A265611 a(n) = a(n-1) + floor((n-1)/2) - (-1)^n + 2 for n>=2, a(0)=1, a(1)=3.

1, 3, 4, 8, 10, 15, 18, 24, 28, 35, 40, 48, 54, 63, 70, 80, 88, 99, 108, 120, 130, 143, 154, 168, 180, 195, 208, 224, 238, 255, 270, 288, 304, 323, 340, 360, 378, 399, 418, 440, 460, 483, 504, 528, 550, 575, 598, 624, 648, 675, 700, 728, 754, 783, 810, 840

Offset: 0

Author

Peter Luschny, Dec 17 2015

Keywords

• nonn
• easy

Links

• Enrique Pérez Herrero, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
• Peter Luschny, Looking for an interpretation, seqfan mailing list.

Crossrefs

Cf. A002620, A005563, A028552, A052928, A132411, A198442, A217748.

Cf. A084964 and A097065, after the first 3: a(n+1) - a(n) for n>0.

Cf. A055998, after 3: a(n+1) + a(n) for n>0.

Cf. A063929: a(2*n+1) gives the second column of the triangle; for n>0, a(2*n) gives the third column.

Programs

Magma

[1] cat [(2*n*(n+6)-5*(-1)^n+5)/8: n in [1..60]]; // Bruno Berselli, Dec 18 2015

Maple

A265611 := proc(n) iquo(n+1,2); %*(%+irem(n+1,2)+2)+0^n end:
seq(A265611(n), n=0..55);

Mathematica

Join[{1}, Table[(2 n (n + 6) - 5 (-1)^n + 5)/8, {n, 1, 60}]] (* Bruno Berselli, Dec 18 2015 *)

PARI

Vec((x^4-2*x^3+2*x^2-x-1)/(x^4-2*x^3+2*x-1) + O(x^1000)) \\ Altug Alkan, Dec 18 2015

Sage

# The initial values x, y = 0, 1 give the quarter-squares A002620.
def A265611():
    x, y = 1, 2
    while True:
       yield x
       x, y = x + y, x//y + 1
a = A265611(); print([next(a) for i in range(60)])

Formula

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

E.g.f.: 1-(5/8)*exp(-x)+(1/8)*(5+14*x+2*x^2)*exp(x).

a(2*n) = n*(n+3) + 0^n = A028552(n) + 0^n. [Here 0^0 = 1, otherwise 0^s = 0. - N. J. A. Sloane, Aug 26 2022]

a(2*n+1) = (n+1)*(n+3) = A005563(n+1).

a(n+1) - a(n) = floor(n/2) + 2 + (-1)^n - 0^n.

a(n) = a(-n-6) = (2*n*(n+6) - 5*(-1)^n + 5)/8 for n>0, a(0)=1. [Bruno Berselli, Dec 18 2015]

For n>0, a(n) = n + 1 + Sum_{i=1..n+1} floor(i/2) + (-1)^i = n + floor((n+1)^2/4) + (1 - (-1)^n)/2. - Enrique Pérez Herrero, Dec 18 2015

Sum_{n>=0} 1/a(n) = 85/36. - Enrique Pérez Herrero, Dec 18 2015

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>5. - R. H. Hardin, Dec 21 2015, proved by Susanne Wienand for the algorithm sent to the seqfan mailing list and used in the Sage script below.

a(n) = A002620(n+1) + A052928(n+1) for n>=1. (Note A198442(n) = A002620(n+2) - A052928(n+2) for n>=1.) - Peter Luschny, Dec 22 2015

a(n) = (floor((n+3)/2)-1)*(ceiling((n+3)/2)+1) for n>0. - Wesley Ivan Hurt, Mar 30 2017