A084624 a(n) = floor(C(n+5,5)/C(n+2,2)).

1, 2, 3, 5, 8, 12, 16, 22, 28, 36, 45, 56, 68, 81, 96, 114, 133, 154, 177, 202, 230, 260, 292, 327, 365, 406, 449, 496, 545, 598, 654, 714, 777, 843, 913, 988, 1066, 1148, 1234, 1324, 1419, 1518, 1621, 1729, 1842, 1960, 2082, 2210, 2342, 2480, 2623, 2772, 2926

Offset: 0

Author

Paul Barry, Jun 01 2003

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-3,3,-1).

Crossrefs

Cf. A011865, A084625, A084626, A084627, A084628, A084630, A084631.

Programs

Magma

[Floor(Binomial(n+5,3)/10): n in [0..60]]; // G. C. Greubel, Mar 24 2023

Mathematica

LinearRecurrence[{3,-3,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-3,3,-1},{1,2,3, 5,8,12,16,22,28,36,45,56,68,81,96,114,133,154,177,202,230, 260,292},53] (* Ray Chandler, Jul 17 2015 *)
Table[Floor[Binomial[n+5,5]/Binomial[n+2,2]],{n,0,60}] (* or *) Table[ Floor[((3+n)(4+n)(5+n))/60],{n,0,60}] (* Harvey P. Dale, Sep 04 2017 *)
Floor[Binomial[Range[5,65],3]/10] (* G. C. Greubel, Mar 24 2023 *)

SageMath

[(binomial(n+5,3)//10) for n in range(61)] # G. C. Greubel, Mar 24 2023

Formula

a(n) = 1 + floor( n*(n^2 + 12*n + 47)/60 ).

From G. C. Greubel, Mar 24 2023: (Start)

a(n) = floor( binomial(n+5,3)/10 ).

G.f.: (1 -x +x^3 -x^6 +2*x^7 -2*x^8 +2*x^9 -x^10 +x^11 -x^12 +x^14 +x^15 -2*x^16 +x^17)/((1-x)^3*(1-x^20)). (End)