A107242 Sum of squares of tetranacci numbers (A001630).
0, 0, 1, 5, 14, 50, 194, 723, 2659, 9884, 36780, 136636, 507517, 1885793, 7006962, 26034006, 96728470, 359395319, 1335332919, 4961420008, 18434129192, 68491926888, 254481427113, 945524491213, 3513091674982, 13052875206698
Offset: 0
Examples
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
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).
Programs
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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 *)
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]
Extensions
a(13) and a(23) corrected by R. J. Mathar, Aug 11 2009