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4-wave sequence.
12

%I #22 Feb 21 2023 09:43:23

%S 1,1,1,1,2,3,4,7,9,10,19,26,30,56,75,85,160,216,246,462,622,707,1329,

%T 1791,2037,3828,5157,5864,11021,14849,16886,31735,42756,48620,91376,

%U 123111,139997,263108,354484,403104,757588,1020696,1160693,2181389

%N 4-wave sequence.

%C This sequence is related to the nonagon or 9-gon.

%H F. v. Lamoen, <a href="http://home.wxs.nl/~lamoen/wiskunde/wave.htm">Wave sequences</a>

%H P. Steinbach, <a href="http://www.jstor.org/stable/2691048">Golden fields: a case for the heptagon</a>, Math. Mag. 70 (1997), no. 1, 22-31.

%H Eric Weisstein's World of Mathematics, <a href="http://www.mathworld.wolfram.com/Nonagon.html">Nonagon</a>.

%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,3,-3,3,0,0,0,-1,1,-1).

%F a(n) = a(n-1)+a(n-2) if n=3*m+1, a(n) = a(n-1)+a(n-4) if n=3*m+2, a(n) = a(n-1)+a(n-6) if n=3*m. Also: a(n) = 2*a(n-3)+3*a(n-6)-a(n-9)-a(n-12).

%F G.f.: -(-1-x-x^2+x^3-x^5+x^6)/(1-2*x^3-3*x^6+x^9+x^12)

%F a(n-1) = sequence(sequence(T(n,k), k=2..4), n>=2) with a(0)=1; T(n,k) = sum(T(n-1,k1), k1 = 5-k..4) with T(1,1) = T(1,2) = T(1,3) = 0 and T(1,4) = 1; n>=1 and 1 <= k <= 4. [Steinbach]

%e The first few rows of the T(n,k) array are, n>=1, 1 <= k <=4:

%e 0, 0, 0, 1

%e 1, 1, 1, 1

%e 1, 2, 3, 4

%e 4, 7, 9, 10

%e 10, 19, 26, 30

%e 30, 56, 75, 85

%e 85, 160, 216, 246

%p m:=4: nmax:=15: for k from 1 to m-1 do T(1,k):=0 od: T(1,m):=1: for n from 2 to nmax do for k from 1 to m do T(n,k):= add(T(n-1,k1), k1=m-k+1..m) od: od: for n from 1 to nmax/2 do seq(T(n,k), k=1..m) od; a(0):=1: Tx:=1: for n from 2 to nmax do for k from 2 to m do a(Tx):= T(n,k): Tx:=Tx+1: od: od: seq(a(n), n=0..Tx-1); # _Johannes W. Meijer_, Aug 03 2011

%t LinearRecurrence[{1,-1,3,-3,3,0,0,0,-1,1,-1},{1,1,1,1,2,3,4,7,9,10,19},50] (* _Harvey P. Dale_, Oct 02 2015 *)

%Y The a(3*n) lead to A006357; The T(n,k) lead to A076264 and A091024.

%Y Cf. A038196, A038201.

%Y Cf. A120747 (m = 5: hendecagon or 11-gon)

%K easy,nonn

%O 0,5

%A _Floor van Lamoen_

%E Edited by _Floor van Lamoen_, Feb 05 2002

%E Edited and information added by _Johannes W. Meijer_, Aug 03 2011