A023857 a(n) = 1*(n+3-1) + 2*(n+3-2) + .... + k*(n+3-k), where k=floor((n+1)/2).

3, 4, 13, 16, 34, 40, 70, 80, 125, 140, 203, 224, 308, 336, 444, 480, 615, 660, 825, 880, 1078, 1144, 1378, 1456, 1729, 1820, 2135, 2240, 2600, 2720, 3128, 3264, 3723, 3876, 4389, 4560, 5130, 5320, 5950, 6160, 6853, 7084, 7843, 8096, 8924, 9200, 10100, 10400, 11375, 11700

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Crossrefs

Cf. A023855, A023856, A024305, A024854.

Programs

GAP

List([1..60], n-> (4*n^3 +27*n^2 +50*n +21 -3*(n^2+6*n+7)*(-1)^n)/48) # G. C. Greubel, Jun 12 2019

Magma

[(4*n^3 +27*n^2 +50*n +21 -3*(n^2+6*n+7)*(-1)^n)/48: n in [1..60]]; // G. C. Greubel, Jun 12 2019

Maple

seq(sum(i*(n-i+3), i=1..ceil(n/2)), n=1..60); # Wesley Ivan Hurt, Sep 20 2013

Mathematica

Table[-Ceiling[n/2]*(Ceiling[n/2]+1)*(2*Ceiling[n/2]-3n-8)/6, {n,60}] (* Wesley Ivan Hurt, Sep 20 2013 *)
LinearRecurrence[{1,3,-3,-3,3,1,-1},{3,4,13,16,34,40,70},60] (* Harvey P. Dale, Feb 13 2018 *)

PARI

a(n) = (4*n^3 +27*n^2 +50*n +21 -3*(n^2+6*n+7)*(-1)^n)/48; \\ G. C. Greubel, Jun 12 2019

Sage

[(4*n^3 +27*n^2 +50*n +21 -3*(n^2+6*n+7)*(-1)^n)/48 for n in (1..60)] # G. C. Greubel, Jun 12 2019

Formula

a(n) = Sum_{i=1..ceiling(n/2)} i*(n-i+3) = -ceiling(n/2)*(ceiling(n/2)+1)*(2*ceiling(n/2) - 3*n - 8)/6. - Wesley Ivan Hurt, Sep 20 2013

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

a(n) = 3*A058187(n-1) + A058187(n-2). - R. J. Mathar, Sep 25 2013

a(n) = (4*n^3 + 27*n^2 + 50*n + 21 - 3*(n^2 + 6*n + 7)*(-1)^n)/48. - Luce ETIENNE, Nov 21 2014

E.g.f.: (x*(51 + 18*x + 2*x^2)*cosh(x) + (21 + 30*x + 21*x^2 + 2*x^3)*sinh(x))/24. - G. C. Greubel, Jun 12 2019

Extensions

Title simplified by Sean A. Irvine, Jun 12 2019