, 42, 42, 49, 49, 56, 56, 64, 64, 72, 72, 81, 81, 90, 90, 100, 100, 110, 110, 121, 121, 132, 132, 144, 144, 156, 156, 169, 169, 182, 182, 196, 196, 210, 210, 225, 225 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The area of the largest rectangle whose perimeter is not greater than n. - Dmitry Kamenetsky, Aug 30 2006

Also number of partitions of n into parts 1, 2 or 4. - Reinhard Zumkeller, Aug 12 2011

Let us consider a rectangle composed of unit squares. Then count how many squares are necessary to surround this rectangle by a layer whose width is 1 unit. And repeat this surrounding ad libitum. This sequence, prepended by 4 zeros and with offset 0, gives the number of rectangles that need 2*n unit squares in one of their surrounding layers. - Michel Marcus, Sep 19 2015

a(n) is the number of nonnegative integer solutions (x,y,z) for n-2 <= 2*x + 3*y + 4*z <= n. For example, the two solutions for 1 <= 2*x + 3*y + 4*z <= 3 are (1,0,0) and (0,1,0). - Ran Pan, Oct 07 2015

Conjecture: Consider the number of compositions of n>=4*k+8 into odd parts, where the order of the parts 1,3,..,2k+1 does not count. Then, as k approaches infinity, a(n-4*k-8) is equal to the number of these restricted compositions minus A000009(n), the number of strict partitions of n. - Gregory L. Simay, Aug 12 2016

REFERENCES

D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 105.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 112, D(n).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Ran Pan, Exercise U, Project P.

Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1,-1,1).

FORMULA

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

a(n) = (2*n^2 + 14*n + 21 + (2*n + 7)*(-1)^n)/32 + ((1 + (-1)^n)/2 - (1 - (-1)^n)*i/2)*i^n/8, with i = sqrt(-1).

a(n) = floor(((n+1)*((-1)^n+n+6)+9)/16). - Tani Akinari, Jun 16 2013

a(n) = Sum_{i=1..floor((n+6)/2)} floor((n+6-2*i-(n mod 2))/4). - Wesley Ivan Hurt, Mar 31 2014

a(0)=1, a(1)=1, a(2)=2, a(3)=2, a(4)=4, a(5)=4, a(6)=6; for n>6, a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-6) + a(n-7). - Harvey P. Dale, Jun 03 2015

a(n) = floor(floor(n/2+2)^2/4) = floor(floor(n/2+2)^2/2)/2. - Bruno Berselli, Mar 03 2016

MAPLE

seq((7/8+(-1)^k/8 + k + k^2/4)$2, k=0..100); # Robert Israel, Oct 08 2015

MATHEMATICA

CoefficientList[Series[1/((1 - x) (1 - x^2) (1 - x^4)), {x, 0, 60}], x] (* Vincenzo Librandi, Apr 02 2014 *)

LinearRecurrence[{1, 1, -1, 1, -1, -1, 1}, {1, 1, 2, 2, 4, 4, 6}, 60] (* Harvey P. Dale, Jun 03 2015 *)

Table[Floor[((n + 1) ((-1)^n + n + 6) + 9)/16], {n, 0, 57}] (* Michael De Vlieger, Aug 14 2016 *)

PROG

(PARI) Vec(1/((1-x)*(1-x^2)*(1-x^4)) + O(x^70)) \\ Michel Marcus, Mar 31 2014

(PARI) vector(100, n, n--; floor(((n+1)*((-1)^n+n+6)+9)/16)) \\ Altug Alkan, Oct 08 2015

(MAGMA) [Floor(((n+1)*((-1)^n+n+6)+9)/16): n in [0..60]]; // Vincenzo Librandi, Apr 02 2014

(Sage) [floor(floor(n/2+2)^2/2)/2 for n in (0..60)] # Bruno Berselli, Mar 03 2016

CROSSREFS

Cf. A002620.

Sequence in context: A259881 A238132 A278296 * A001364 A029010 A060027

Adjacent sequences:  A008639 A008640 A008641 * A008643 A008644 A008645

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified September 26 06:54 EDT 2017. Contains 292502 sequences.