A120747 -id:A120747 - cp's OEIS frontend

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

Floor van Lamoen

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

			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

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).

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)

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

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 *)

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]

Edited by Floor van Lamoen, Feb 05 2002

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

A038201 5-wave sequence.

Original entry on oeis.org

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

Floor van Lamoen

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

			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 + ...

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

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).

Cf. A038196, A038197.

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

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

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 *)

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

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]

Edited by Floor van Lamoen, Feb 05 2002

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

A069009 Let M denote the 6 X 6 matrix with rows / 1,1,1,1,1,1 / 1,1,1,1,1,0 / 1,1,1,1,0,0 / 1,1,1,0,0,0 / 1,1,0,0,0,0 / 1,0,0,0,0,0 / and A(n) the vector (x(n),y(n),z(n),t(n),u(n),v(n)) = M^n*A where A is the vector (1,1,1,1,1,1); then a(n) = t(n).

Original entry on oeis.org

1, 3, 15, 59, 250, 1030, 4283, 17752, 73658, 305513, 1267344, 5257031, 21806850, 90457205, 375227042, 1556484658, 6456477531, 26782210229, 111095686086, 460837670465, 1911607611040, 7929568022610, 32892759309540

Offset: 0

Benoit Cloitre, Apr 02 2002

This sequence is related to the tridecagon or triskaidecagon (13-gon).

The lengths of the diagonals of the regular tridecagon are r[k] = sin(k*Pi/13)/sin(Pi/13), 1 <= k <= 6, where r[1] = 1 is the length of the edge.

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
Index entries for linear recurrences with constant coefficients, signature (3, 6, -4, -5, 1, 1).

Cf. A066170.
Cf. A006359, A069007, A069008, A069009, A070778, A006359 (offset), for x(n), y(n), z(n), t(n), u(n), v(n).
Cf. A120747 (m = 5: hendecagon or 11-gon)

nmax:=22: with(LinearAlgebra): M:=Matrix([[1,1,1,1,1,1], [1,1,1,1,1,0], [1,1,1,1,0,0], [1,1,1,0,0,0], [1,1,0,0,0,0], [1,0,0,0,0,0]]): v:= Vector[row]([1,1,1,1,1,1]): for n from 0 to nmax do b:=evalm(v&*M^n); a(n):=b[4] od: seq(a(n), n=0..nmax); # Johannes W. Meijer, Aug 03 2011
nmax:=24: m:=6: 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/3 do seq(T(n,k), k=1..m) od; for n from 2 to nmax do a(n-2):=T(n,3) od: seq(a(n), n=0..nmax-2); # Johannes W. Meijer, Aug 03 2011

b = {1, -3, -6, 4, 5, -1, -1}; p[x_] := Sum[x^(n - 1)*b[[8 - n]], {n, 1, 7}] q[x_] := ExpandAll[x^6*p[1/x]] Table[ SeriesCoefficient[ Series[x/q[x], {x, 0, 30}], n], {n, 0, 30}] (* Roger L. Bagula and Gary W. Adamson, Sep 19 2006 *)
CoefficientList[Series[1/(1 - 3 x - 6 x^2 + 4 x^3 + 5 x^4 - x^5 - x^6), {x, 0, 33}], x] (* Vincenzo Librandi, Sep 19 2015 *)

Vec(1/(1-3*x-6*x^2+4*x^3+5*x^4-x^5-x^6)+O(x^33)) \\ Joerg Arndt, Sep 19 2015

G.f.: 1/(1-3*x-6*x^2+4*x^3+5*x^4-x^5-x^6). - Roger L. Bagula and Gary W. Adamson, Sep 19 2006
a(n-2) = T(n,3) with T(n,k) = sum(T(n-1,k1), k1=7-k..6), T(1,1) = T(1,2) = T(1,3) = T(1,4) = T(1,5) = 0 and T(1,6) = 1, n>=1 and 1 <= k <= 6. [Steinbach]
sum(T(n,k)*r[k], k=1..6) = r[6]^n, n>=1, with r[k] = sin(k*Pi/13)/sin(Pi/13). [Steinbach]

Edited by Henry Bottomley, May 06 2002

Information added by Johannes W. Meijer, Aug 03 2011

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A038197 4-wave sequence.

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A038201 5-wave sequence.

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A069009 Let M denote the 6 X 6 matrix with rows / 1,1,1,1,1,1 / 1,1,1,1,1,0 / 1,1,1,1,0,0 / 1,1,1,0,0,0 / 1,1,0,0,0,0 / 1,0,0,0,0,0 / and A(n) the vector (x(n),y(n),z(n),t(n),u(n),v(n)) = M^n*A where A is the vector (1,1,1,1,1,1); then a(n) = t(n).

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