A038197 4-wave sequence.

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

Author

Floor van Lamoen

Keywords

• easy
• nonn

Comments

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

Examples

			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
		

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.
• Index entries for linear recurrences with constant coefficients, signature (1,-1,3,-3,3,0,0,0,-1,1,-1).

Crossrefs

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

Cf. A038196, A038201.

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

Programs

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 *)

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]

Extensions

Edited by Floor van Lamoen, Feb 05 2002

Edited and information added by Johannes W. Meijer, Aug 03 2011