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URL: https://oeis.org/A217278

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A217278
Sequences A124174 and A006454 interlaced.
3
0, 0, 1, 3, 10, 15, 45, 120, 351, 528, 1540, 4095, 11935, 17955, 52326, 139128, 405450, 609960, 1777555, 4726275, 13773376, 20720703, 60384555, 160554240, 467889345, 703893960, 2051297326, 5454117903, 15894464365, 23911673955, 69683724540, 185279454480
OFFSET
0,4
COMMENTS
a(2n) and 2*a(2n) + 1 are triangular.
a(2n + 1) is triangular and a(2n + 1)/2 is the harmonic mean of consecutive triangular numbers (therefore, a(2n + 1) + 1 is square).
FORMULA
a(n) = 35*(a(n-4) - a(n-8)) + a(n-12).
lim n --> infinity a(2n)/a(2n - 1) = (3 + sqrt(8))/2.
From Raphie Frank, Dec 21 2015: (Start)
a(2*n + 1) = 1/64*(((4 + sqrt(2)) * (1 - (-1)^(n+1) * sqrt(2))^(2*floor((n+1)/2)) + (4 - sqrt(2)) * (1+(-1)^(n+1) * sqrt(2))^(2*floor((n+1)/2))))^2 - 1.
a(2*n + 2) = 1/2*(3*(a(2*n + 1)) + sqrt((a(2*n + 1)) + 1) * sqrt(8*(a(2*n + 1)) + 1) + 1).
(End)
EXAMPLE
a(18) = 35*(52326 - 1540) + 45 = 1777555,
a(19) = 35*(139128 - 4095) + 120 = 4726275.
MATHEMATICA
LinearRecurrence[{0, 1, 0, 34, 0, -34, 0, -1, 0, 1}, {0, 0, 1, 3, 10, 15, 45, 120, 351, 528}, 40] (* Harvey P. Dale, Aug 04 2019 *)
PROG
(PARI) concat([0, 0], Vec(-x^2*(3*x^5+x^4+12*x^3+9*x^2+3*x+1)/((x-1)*(x+1)*(x^2-2*x-1)*(x^2+2*x-1)*(x^4+6*x^2+1)) + O(x^100))) \\ Colin Barker, Jun 23 2015
CROSSREFS
Cf. (sqrt(8a(2n) + 1) - 1)/2 = A216134(n) = A216162(2n + 1).
Cf. sqrt(a(2n+1) + 1) = A006452(n + 1) = A216162(2n + 2).
Cf. (sqrt(8a(2n+1) + 1) - 1)/2 = A006451(n).
Sequence in context: A020330 A023861 A037345 * A175336 A370437 A259877
KEYWORD
nonn,easy
AUTHOR
Raphie Frank, Sep 29 2012
STATUS
approved