A135836 - OEIS

%I #17 Jun 04 2024 12:12:55

%S 3,22,82,254,677,1692,3972,9052,19975,43394,92534,195546,408489,

%T 848584,1749544,3594104,7345547,14976366,30424986,61706038,124829101,

%U 252226676,508704716,1025115156,2062984719,4149086938,8336437438,16742227730,33599246513,67406551968

%N Column three of the triangular matrix in A135835.

%C Column two of the associated matrix is A005803.

%H G. C. Greubel, <a href="/A135836/b135836.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,-10,15,-6).

%F From _G. C. Greubel_, Feb 07 2022: (Start)

%F a(n) = (1/4)*(110 + 26*n + 2^(n+8) - (1 - (-1)^n)*106*3^((n+1)/2) - (1 + (-1)^n)*61*3^(1+n/2)).

%F a(2*n) = (1/2)*(55 + 26*n + 2^(2*n+7) - 61*3^(n+1)).

%F a(2*n+1) = (1/2)*(68 + 26*n + 4^(n+4) - 106*3^(n+1)).

%F G.f.: x*(3 + 10*x)/((1-x)^2*(1 - 2*x - 3*x^2 + 6*x^3)).

%F E.g.f.: (1/2)*( (55 + 13*x)*exp(x) + 128*exp(2*x) - 183*cosh(sqrt(3)*x) - 106*sqrt(3)*sinh(sqrt(3)*x) ). (End)

%t LinearRecurrence[{4,-2,-10,15,-6}, {3,22,82,254,677}, 40] (* _G. C. Greubel_, Feb 07 2022 *)

%o (Magma) [(1/12)*(330 +78*n +3*2^(n+8) -(1-(-1)^n)*106*3^((n+3)/2) -(1+(-1)^n)*61*3^(2 +n/2)): n in [1..40]]; // _G. C. Greubel_, Feb 07 2022

%o (SageMath)

%o def a(n):

%o     if (n%2==0): return (1/2)*(55 + 13*n + 2^(n+7) -61*3^(n/2+1))

%o     else: return (1/2)*(55 + 13*n + 2^(n+7) - 106*3^((n+1)/2))

%o [a(n) for n in (1..40)] # _G. C. Greubel_, Feb 07 2022

%Y Cf. A005803, A135835.

%K nonn,easy

%O 1,1

%A _John W. Layman_, Nov 30 2007

%E Terms a(14) onward added by _G. C. Greubel_, Feb 07 2022