A086500 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.

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

Offset

1,2

Comments

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.

Links

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).

Formula

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)

Mathematica

Table[n*(n + 1)*(2*n + 1 + (-1)^n)/4, {n, 1, 40}] (* Amiram Eldar, Feb 22 2022 *)

Prog

(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

Crossrefs

Cf. A001082, A022998, A063196, A181900.

Sequence in context: A153185 A325450 A212345 * A022669 A107313 A232921

Adjacent sequences: A086497 A086498 A086499 * A086501 A086502 A086503

Keyword

nonn,easy

Author

Amarnath Murthy, Jul 28 2003

Extensions

More terms from Ray Chandler, Sep 17 2003

Status

approved