A216162 Sequences A006452 and A216134 interlaced.

1, 0, 1, 1, 2, 4, 4, 9, 11, 26, 23, 55, 64, 154, 134, 323, 373, 900, 781, 1885, 2174, 5248, 4552, 10989, 12671, 30590, 26531, 64051, 73852, 178294, 154634, 373319, 430441, 1039176, 901273, 2175865, 2508794, 6056764, 5253004, 12681873, 14622323, 35301410

Offset: 0

Author

Raphie Frank, Sep 07 2012

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,1,0,6,0,-6,0,-1,0,1)

Crossrefs

Cf. A000129.

For some k in n:

a(2n) = A006452 (k^2 - 1 is triangular).

a(2n + 1) = A216134 (T_k and 2T_k + 1 are triangular).

a(2n + 1) - a(2n) = A006451 (T_k + 1 is square).

a(2n + 1) + a(2n) = A124124 (T_k and (T_k - 1)/2 are triangular).

a(4n + 1) + a(4n + 2) = A001108 (T_k is square).

a(4n + 3) + a(4n + 4) = A001652 (T_k and 2T_k are triangular).

Sum(a(n)) - 1 = A048776 for even n (the second partial summation of the Pell numbers).

Programs

PARI

Vec((-1-x^3+5*x^4-3*x^5-2*x^6+x^7-2*x^8+x^9)/((x-1)*(1+x)*(x^4-2*x^2-1)*(x^4+2*x^2-1))+O(x^99)) \\ Charles R Greathouse IV, Jun 12 2015

Formula

(a(2n) + a(2n - 1)) - (a(2n - 2) + a(2n - 3)) = A000129(n); n>1.

It follows that sqrt(2) = lim n --> infinity ((a(2n + 2) + a(2n + 1)) - (a(2n - 2) + a(2n - 3)))/((a(2n + 2) + a(2n + 1)) - (a(2n) + a(2n - 1))).

G.f. ( -1-x^3+5*x^4-3*x^5-2*x^6+x^7-2*x^8+x^9 ) / ( (x-1)*(1+x)*(x^4-2*x^2-1)*(x^4+2*x^2-1) ). - R. J. Mathar, Sep 08 2012

Extensions

Edited by N. J. A. Sloane, May 24 2021