A275580 Add square root of sum of terms.

1, 2, 3, 5, 8, 12, 17, 23, 31, 41, 52, 65, 81, 99, 119, 142, 168, 197, 229, 264, 303, 346, 392, 442, 497, 556, 619, 687, 760, 838, 921, 1009, 1103, 1203, 1308, 1419, 1537, 1661, 1791, 1928, 2072, 2223, 2381, 2546, 2719, 2900, 3088

Offset: 0

Author

Hugo van der Sanden, Aug 02 2016

Keywords

• easy
• nonn

Comments

a(0) = 1; a(n) = a(n-1) + floor(sqrt(Sum_{i=0..n-1} a(i))).

This appears to give asymptotically a(n) = n^3/36, sum of terms n^4/144, regardless of the starting value a(0).

Examples

			a(3) = a(2) + floor(sqrt(1 + 2)) = 2 + 1 = 3;
a(4) = a(3) + floor(sqrt(1 + 2 + 3)) = 3 + 2 = 5.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Robert Israel, Formulas for A275580
• Christian Krause, LODA
• Index entries for linear recurrences with constant coefficients, signature (3,-4,5,-6,5,-4,3,-1).

Programs

Maple

G:= (x^4-x^3+x^2-x+1)/((x^5+x^3-x^2-1)*(x-1)^3):
S:= series(G,x,101):
seq(coeff(S,x,j),j=0..100); # Robert Israel, Aug 09 2016

Mathematica

a = {1}; Do[AppendTo[a, a[[k]] + Floor@ Sqrt@ Total@ a], {k, 46}]; a (* Michael De Vlieger, Aug 03 2016 *)

PARI

lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, va[n] = va[n-1] + floor(sqrt(sum(k=1, n-1, va[k])));); va;} \\ Michel Marcus, Aug 02 2016

Formula

G.f.: (1-x+x^2-x^3+x^4)/((1-x)^3(1+x^2-x^3-x^5)). See link "Formulas for A275580". - Robert Israel, Aug 09 2016

a(n) = n + 1 + Sum_{i=0..n} floor((floor(i^2 / 3) + i) / 4); derived from corresponding LODA program (see link). - Hugo van der Sanden, Feb 24 2021