

A290168


If n is even then a(n) = n^2*(n+2)/8, otherwise a(n) = (n1)*n*(n+1)/8.


1



0, 0, 2, 3, 12, 15, 36, 42, 80, 90, 150, 165, 252, 273, 392, 420, 576, 612, 810, 855, 1100, 1155, 1452, 1518, 1872, 1950, 2366, 2457, 2940, 3045, 3600, 3720, 4352, 4488, 5202, 5355, 6156, 6327, 7220, 7410, 8400
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,3


COMMENTS

Bisection of a(n) [0, 2, 12, 36, 80, 150, 252, ...] is A011379.
Bisection [0, 3, 15, 42, 90, 165, 273, ...] is A059270.
Considering s(n) = [0, 0, 0, 0, 1, 1, 3, 3, 6, 6, 10, 10, 15, 15, ...] (triangular numbers repeated  see A008805), a(n) = n*s(n+2) holds.
Considering the first differences of a(n), b(n) = [0, 2, 1 , 9, 3, 21, 6, 38, 10, 60, 15, 87, ...], b(n) shows bisections A000217 and A005476. In addition, b(n) begins like A249264 up to 12th term, and is an alternation of 4 multiples of 3 and 2 not multiples; b(n) is also such that b(2n) + b(2n+1) = A049450(n).
Considering the second differences c(n), c(n) shows bisections A001105(n+1) and A000384(n+1), c(n) has 3 consecutive terms multiples of 3 alternating with 3 not multiples; in addition, c(2n) + c(2n+1) = A000027(n).
Considering a(n)/c(n) = [0, 0, 1/4, 1/2, 2/3, 1, 9/8, 3/2, 8/5, 2, 25/12, 5/2, ...], it appears that it is A129194(n)/A022998(n+1) and A026741(n)/A000034(n) alternating.


LINKS

Table of n, a(n) for n=0..40.
Index entries for linear recurrences with constant coefficients, signature (1,3,3,3,3,1,1).


FORMULA

G.f.: x^2*(2 + x + 3*x^2)/((x1)^4*(x+1)^3).
a(n) = (1/16)*(1)^n*n*(1 + (1)^(n+1) + 2*(1 + (1)^n)*n + 2*(1)^n*n^2).


MATHEMATICA

a[n_] := If[EvenQ[n], n^2*(n + 2)/8, (n  1)*n*(n + 1)/8]; Table[a[n], {n, 0, 40}]


PROG

(PARI) a(n) = if(n%2==0, n^2*(n+2)/8, (n1)*n*(n+1)/8) \\ Felix Fröhlich, Jul 23 2017


CROSSREFS

Cf. A000027, A000034, A000217, A000384, A001105, A005476, A008805, A011379, A022998, A026741, A049450, A059270, A129194, A135713, A161680, A249264.
Sequence in context: A076175 A289870 A067780 * A124486 A260908 A123761
Adjacent sequences: A290165 A290166 A290167 * A290169 A290170 A290171


KEYWORD

nonn


AUTHOR

JeanFrançois Alcover and Paul Curtz, Jul 23 2017


STATUS

approved



