A272549 - OEIS

%I #12 May 07 2016 00:46:01

%S 0,1,6,3,10,15,28,21,36,45,66,55,78,91,120,105,136,153,190,171,210,

%T 231,276,253,300,325,378,351,406,435,496,465,528,561,630,595,666,703,

%U 780,741,820,861,946,903,990,1035,1128,1081,1176,1225,1326,1275,1378,1431,1540,1485,1596,1653,1770

%N Expansion of x*(1 + 5*x - 3*x^2 + 7*x^3 + 3*x^4 + 3 *x^5 - x^6 + x^7)/((1 - x)^3*(1 + x + x^2 + x^3)^2).

%C Permutation of triangular numbers.

%C Consecutive alternating even and odd triangular numbers.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangularNumber.html">Triangular Number</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,2,-2,0,0,-1,1)

%F O.g.f.: x*(1 + 5*x - 3*x^2 + 7*x^3 + 3*x^4 + 3 *x^5 - x^6 + x^7)/((1 - x)^3*(1 + x + x^2 + x^3)^2).

%F E.g.f.: (1/2)*((x^2 + x + 1)*cosh(x) + x*sin(x) + (x - 1)*cos(x) + x*(x + 3)*sinh(x)).

%F a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5) - a(n-8) + a(n-9).

%F a(n) = (1/8)*(2*n + sin((Pi*n)/2) - cos((Pi*n)/2) + (-1)^n) *(2*n + sin((Pi*n)/2) - cos((Pi*n)/2) + (-1)^n + 2).

%F a(n) = A000217(A116966(n-1)), n>0.

%F a(n) mod 2 = A000035(n)

%F Sum_{n>=1} 1/a(n) = 2.

%e a(0) = 0;

%e a(1) = 1;

%e a(2) = 1 + 2 + 3 = 6;

%e a(3) = 1 + 2 = 3;

%e a(4) = 1 + 2 + 3 + 4 = 10;

%e a(5) = 1 + 2 + 3 + 4 + 5 = 15;

%e a(6) = 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28;

%e a(7) = 1 + 2 + 3 + 4 + 5 + 6 = 21;

%e a(8) = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36, etc.

%e Illustration of initial terms:

%e -------------------------------------------------------------------

%e                                          o

%e                                          o o             o

%e                              o           o o o           o o

%e                    o         o o         o o o o         o o o

%e      o             o o       o o o       o o o o o       o o o o

%e      o o     o     o o o     o o o o     o o o o o o     o o o o o

%e o    o o o   o o   o o o o   o o o o o   o o o o o o o   o o o o o o

%e --------------------------------------------------------------------

%e --------------------------------------------------------------------

%e n=1   n=2    n=3     n=4       n=5           n=6             n=7

%t LinearRecurrence[{1, 0, 0, 2, -2, 0, 0, -1, 1}, {0, 1, 6, 3, 10, 15, 28, 21, 36}, 59]

%t Table[(1/8) (2 n + Sin[(Pi n)/2] - Cos[(Pi n)/2] + (-1)^n) (2 n + Sin[(Pi n)/2] - Cos[(Pi n)/2] + (-1)^n + 2), {n, 0, 58}]

%t Table[(1/8) (2 n - (-1)^(n - 1) + I^((n - 2) (n - 1))) (2 n - (-1)^(n - 1) + I^((n - 2) (n - 1)) + 2), {n, 0, 58}]

%Y Cf. A000035, A000217, A116966.

%K nonn,easy

%O 0,3

%A _Ilya Gutkovskiy_, May 02 2016