A272549 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).
0, 1, 6, 3, 10, 15, 28, 21, 36, 45, 66, 55, 78, 91, 120, 105, 136, 153, 190, 171, 210, 231, 276, 253, 300, 325, 378, 351, 406, 435, 496, 465, 528, 561, 630, 595, 666, 703, 780, 741, 820, 861, 946, 903, 990, 1035, 1128, 1081, 1176, 1225, 1326, 1275, 1378, 1431, 1540, 1485, 1596, 1653, 1770
Offset: 0
Examples
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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n=1 n=2 n=3 n=4 n=5 n=6 n=7
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)
Programs
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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}]
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.
Comments