A023856 a(n) = 1*(n+1-1) + 2*(n+1-2) + ... + k*(n+1-k), where k = floor((n+1)/2).

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

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

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

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Emanuele Munarini, Topological indices for the antiregular graphs, Le Mathematiche (2021) Vol. 76, No. 1, see p. 302.
• Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Crossrefs

Cf. A000292, A023855, A023857, A024305, A024854.

Programs

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

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},60] (* Harvey P. Dale, Jan 09 2017 *)

PARI

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

SageMath

[(n+2)*(4*n^2 +13*n +6 -3*(n+2)*(-1)^n)/48 for n in (1..60)] # G. C. Greubel, Jul 12 2022

Formula

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

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*(1-x)^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

E.g.f.: (1/24)*( x*(36 + 15*x + 2*x^2)*cosh(x) + (12 + 21*x + 18*x^2 + 2*x^3)*sinh(x) ). - G. C. Greubel, Jul 12 2022

Extensions

Title simplified by Sean A. Irvine, Jun 12 2019