A217278 Sequences A124174 and A006454 interlaced.

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).

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,1,0,34,0,-34,0,-1,0,1).

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

Adjacent sequences: A217275 A217276 A217277 * A217279 A217280 A217281

Keyword

nonn,easy

Author

Raphie Frank, Sep 29 2012

Status

approved