A038201 5-wave sequence.

1, 1, 1, 1, 1, 2, 3, 4, 5, 9, 12, 14, 15, 29, 41, 50, 55, 105, 146, 175, 190, 365, 511, 616, 671, 1287, 1798, 2163, 2353, 4516, 6314, 7601, 8272, 15873, 22187, 26703, 29056, 55759, 77946, 93819, 102091, 195910, 273856, 329615, 358671, 688286, 962142

Offset: 0

Author

Floor van Lamoen

Keywords

• easy
• nonn

Comments

This sequence is related to the hendecagon or 11-gon, see A120747.

Sequence of perfect distributions for a cascade merge with six tapes according to Knuth. - Michael Somos, Feb 07 2004

Examples

			The first few rows of the T(n,k) array are, n>=1, 1 <= k <=5:
  0,   0,   0,   0,   1
  1,   1,   1,   1,   1
  1,   2,   3,   4,   5
  5,   9,   12,  14,  15
  15,  29,  41,  50,  55
  55,  105, 146, 175, 190
  190, 365, 511, 616, 671
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 + ...
		

References

• D. E. Knuth, Art of Computer Programming, Vol. 3, Sect. 5.4.3, Eq. (1).

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, Hendecagon.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,3,0,0,0,-4,0,0,0,-1,0,0,0,1).

Crossrefs

Cf. A038196, A038197.

The a(4*n) values (column 0) lead to A006358; the T(n,k) lead to A069006, A038342 and A120747.

Programs

Maple

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

Mathematica

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 *)

PARI

{a(n) = local(m); if( n<=0, n==0, m = (n-1)\4 * 4; sum(k=2*m - n, m, a(k)))};

Formula

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.

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).

a(n) = 3*a(n-4)+3*a(n-8)-4*a(n-12)-a(n-16)+a(n-20).

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]

Extensions

Edited by Floor van Lamoen, Feb 05 2002

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