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A300296
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Numerators of n*(5 + 3*n)/(8*(1 + 3*n)*(4 + 3*n)), n >= 0.
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1
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0, 1, 11, 21, 17, 25, 69, 91, 29, 9, 175, 209, 123, 143, 329, 375, 53, 119, 531, 589, 325, 357, 781, 851, 231, 125, 1079, 1161, 623, 667, 1425, 1519, 101, 429, 1819, 1925, 1017, 1073, 2261, 2379, 625, 164, 2751, 2881, 1507
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OFFSET
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0,3
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COMMENTS
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The denominators are given in A300297.
The rational r(n) = a(n)/A300297(n) = (1/8)*n*(5 + 3*n)/(A(n)*A(n+1)) = 1/24 -(1/6)/(A(n)*A(n+1)), with the arithmetic progression A(k) = 1 + 3*k = A016777(k) is the value of the sum Sum_{k=0..n-1} 1/(A(k)*A(k+1)*A(k+2), for n >= 1, and r(0) = 0. See the Jolley reference, pp. 40-41, (210), and the general remark (201) on p. 38.
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REFERENCES
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L. B. W. Jolley, Summation of Series, Dover Publications, 2nd rev. ed., 1961, pp. 38, 40, 41.
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LINKS
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FORMULA
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a(n) = numerator(r(n)), with r(n) = n*(5 + 3*n)/(8*(1 + 3*n)*(4 + 3*n)).
a(n) = n*(5 + 3*n)/32 if n == 0 or 9 (mod 32), a(n) = n*(5 + 3*n)/16 if n == 16 or 25 (mod 32), a(n) = n*(5 + 3*n)/8 if n == 1 or 8 or 17 or 24 (mod 32), and for other n one has a(n) = n*(5 + 3*n)/4 if n == 0 or 1 (mod 4) and a(n) = n*(5 + 3*n)/2 if n == 2 or 3 (mod 4).
G.f.: G(x) = (1/24)*(1 - hypergeometric([1, 2], [7/3], -x/(1-x)))/(1-x).
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EXAMPLE
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The series begins: 1/(1*4*7) + 1/(4*7*10) + 1/(7*10*13) + ...
The partial sums are r(n) = a(n)/A300297(n), n >= 1, and with r(0) = 0 they begin with 0/1, 1/28, 11/280, 21/520, 17/416, 25/608, 69/1672, 91/2200, 29/700, 9/217, 175/4216, 209/5032, 123/2960, 143/3440, 329/7912, 375/9016, ...
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PROG
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(PARI) a(n) = numerator(n*(5 + 3*n)/(8*(1 + 3*n)*(4 + 3*n))); \\ Altug Alkan, Mar 18 2018
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CROSSREFS
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KEYWORD
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nonn,frac,easy
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AUTHOR
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STATUS
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approved
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