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%I #21 Jan 05 2025 19:51:38
%S 0,0,0,0,1,2,6,22,86,342,1303,5024,19424,75120,290416,1122160,4337009,
%T 16762634,64787534,250400910,967783566,3740437902,14456621263,
%U 55874162432,215950971648,834640190272,3225844698176,12467736540480
%N Sum of squares of pentanacci numbers (A001591).
%H W. C. Lynch, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/8-1/lynch.pdf">The t-Fibonacci numbers and polyphase sorting</a>, Fib. Quart., 8 (1970), pp. 6ff.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Fibonacci n-Step Number.</a>
%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (3, 2, 3, 7, 14, -32, -2, 6, -4, -6, 10, 1, -1, 0, 1, -1).
%F a(n) = F_5(1)^2 + F_5(1)^2 + F_5(2)^2 + ... F_5(n)^2 where F_5(n) = A001591(n). a(0) = 0, a(n+1) = a(n) + A001591(n)^2.
%F a(n)= 3*a(n-1) +2*a(n-2) +3*a(n-3) +7*a(n-4) +14*a(n-5) -32*a(n-6) -2*a(n-7) +6*a(n-8) -4*a(n-9) -6*a(n-10) +10*a(n-11) +a(n-12) -a(n-13) +a(n-15) -a(n-16). [_R. J. Mathar_, Aug 11 2009]
%F G.f.: x^4*(x^10 +x^9 +x^7 +x^6 -6*x^5 -5*x^4 -3*x^3 -2*x^2 -x +1) / ((x -1)*(x^5 +x^4 +x^3 +3*x^2 +3*x -1)*(x^10 -x^9 -x^7 +x^6 -6*x^5 +3*x^4 +3*x^3 +2*x^2 +x +1)). - _Colin Barker_, May 08 2013
%e a(0) = 0 = 0^2 since F_5(0) = A001591(0) = 0.
%e a(1) = 0 = 0^2 + 0^2
%e a(2) = 0 = 0^2 + 0^2 + 0^2
%e a(3) = 0 = 0^2 + 0^2 + 0^2 + 0^2
%e a(4) = 1 = 0^2 + 0^2 + 0^2 + 0^2 + 1^2
%e a(5) = 2 = 0^2 + 0^2 + 0^2 + 0^2 + 1^2 + 1^2
%e a(6) = 6 = 0^2 + 0^2 + 0^2 + 0^2 + 1^2 + 1^2 + 2^2
%e a(7) = 22 = 0^2 + 0^2 + 0^2 + 0^2 + 1^2 + 1^2 + 2^2 + 4^2
%e a(8) = 86 = 8^2 + 22
%e a(9) = 342 = 16^2 + 86
%t Accumulate[LinearRecurrence[{1,1,1,1,1},{0,0,0,0,1},30]^2] (* _Harvey P. Dale_, Jan 04 2015 *)
%t LinearRecurrence[{3, 2, 3, 7, 14, -32, -2, 6, -4, -6, 10, 1, -1, 0, 1, -1},{0, 0, 0, 0, 1, 2, 6, 22, 86, 342, 1303, 5024, 19424, 75120, 290416, 1122160},28] (* _Ray Chandler_, Aug 02 2015 *)
%Y Cf. A001591, A107239, A107242, A107244, A107245, A107246, A107247, A107248.
%K easy,nonn,changed
%O 0,6
%A _Jonathan Vos Post_, May 19 2005
%E a(26) and a(27) corrected by _R. J. Mathar_, Aug 11 2009