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A038197
4-wave sequence.
12
1, 1, 1, 1, 2, 3, 4, 7, 9, 10, 19, 26, 30, 56, 75, 85, 160, 216, 246, 462, 622, 707, 1329, 1791, 2037, 3828, 5157, 5864, 11021, 14849, 16886, 31735, 42756, 48620, 91376, 123111, 139997, 263108, 354484, 403104, 757588, 1020696, 1160693, 2181389
OFFSET
0,5
COMMENTS
This sequence is related to the nonagon or 9-gon.
LINKS
F. v. Lamoen, Wave sequences
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
Eric Weisstein's World of Mathematics, Nonagon.
FORMULA
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).
G.f.: -(-1-x-x^2+x^3-x^5+x^6)/(1-2*x^3-3*x^6+x^9+x^12)
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]
EXAMPLE
The first few rows of the T(n,k) array are, n>=1, 1 <= k <=4:
0, 0, 0, 1
1, 1, 1, 1
1, 2, 3, 4
4, 7, 9, 10
10, 19, 26, 30
30, 56, 75, 85
85, 160, 216, 246
MAPLE
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
MATHEMATICA
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 *)
CROSSREFS
The a(3*n) lead to A006357; The T(n,k) lead to A076264 and A091024.
Cf. A120747 (m = 5: hendecagon or 11-gon)
Sequence in context: A138576 A259483 A187503 * A187506 A348443 A023546
KEYWORD
easy,nonn
EXTENSIONS
Edited by Floor van Lamoen, Feb 05 2002
Edited and information added by Johannes W. Meijer, Aug 03 2011
STATUS
approved