A084630 a(n) = floor(C(n+7,7)/C(n+5,5)).

1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 22, 23, 25, 26, 28, 30, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 56, 58, 60, 63, 65, 68, 70, 73, 76, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 115, 118, 121, 125, 128, 132, 135

Offset: 0

Author

Paul Barry, Jun 01 2003

Keywords

• easy
• nonn

Comments

The general Somos-6 sequence terms s(n), with general coefficients and initial values s(0)..s(5), are Laurent polynomials with denominators a product of initial values raised to powers being entries in this sequence. Thus, the denominator of s(n) = Product_{k=0..5} s(k)^a(n-k-6). - Michael Somos, Apr 10 2020

Examples

			G.f. = 1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + ... - _Michael Somos_, Apr 10 2020
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A084628, A084631.

Programs

Magma

[Floor(Binomial(n+7,2)/21): n in [0..80]]; // G. C. Greubel, Mar 23 2023

Mathematica

a[n_]:= Quotient[n(n+13), 42] + 1; (* Michael Somos, Apr 10 2020 *)
Floor[Binomial[Range[0,100]+7,2]/21] (* G. C. Greubel, Mar 23 2023 *)

PARI

{a(n) = n*(n + 13)\42 + 1}; /* Michael Somos, Apr 10 2020 */

SageMath

[binomial(n+7,2)//21 for n in range(81)] # G. C. Greubel, Mar 23 2023

Formula

a(n) = 1 + floor( n*(n+13)/42 ).

From Michael Somos, Apr 10 2020: (Start)

G.f.: (1-x+x^3-x^4+x^5-x^6+x^7-x^9+x^10)/((1-x)^2*(1-x^21)).

a(n) = a(-13-n).

a(n) = a(n-21) + n + 4 for all n in Z.

0 = +a(n)*(a(n+1) -a(n+3) -a(n+4) +a(n+6)) + a(n+1)*(-a(n+1) +a(n+3) +a(n+4) -a(n+5)) + a(n+2)*(-a(n+3) +a(n+4) +a(n+5) -a(n+6)) + a(n+3)*(+a(n+3) -a(n+5) +a(n+6) -a(n+6)) + a(n+5)*(-a(n+5) +a(n+6)) for all n in Z. (End)

a(n) = floor(binomial(n+7,2)/21). - G. C. Greubel, Mar 23 2023