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A023856 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n+1-k), where k = floor((n+1)/2), s = (natural numbers), t = (natural numbers >= 2). 9
2, 3, 10, 13, 28, 34, 60, 70, 110, 125, 182, 203, 280, 308, 408, 444, 570, 615, 770, 825, 1012, 1078, 1300, 1378, 1638, 1729, 2030, 2135, 2480, 2600, 2992, 3128, 3570, 3723, 4218, 4389, 4940, 5130, 5740, 5950, 6622, 6853, 7590, 7843, 8648, 8924, 9800, 10100, 11050, 11375 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Or, a(n) = s(1)s(n) + s(2)s(n-1) + ... + s(k)s(n-k+1), where k = floor( n/2 ) and s = (natural numbers).

Sum of the areas of the distinct rectangles with positive integer length and width such that L + W = n + 2, W < L. For example, a(5) = 28; the rectangles are 1 X 6, 2 X 5 and 3 X 4. The sum of the areas is then 1*6 + 2*5 + 3*4 = 28. - Wesley Ivan Hurt, Nov 12 2017

LINKS

Table of n, a(n) for n=1..50.

Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

FORMULA

a(n) = (n+2)*(4*n^2 + 13*n + 6 - 3(n+2)(-1)^n)/48.

a(n) = Sum_{k=1..(2*n+1+(-1)^(n+1))/4} (n-k+1)*(k-1), with n >= 3. - Paolo P. Lava, Jan 31 2007

a(n) = Sum_{i=1..ceiling(n/2)} i*(n-i+2) = -ceiling(n/2)*(ceiling(n/2) + 1)*(2*ceiling(n/2) - 3n - 5)/6. - Wesley Ivan Hurt, Sep 20 2013

G.f.: x*(2+x+x^2) / ( (1+x)^3*(x-1)^4 ). - R. J. Mathar, Sep 25 2013

a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7). - Wesley Ivan Hurt, Dec 01 2017

a(n - 1) = (A000292(n) - (n mod 2) * (ceiling(n / 2)) ^ 2) / 2. - Luc Rousseau, Feb 25 2018

MAPLE

seq(add(i*(k-i+2), i=1..ceil(k/2)), k=1..70); # Wesley Ivan Hurt, Sep 20 2013

MATHEMATICA

Table[-Ceiling[n/2]*(Ceiling[n/2]+1)*(2*Ceiling[n/2]-3n-5)/6, {n, 100}] (* Wesley Ivan Hurt, Sep 20 2013 *)

LinearRecurrence[{1, 3, -3, -3, 3, 1, -1}, {2, 3, 10, 13, 28, 34, 60}, 50] (* Harvey P. Dale, Jan 09 2017 *)

PROG

(MAGMA) [(n+2)*(4*n^2 + 13*n + 6 - 3*(n+2)*(-1)^n)/48 : n in [1..80]]; // Wesley Ivan Hurt, Nov 29 2017

(PARI) x='x+O('x^99); Vec(x*(2+x+x^2)/((1+x)^3*(x-1)^4)) \\ Altug Alkan, Dec 17 2017

CROSSREFS

Cf. A023855, A023857, A024305, A024854.

Sequence in context: A256414 A004688 A024852 * A267008 A129315 A171126

Adjacent sequences:  A023853 A023854 A023855 * A023857 A023858 A023859

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified May 27 23:50 EDT 2018. Contains 304726 sequences. (Running on oeis4.)