A279169 a(n) = floor( 4*n^2/5 ).

0, 0, 3, 7, 12, 20, 28, 39, 51, 64, 80, 96, 115, 135, 156, 180, 204, 231, 259, 288, 320, 352, 387, 423, 460, 500, 540, 583, 627, 672, 720, 768, 819, 871, 924, 980, 1036, 1095, 1155, 1216, 1280, 1344, 1411, 1479, 1548, 1620, 1692, 1767, 1843, 1920, 2000, 2080, 2163, 2247

Offset: 0

Author

Bruno Berselli, Dec 07 2016

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Crossrefs

Cf. A090223: floor(4*n/5).

Subsequence of A008728, A014601, A118015, A131242.

Cf. similar sequences with closed form floor(k*n^2/5): A118015 (k=1), A033437 (k=2), A184535 (k=3).

Programs

Magma

[4*n^2 div 5: n in [0..60]];

Mathematica

Table[Floor[4 n^2/5], {n, 0, 60}]
LinearRecurrence[{2,-1,0,0,1,-2,1},{0,0,3,7,12,20,28},60] (* Harvey P. Dale, Nov 07 2020 *)

PARI

vector(60, n, n--; floor(4*n^2/5))

Python

[int(4*n**2/5) for n in range(60)]

Sage

[floor(4*n^2/5) for n in range(60)]

Formula

O.g.f.: x^2*(3 + x + x^2 + 3*x^3)/((1 - x)^3*(1 + x + x^2 + x^3 + x^4)).

a(n) = a(-n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7).

a(5*m+r) = 4*m*(5*m + 2*r) + a(r), where m >= 0 and 0 <= r < 5. Example: for m=4 and r=3, a(5*4+3) = a(23) = 4*4*(5*4 + 2*3) + a(3) = 416 + 7 = 423.

a(n) = A118015(2*n) = A008728(4*n+2) = A131242(4*n+4) = A014601(floor(2*n^2/5)).

Sum_{n>=2} 1/a(n) = Pi^2/120 + sqrt(29 - 62/sqrt(5))*Pi/8 + 5/16. - Amiram Eldar, Sep 26 2022