OFFSET
0,4
LINKS
W. C. Lynch, The t-Fibonacci numbers and polyphase sorting, Fib. Quart., 8 (1970), pp. 6ff.
Eric Weisstein's World of Mathematics, Tetranacci Number.
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.
Index entries for linear recurrences with constant coefficients, signature (3, 2, 2, 6, -16, -2, 6, -2, 2, 1, -1).
FORMULA
a(n) = F_4(1)^2 + F_4(1)^2 + F_4(2)^2 + ... F_4(n)^2 where F_4(n) = A001630(n). a(0) = 0, a(n+1) = a(n) + A001630(n)^2.
a(n)= 3*a(n-1) +2*a(n-2) +2*a(n-3) +6*a(n-4) -16*a(n-5) -2*a(n-6) +6*a(n-7) -2*a(n-8) +2*a(n-9) +a(n-10) -a(n-11). G.f.: x^2*(1+x)*(x^6-x^5-4*x^2+x+1)/((x-1) *(x^4+x^3-3*x^2-3*x+1) *(x^6-x^5+2*x^4-\ 2*x^3-2*x^2-x-1)). [R. J. Mathar, Aug 11 2009]
EXAMPLE
a(0) = 0 = 0^2,
a(1) = 0 = 0^2 + 0^2
a(2) = 1 = 0^2 + 0^2 + 1^2
a(3) = 5 = 0^2 + 0^2 + 1^2 + 2^2
a(4) = 14 = 0^2 + 0^2 + 1^2 + 2^2 + 3^2
a(5) = 50 = 0^2 + 0^2 + 1^2 + 2^2 + 3^2 + 6^2
a(6) = 194 = 0^2 + 0^2 + 1^2 + 2^2 + 3^2 + 6^2 + 12^2
a(7) = 723 = 0^2 + 0^2 + 1^2 + 2^2 + 3^2 + 6^2 + 12^2 + 23^2
a(8) = 2659 = 0^2 + 0^2 + 1^2 + 2^2 + 3^2 + 6^2 + 12^2 + 23^2 + 44^2
MATHEMATICA
Accumulate[LinearRecurrence[{1, 1, 1, 1}, {0, 0, 1, 2}, 40]^2] (* or *) LinearRecurrence[{3, 2, 2, 6, -16, -2, 6, -2, 2, 1, -1}, {0, 0, 1, 5, 14, 50, 194, 723, 2659, 9884, 36780}, 40] (* Harvey P. Dale, Aug 25 2013 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, May 18 2005
EXTENSIONS
a(13) and a(23) corrected by R. J. Mathar, Aug 11 2009
STATUS
approved