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A056944 Amount by which used area of rectangle needed to enclose a non-touching spiral of length n on a square lattice exceeds unused area. 7
0, 1, 2, 2, 2, 4, 3, 2, 4, 6, 4, 2, 4, 6, 8, 5, 2, 4, 6, 8, 10, 6, 2, 4, 6, 8, 10, 12, 7, 2, 4, 6, 8, 10, 12, 14, 8, 2, 4, 6, 8, 10, 12, 14, 16, 9, 2, 4, 6, 8, 10, 12, 14, 16, 18, 10, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 11, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 12, 2, 4, 6, 8, 10, 12, 14, 16 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

m (when n is m-th triangular number) followed by m even numbers from 2 through 2m.

LINKS

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

FORMULA

a(n) = 2n - floor((sqrt(8n+1)-1)/2)*ceiling((sqrt(8n+1)-1)/2) = 2n - A002024(n)*A003056(n) = 2n - A056942(n) = n -A056943(n). If n = t(t+1)/2 then a(n)=t; if n = t(t+1)/2+k with 0 < k <= t then a(n)=2k.

EXAMPLE

a(9)=6 since spiral is as marked by 9 X's in 4*3 = 12 rectangle, with 12-9 = 3 spaces unused and a used-unused difference of 9-3 = 6:

X.XX

X..X

XXXX

As a triangle, the first few rows are: 1; 2, 2; 2, 4, 3; 2, 4, 6, 4; 2, 4, 6, 8, 5; 2, 4, 6, 8, 10, 6; 2, 4, 6, 8, 10, 12, 7; ... (= reversal of triangle A143595). Row sums = n^2. [Gary W. Adamson, Aug 26 2008]

MATHEMATICA

uar[n_]:=Module[{c=(Sqrt[8n+1]-1)/2}, 2n-Floor[c]Ceiling[c]]; Array[uar, 90, 0] (* Harvey P. Dale, Aug 14 2013 *)

PROG

(MAGMA) [2*n-Floor((Sqrt(8*n+1)-1)/2)*Ceiling((Sqrt(8*n+1)-1)/2): n in [0..90]]; // Vincenzo Librandi, Aug 06 2017

CROSSREFS

Cf. A002024, A003056, A056942, A056943.

Cf. A143595. [Gary W. Adamson, Aug 26 2008]

Sequence in context: A273875 A054709 A121806 * A283681 A222819 A194319

Adjacent sequences:  A056941 A056942 A056943 * A056945 A056946 A056947

KEYWORD

easy,nonn,nice

AUTHOR

Henry Bottomley, Jul 13 2000

STATUS

approved

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Last modified January 22 08:47 EST 2018. Contains 298042 sequences.