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, 3284, 3489, 3702, 3923, 4153

Offset

0,2

Comments

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

a(n) is also the genus of the (n+5)-triangular honeycomb rook graph. - Eric W. Weisstein, Mar 10 2026

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Robert Israel, Formulas for A275580

Christian Krause, LODA

Eric Weisstein's World of Mathematics, Graph Genus.

Eric Weisstein's World of Mathematics, Triangular Honeycomb Rook Graph.

Index entries for linear recurrences with constant coefficients, signature (3,-4,5,-6,5,-4,3,-1).

Formula

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

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

Example

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

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 *)
RecurrenceTable[{a[0] == s[0] == 1, a[n] == a[n - 1] + Floor[Sqrt[s[n - 1]]], s[n] == s[n - 1] + a[n]}, a, {n, 0, 20}, DependentVariables -> {a, s}] (* Eric W. Weisstein, Mar 10 2026 *)
Table[((n + 2) (n^2 + 4 n + 16) + 9 Sin[n Pi/2])/36 - 2 Sqrt[3] Sin[2 Pi (n + 5)/3]/27, {n, 0, 20}] (* Eric W. Weisstein, Mar 10 2026 *)
CoefficientList[Series[(1 - x + x^2 - x^3 + x^4)/((-1 + x)^4 (1 + x + 2 x^2 + x^3 + x^4)), {x, 0, 20}], x] (* Eric W. Weisstein, Mar 10 2026 *)
LinearRecurrence[{3, -4, 5, -6, 5, -4, 3, -1}, {1, 2, 3, 5, 8, 12, 17, 23}, 20] (* Eric W. Weisstein, Mar 10 2026 *)

Prog

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

Crossrefs

Sequence in context: A022856 A089071 A177205 * A175829 A241552 A175827

Adjacent sequences: A275577 A275578 A275579 * A275581 A275582 A275583

Keyword

easy,nonn

Author

Hugo van der Sanden, Aug 02 2016

Status

approved