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A107244 Sum of squares of hexanacci numbers (A001592, Fibonacci 6-step numbers).

%I A107244 #13 Feb 16 2025 08:32:57
%S A107244 0,0,0,0,0,1,2,6,22,86,342,1366,5335,20960,82464,324528,1277104,
%T A107244 5025200,19770800,77789489,306071370,1204272270,4738336974,
%U A107244 18643463374,73354544590,288620849614,1135607911375,4468164041216,17580442344960
%N A107244 Sum of squares of hexanacci numbers (A001592, Fibonacci 6-step numbers).
%C A107244 Primes include: a(6) = 2. Semiprimes include a(7) = 6 = 2 * 3, a(8) = 22 = 2 * 11, a(9) = 86 = 2 * 43, a(11) = 1366 = 2 * 683, a(19) = 77789489 = 3989 * 19501, a(23) = 18643463374 = 2 * 9321731687,
%H A107244 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Fibonacci n-Step Number.</a>
%H A107244 <a href="/index/Rec#order_22">Index entries for linear recurrences with constant coefficients</a>, signature (3, 2, 4, 6, 14, 28, -67, -9, -8, 28, -8, -12, 20, 5, 5, -10, 0, 2, -2, 0, -1, 1).
%F A107244 a(n) = F_6(0)^2 + F_6(1)^2 + ... F_6(n)^2, where F_6(n) = A001592(n). a(0) = 0, a(n+1) = a(n) + A001592(n).
%F A107244 a(n)= 3*a(n-1) +2*a(n-2) +4*a(n-3) +6*a(n-4) +14*a(n-5) +28*a(n-6) -67*a(n-7) -9*a(n-8) -8*a(n-9) +28*a(n-10) -8*a(n-11) -12*a(n-12) +20*a(n-13) +5*a(n-14) +5*a(n-15) -10*a(n-16) +2*a(n-18) -2*a(n-19) -a(n-21) +a(n-22). [From _R. J. Mathar_, Aug 11 2009]
%e A107244 a(0) = 0 = 0^2
%e A107244 a(1) = 0 = 0^2 + 0^2
%e A107244 a(2) = 0 = 0^2 + 0^2 + 0^2
%e A107244 a(3) = 0 = 0^2 + 0^2 + 0^2 + 0^2
%e A107244 a(4) = 0 = 0^2 + 0^2 + 0^2 + 0^2 + 0^2
%e A107244 a(5) = 1 = 0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 1^2
%e A107244 a(6) = 2 = 0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 1^2 + 1^2
%e A107244 a(7) = 6 = 0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 1^2 + 1^2 + 2^2
%e A107244 a(8) = 22 = 0^2 + 0^2 +0^2 + 0^2 + 0^2 + 1^2 + 1^2 + 2^2 + 4^2
%t A107244 Accumulate[LinearRecurrence[{1,1,1,1,1,1},{0,0,0,0,0,1},50]^2] (* _Harvey P. Dale_, Jan 19 2012 *)
%t A107244 LinearRecurrence[{3, 2, 4, 6, 14, 28, -67, -9, -8, 28, -8, -12, 20, 5, 5, -10, 0, 2, -2, 0, -1, 1},{0, 0, 0, 0, 0, 1, 2, 6, 22, 86, 342, 1366, 5335, 20960, 82464, 324528, 1277104, 5025200, 19770800, 77789489, 306071370, 1204272270},29] (* _Ray Chandler_, Aug 02 2015 *)
%Y A107244 Cf. A001592, A107239-A107243, A107245-A107248.
%K A107244 easy,nonn
%O A107244 0,7
%A A107244 _Jonathan Vos Post_, May 19 2005