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A086500
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Group the natural numbers such that the n-th group sum is divisible by the n-th triangular number: (1), (2, 3, 4), (5, 6, 7), (8, 9, 10, 11, 12), (13, 14, 15, 16, 17), (18, 19, 20, 21, 22, 23, 24), ... Sequence contains the group sum.
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3
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1, 9, 18, 50, 75, 147, 196, 324, 405, 605, 726, 1014, 1183, 1575, 1800, 2312, 2601, 3249, 3610, 4410, 4851, 5819, 6348, 7500, 8125, 9477, 10206, 11774, 12615, 14415, 15376, 17424, 18513, 20825, 22050, 24642, 26011, 28899, 30420, 33620, 35301
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OFFSET
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1,2
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COMMENTS
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The number of terms in the groups is given by A063196. i.e., 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, ...
Also the arithmetic mean of the n-th group is T(n), the n-th triangular number.
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LINKS
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Table of n, a(n) for n=1..41.
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).
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FORMULA
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a(n) = n*(n+1)*(2*n+1+(-1)^n)/4. - Wesley Ivan Hurt, Sep 19 2014
a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7) for n>7. - Colin Barker, Sep 19 2014
G.f.: x*(x^4+8*x^3+6*x^2+8*x+1) / ((x-1)^4*(x+1)^3). - Colin Barker, Sep 19 2014
From Amiram Eldar, Feb 22 2022: (Start)
Sum_{n>=1} 1/a(n) = 4*(1-log(2)).
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/2 - 4. (End)
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MATHEMATICA
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Table[n*(n + 1)*(2*n + 1 + (-1)^n)/4, {n, 1, 40}] (* Amiram Eldar, Feb 22 2022 *)
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PROG
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(Haskell)
a086500 n = a086500_list !! (n-1)
a086500_list = scanl1 (+) $ tail a181900_list
-- Reinhard Zumkeller, Mar 31 2012
(PARI) Vec(x*(x^4+8*x^3+6*x^2+8*x+1)/((x-1)^4*(x+1)^3) + O(x^100)) \\ Colin Barker, Sep 20 2014
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CROSSREFS
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Cf. A001082, A022998, A063196, A181900.
Sequence in context: A153185 A325450 A212345 * A022669 A107313 A232921
Adjacent sequences: A086497 A086498 A086499 * A086501 A086502 A086503
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KEYWORD
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nonn,easy
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AUTHOR
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Amarnath Murthy, Jul 28 2003
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EXTENSIONS
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More terms from Ray Chandler, Sep 17 2003
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STATUS
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approved
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