OFFSET
0,5
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)
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
PROG
(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
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).
KEYWORD
nonn,easy
AUTHOR
Raphie Frank, Sep 07 2012
EXTENSIONS
Edited by N. J. A. Sloane, May 24 2021
STATUS
approved