A135836 Column three of the triangular matrix in A135835.
3, 22, 82, 254, 677, 1692, 3972, 9052, 19975, 43394, 92534, 195546, 408489, 848584, 1749544, 3594104, 7345547, 14976366, 30424986, 61706038, 124829101, 252226676, 508704716, 1025115156, 2062984719, 4149086938, 8336437438, 16742227730, 33599246513, 67406551968
Offset: 1
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000
- Index entries for linear recurrences with constant coefficients, signature (4,-2,-10,15,-6).
Programs
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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
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Mathematica
LinearRecurrence[{4,-2,-10,15,-6}, {3,22,82,254,677}, 40] (* G. C. Greubel, Feb 07 2022 *) -
SageMath
def a(n): if (n%2==0): return (1/2)*(55 + 13*n + 2^(n+7) -61*3^(n/2+1)) else: return (1/2)*(55 + 13*n + 2^(n+7) - 106*3^((n+1)/2)) [a(n) for n in (1..40)] # G. C. Greubel, Feb 07 2022
Formula
From G. C. Greubel, Feb 07 2022: (Start)
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)).
a(2*n) = (1/2)*(55 + 26*n + 2^(2*n+7) - 61*3^(n+1)).
a(2*n+1) = (1/2)*(68 + 26*n + 4^(n+4) - 106*3^(n+1)).
G.f.: x*(3 + 10*x)/((1-x)^2*(1 - 2*x - 3*x^2 + 6*x^3)).
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)
Extensions
Terms a(14) onward added by G. C. Greubel, Feb 07 2022
Comments