

A184535


a(n) = floor(3/5 * n^2), with a(1)=1.


12



1, 2, 5, 9, 15, 21, 29, 38, 48, 60, 72, 86, 101, 117, 135, 153, 173, 194, 216, 240, 264, 290, 317, 345, 375, 405, 437, 470, 504, 540, 576, 614, 653, 693, 735, 777, 821, 866, 912, 960, 1008, 1058, 1109, 1161, 1215, 1269, 1325, 1382, 1440, 1500, 1560, 1622, 1685, 1749, 1815, 1881, 1949, 2018, 2088, 2160, 2232, 2306, 2381
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OFFSET

1,2


COMMENTS

Apart from the initial term this is the elliptic troublemaker sequence R_n(2,5) in the notation of Stange (see Table 1, p.16). For other elliptic troublemaker sequences see the cross references below.  Peter Bala, Aug 08 2013


LINKS

Ray Chandler, Table of n, a(n) for n = 1..10000
K. E. Stange, Integral points on elliptic curves and explicit valuations of division polynomials arXiv:1108.3051v3 [math.NT]
Wikipedia, Maximum number of deciamonds in a hexagon.
Index entries for linear recurrences with constant coefficients, signature (2, 1, 0, 0, 1, 2, 1).


FORMULA

a(n) = floor(1/{(5+n^3)^(1/3)}), where {}=fractional part.
a(n)= +2*a(n1) a(n2) +a(n5) 2*a(n6) +a(n7), for n>8, with g.f. 1x^2*(1+x)*(2*x^2x+2)/ ((x^4+x^3+x^2+x+1) *(x1)^3), so a(n) is (3n^22)/5 plus a fifth of A164116 for n>1. [Bruno Berselli, Jan 30 2011. See the following Bala's comment.]
From Peter Bala, Aug 08 2013: (Start)
a(n) = floor(3/5*n^2) for n >= 2.
The sequence b(n) := floor(3/5*n^2)  3/5*n^2, n >= 1, is periodic with period [3/5, 2/5, 2/5, 3/5, 0] of length 5. The generating function and recurrence equation given above easily follow from these observations.
The sequence c(n) := 5/2*( (2*n/5  floor(2*n/5))^2  (2*n/5  floor(2*n/5)) ) is also periodic with period 5, and calculation shows it has the same period as the sequence b(n). Thus b(n) = c(n), yielding the alternative formula a(n) = 3/5*n^2 + 5/2*( (2*n/5  floor(2*n/5))^2  (2*n/5  floor(2*n/5)) ), which is one of the formulas for the elliptic troublemaker sequence R_n(2,5) given in Stange (see Section 7, equation (21)). (End)


MAPLE

1, seq(floor(3/5*n^2), n=2..10^3); # Muniru A Asiru, Feb 04 2018


MATHEMATICA

p[n_] := FractionalPart[(n^3 + 5)^(1/3)]; q[n_] := Floor[1/p[n]]; Table[q[n], {n, 1, 120}]
Join[{1}, LinearRecurrence[{2, 1, 0, 0, 1, 2, 1}, {2, 5, 9, 15, 21, 29, 38}, 62]] (* Ray Chandler, Aug 31 2015 *)


PROG

(GAP) Concatenation([1], List([2..10^3], n>Int(3/5 * n^2))); # Muniru A Asiru, Feb 04 2018
(PARI) a(n) = if(n==1, 1, 3*n^2\5); \\ Altug Alkan, Mar 03 2018


CROSSREFS

Cf. A183532, A279169.
Elliptic troublemaker sequences: A000212 (= R_n(1,3) = R_n(2,3)), A002620 (= R_n(1,2)), A007590 (= R_n(2,4)), A030511 (= R_n(2,6) = R_n(4,6))), A033436 (= R_n(1,4) = R_n(3,4)), A033437 (= R_n(1,5) = R_n(4,5)), A033438 (= R_n(1,6) = R_n(5,6)), A184535 (= R_n(2,5) = R_n(3,5)).
Sequence in context: A006599 A013933 A101201 * A033096 A195014 A152738
Adjacent sequences: A184532 A184533 A184534 * A184536 A184537 A184538


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Jan 16 2011


EXTENSIONS

Better name from Peter Bala, Aug 08 2013


STATUS

approved



