A178730 Partial sums of floor(7^n/8)/6.

0, 1, 8, 58, 408, 2859, 20016, 140116, 980816, 6865717, 48060024, 336420174, 2354941224, 16484588575, 115392120032, 807744840232, 5654213881632, 39579497171433, 277056480200040, 1939395361400290, 13575767529802040, 95030372708614291, 665212608960300048, 4656488262722100348, 32595417839054702448

Offset: 1

Author

Mircea Merca, Dec 26 2010

Keywords

• nonn
• less

Comments

Partial sums of A033117.

Examples

			a(3) = (1/6)*(floor(7/8) + floor(7^2/8) + floor(7^3/8)) = (1/6)*(0+6+42) = 8.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
• Index entries for linear recurrences with constant coefficients, signature (8,-6,-8,7).

Crossrefs

Column k=7 of A368296.

Cf. A033117.

Programs

Magma

[Floor((7*7^n-24*n-7)/48)/6: n in [1..30]]; // Vincenzo Librandi, Jun 21 2011

Maple

A178730 := proc(n) add( floor(7^i/8)/6,i=0..n) ; end proc:

Mathematica

CoefficientList[Series[x/((1+x)(1-7x)(1-x)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 26 2014 *)

PARI

vector(30, n, (((7^(n+1)-24*n-7)/48)\1)/6) \\ G. C. Greubel, Jan 24 2019

Sage

[floor((7^(n+1)-24*n-7)/48)/6 for n in (1..30)] # G. C. Greubel, Jan 24 2019

Formula

6*a(n) = round((7*7^n - 24*n - 16)/48).

6*a(n) = floor((7*7^n - 24*n - 7)/48).

6*a(n) = ceiling((7*7^n - 24*n - 25)/48).

6*a(n) = round((7*7^n - 24*n - 7)/48).

a(n) = a(n-2) + (7^(n-1) - 1)/6, n > 2.

a(n) = 8*a(n-1) - 6*a(n-2) - 8*a(n-3) + 7*a(n-4), n > 4.

G.f.: x^2/((1+x)*(1-7*x)*(1-x)^2).

a(n) = (7^(n+1) - 24*n + 9*(-1)^n - 16)/288. - Bruno Berselli, Jan 11 2011

a(n) = (floor(7^(n+1)/48) - floor((n+1)/2))/6. - Seiichi Manyama, Dec 22 2023