OFFSET
0,3
COMMENTS
Permutation of triangular numbers.
Consecutive alternating even and odd triangular numbers.
LINKS
Eric Weisstein's World of Mathematics, Triangular Number
Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1)
FORMULA
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).
E.g.f.: (1/2)*((x^2 + x + 1)*cosh(x) + x*sin(x) + (x - 1)*cos(x) + x*(x + 3)*sinh(x)).
a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5) - a(n-8) + a(n-9).
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).
a(n) mod 2 = A000035(n)
Sum_{n>=1} 1/a(n) = 2.
EXAMPLE
a(0) = 0;
a(1) = 1;
a(2) = 1 + 2 + 3 = 6;
a(3) = 1 + 2 = 3;
a(4) = 1 + 2 + 3 + 4 = 10;
a(5) = 1 + 2 + 3 + 4 + 5 = 15;
a(6) = 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28;
a(7) = 1 + 2 + 3 + 4 + 5 + 6 = 21;
a(8) = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36, etc.
Illustration of initial terms:
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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 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
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
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n=1 n=2 n=3 n=4 n=5 n=6 n=7
MATHEMATICA
LinearRecurrence[{1, 0, 0, 2, -2, 0, 0, -1, 1}, {0, 1, 6, 3, 10, 15, 28, 21, 36}, 59]
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}]
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}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ilya Gutkovskiy, May 02 2016
STATUS
approved