A107241 Sum of squares of first n tetranacci numbers (A000288).

1, 2, 3, 4, 20, 69, 238, 863, 3264, 12100, 44861, 166662, 619591, 2301800, 8551800, 31774561, 118060082, 438649107, 1629796276, 6055504952, 22499207241, 83595676570, 310599326171, 1154030334396, 4287794153932, 15931278338957

Offset

1,2

Comments

Tetranacci numbers are also called Fibonacci 4-step numbers. a(n) is prime for n = 2, 3, 8, 26, ... a(n) is semiprime for n = 4, 6, 11, 13, ... a(10) = 12100 = 94^2 + 3264 = 110^2 = 2^2 * 5^2 * 11^2. For Fibonacci numbers (A000045) F(i) we have Sum_{i=1..n} F(i) = F(n)*F(n+1).

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

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) = Sum_{i=1..n} A000288(i)^2.

From R. J. Mathar, Aug 11 2009: (Start)

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.: (1 - x - 5*x^2 - 11*x^3 - 8*x^4 - x^5 - x^6 - 7*x^7 + x^8 + 4*x^9)/((1 - x)*(1 - 3*x - 3*x^2 + x^3 + x^4)(1 + x + 2*x^2 + 2*x^3 - 2*x^4 + x^5 - x^6)). (End)

Example

a(1) = 1^2 = 1.
a(2) = 1^2 + 1^2 = 1.
a(3) = 1^2 + 1^2 + 1^2 = 3, prime.
a(4) = 1^2 + 1^2 + 1^2 + 1^2 = 4 = 2^2, semiprime.
a(5) = 1^2 + 1^2 + 1^2 + 1^2 + 4^2 = 20.
a(6) = 1^2 + 1^2 + 1^2 + 1^2 + 4^2 + 7^2 = 69 = 3 * 23, semiprime.
a(8) = 1^2 + 1^2 + 1^2 + 1^2 + 4^2 + 7^2 + 13^2 + 25^2 = 863, prime.

Maple

T:= proc(n) option remember;
      if n=0 then 0
    elif n<5 then 1
    else add(T(n-j), j=1..4)
      fi; end:
seq( add(T(k)^2, k=1..n), n=1..30); # G. C. Greubel, Dec 18 2019

Mathematica

Accumulate[LinearRecurrence[{1, 1, 1, 1}, {1, 1, 1, 1}, 30]^2] (* Harvey P. Dale, Feb 14 2012 *)
LinearRecurrence[{3, 2, 2, 6, -16, -2, 6, -2, 2, 1, -1}, {1, 2, 3, 4, 20, 69, 238, 863, 3264, 12100, 44861}, 30] (* Ray Chandler, Aug 02 2015 *)
T[n_]:= T[n]= If[n == 0, 0, If[n < 5, 1, Sum[T[n-j], {j, 4}]]]; a[n_]:= Sum[T[j]^2, {j, n}]; Table[a[n], {n, 30}] (* G. C. Greubel, Dec 18 2019 *)

Prog

(Sage)
@CachedFunction
def T(n):
    if (n==0): return 0
    elif (n<5): return 1
    else: return sum(T(n-j) for j in (1..4))
def a(n): return sum(T(j)^2 for j in (1..n))
[a(n) for n in (1..30)] # G. C. Greubel, Dec 18 2019

Crossrefs

Cf. A000288, A107239, A107240, A107241, A107242, A107243, A107244, A107245, A107246, A107247, A107248.

Sequence in context: A012573 A268349 A333690 * A012576 A012579 A012283

Adjacent sequences: A107238 A107239 A107240 * A107242 A107243 A107244

Keyword

easy,nonn

Author

Jonathan Vos Post, May 14 2005

Status

approved