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%I #29 Feb 23 2023 04:19:23
%S 1,1,1,1,1,2,3,4,5,9,12,14,15,29,41,50,55,105,146,175,190,365,511,616,
%T 671,1287,1798,2163,2353,4516,6314,7601,8272,15873,22187,26703,29056,
%U 55759,77946,93819,102091,195910,273856,329615,358671,688286,962142
%N 5-wave sequence.
%C This sequence is related to the hendecagon or 11-gon, see A120747.
%C Sequence of perfect distributions for a cascade merge with six tapes according to Knuth. - _Michael Somos_, Feb 07 2004
%D D. E. Knuth, Art of Computer Programming, Vol. 3, Sect. 5.4.3, Eq. (1).
%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/Hendecagon.html">Hendecagon</a>.
%H <a href="/index/Rec#order_20">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,3,0,0,0,3,0,0,0,-4,0,0,0,-1,0,0,0,1).
%F a(n) = a(n-1)+a(n-2) if n=4*m+1, a(n) = a(n-1)+a(n-4) if n=4*m+2, a(n) = a(n-1)+a(n-6) if n=4*m+3 and a(n) = a(n-1)+a(n-8) if n=4*m.
%F G.f.: -(1+x+x^2+x^3-2*x^4-x^5+x^7-x^8-x^11+x^12)/(-1+3*x^4+3*x^8-4*x^12-x^16+x^20).
%F a(n) = 3*a(n-4)+3*a(n-8)-4*a(n-12)-a(n-16)+a(n-20).
%F a(n-1) = sequence(sequence(T(n,k), k=2..5), n>=2) with a(0)=1; T(n,k) = sum(T(n-1,k1), k1 = 6-k..5) with T(1,1) = T(1,2) = T(1,3) = T(1,4) = 0 and T(1,5) = 1; n>=1 and 1 <= k <= 5. [Steinbach]
%e The first few rows of the T(n,k) array are, n>=1, 1 <= k <=5:
%e 0, 0, 0, 0, 1
%e 1, 1, 1, 1, 1
%e 1, 2, 3, 4, 5
%e 5, 9, 12, 14, 15
%e 15, 29, 41, 50, 55
%e 55, 105, 146, 175, 190
%e 190, 365, 511, 616, 671
%e G.f. = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 9*x^8 + 12*x^9 + ...
%p m:=5: nmax:=12: 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[{0,0,0,3,0,0,0,3,0,0,0,-4,0,0,0,-1,0,0,0,1},{1,1,1,1,1,2,3,4,5,9,12,14,15,29,41,50,55,105,146,175},50] (* _Harvey P. Dale_, Dec 13 2012 *)
%o (PARI) {a(n) = local(m); if( n<=0, n==0, m = (n-1)\4 * 4; sum(k=2*m - n, m, a(k)))};
%Y Cf. A038196, A038197.
%Y The a(4*n) values (column 0) lead to A006358; the T(n,k) lead to A069006, A038342 and A120747.
%K easy,nonn
%O 0,6
%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
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