A058195 Areas of a sequence of right-angled figures described below.

1, 7, 23, 57, 118, 218, 370, 590, 895, 1305, 1841, 2527, 3388, 4452, 5748, 7308, 9165, 11355, 13915, 16885, 20306, 24222, 28678, 33722, 39403, 45773, 52885, 60795, 69560, 79240, 89896, 101592, 114393, 128367, 143583, 160113, 178030, 197410, 218330, 240870

Offset

1,2

Comments

From the NW corner to the SE corner, going the upper (or right) way, the edges have lengths n, n-1, ..., 2, 1, 1, 2, ..., n-1, n. Going the lower (or left) way, the edges have lengths n,1,n-1,2,...,2,n-1,1,n.

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

Formula

a(n) = [(2n^4+10n^3+13n^2+2n)/24], where [] denotes floor. (For even n there is no need for truncation. For odd n the [] removes 1/8.) A formula without [] is (4n^4+20n^3+26n^2+4n+3+3(-1)^(n+1))/48.

From Colin Barker, Oct 07 2015: (Start)

a(n) = 4*a(n-1)-5*a(n-2)+5*a(n-4)-4*a(n-5)+a(n-6) for n>6.

a(n) = (2*n^4+10*n^3+13*n^2+2*n)/24 for n even.

a(n) = (2*n^4+10*n^3+13*n^2+2*n-3)/24 for n odd.

G.f.: -x*(3*x+1) / ((x-1)^5*(x+1)).

(End)

Example

For n=6 the figure is (assuming the "#" character is square ...):
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.#########
.#########
.###########
.############
.############
...#############
...#############
...#############
...#############
......###############
......###############
......###############
..........###########
..........###########
...............######

Prog

(PARI) Vec(-x*(3*x+1)/((x-1)^5*(x+1)) + O(x^100)) \\ Colin Barker, Oct 07 2015

Crossrefs

Sequence in context: A027116 A151718 A027918 * A213770 A235683 A037165

Adjacent sequences: A058192 A058193 A058194 * A058196 A058197 A058198

Keyword

easy,nonn

Author

Jonas Wallgren, Nov 26 2000

Extensions

More terms from James A. Sellers, Dec 06 2000

Status

approved