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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).

%I A272549 #13 Feb 16 2025 08:33:34
%S A272549 0,1,6,3,10,15,28,21,36,45,66,55,78,91,120,105,136,153,190,171,210,
%T A272549 231,276,253,300,325,378,351,406,435,496,465,528,561,630,595,666,703,
%U A272549 780,741,820,861,946,903,990,1035,1128,1081,1176,1225,1326,1275,1378,1431,1540,1485,1596,1653,1770
%N 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).
%C A272549 Permutation of triangular numbers.
%C A272549 Consecutive alternating even and odd triangular numbers.
%H A272549 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularNumber.html">Triangular Number</a>
%H A272549 <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 A272549 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 A272549 E.g.f.: (1/2)*((x^2 + x + 1)*cosh(x) + x*sin(x) + (x - 1)*cos(x) + x*(x + 3)*sinh(x)).
%F A272549 a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5) - a(n-8) + a(n-9).
%F A272549 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 A272549 a(n) = A000217(A116966(n-1)), n>0.
%F A272549 a(n) mod 2 = A000035(n)
%F A272549 Sum_{n>=1} 1/a(n) = 2.
%e A272549 a(0) = 0;
%e A272549 a(1) = 1;
%e A272549 a(2) = 1 + 2 + 3 = 6;
%e A272549 a(3) = 1 + 2 = 3;
%e A272549 a(4) = 1 + 2 + 3 + 4 = 10;
%e A272549 a(5) = 1 + 2 + 3 + 4 + 5 = 15;
%e A272549 a(6) = 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28;
%e A272549 a(7) = 1 + 2 + 3 + 4 + 5 + 6 = 21;
%e A272549 a(8) = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36, etc.
%e A272549 Illustration of initial terms:
%e A272549 -------------------------------------------------------------------
%e A272549                                          o
%e A272549                                          o o             o
%e A272549                              o           o o o           o o
%e A272549                    o         o o         o o o o         o o o
%e A272549      o             o o       o o o       o o o o o       o o o o
%e A272549      o o     o     o o o     o o o o     o o o o o o     o o o o o
%e A272549 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 A272549 --------------------------------------------------------------------
%e A272549 --------------------------------------------------------------------
%e A272549 n=1   n=2    n=3     n=4       n=5           n=6             n=7
%t A272549 LinearRecurrence[{1, 0, 0, 2, -2, 0, 0, -1, 1}, {0, 1, 6, 3, 10, 15, 28, 21, 36}, 59]
%t A272549 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 A272549 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 A272549 Cf. A000035, A000217, A116966.
%K A272549 nonn,easy
%O A272549 0,3
%A A272549 _Ilya Gutkovskiy_, May 02 2016