A038342 -id:A038342 - cp's OEIS frontend

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

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

A069006 Let M denote the 5 X 5 matrix with rows /1,1,1,1,1/1,1,1,1,0/1,1,1,0,0/1,1,0,0,0/1,0,0,0,0/ and A(n) = vector (x(n),y(n),z(n),t(n),u(n)) = M^n*A where A is the vector (1,1,1,1,1); then a(n) = t(n).

Original entry on oeis.org

1, 2, 9, 29, 105, 365, 1287, 4516, 15873, 55759, 195910, 688286, 2418195, 8495917, 29849041, 104869718, 368442700, 1294463368, 4547886208, 15978257251, 56137003923, 197228218022, 692929213991, 2434493909304, 8553197751125

Offset: 0

Benoit Cloitre, Apr 02 2002

a(n-1) (with a(-1) = 0) appears in the formula for 1/rho(11)^n, n >= 0, with rho(11) = 2*cos(Pi/11) (the length ratio (smallest diagonal)/side in the regular 11-gon), when written in the power basis of the degree 5 number field Q(rho(11)): 1/rho(11)^n = A038342(n)*1 + A230080*rho(11) - A230081(n)*rho(11)^2 - a(n-1)*rho(11)^3 + A038342(n-1)* rho(11)^4, n >= 0, with A038342(-1) = 0. See A230080 with the example for n=4. - Wolfdieter Lang, Nov 04 2013

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

Cf. A006359, A069007, A069008, A069009, A070778, A006359(offset), for x(n), y(n), z(n), t(n), u(n), v(n).

A038342, A230080, A230081 (for powers of 1/rho(11), see a comment above).

LinearRecurrence[{3,3,-4,-1,1},{1,2,9,29,105},30] (* Harvey P. Dale, Apr 16 2015 *)

G.f.:(1-x)/(1-x^5+x^4+4*x^3-3*x^2-3*x). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009

a(n) = 3*a(n-1) + 3*a(n-2) - 4*a(n-3) - a(n-4) + a(n-5), n >= 0, with a(-5)=0, a(-4)=-1, a(-3)=a(-2)=a(-1)=0. - Wolfdieter Lang, Nov 04 2013

Edited by Henry Bottomley, May 06 2002

A120747 Sequence relating to the 11-gon (or hendecagon).

Original entry on oeis.org

0, 1, 4, 14, 50, 175, 616, 2163, 7601, 26703, 93819, 329615, 1158052, 4068623, 14294449, 50221212, 176444054, 619907431, 2177943781, 7651850657, 26883530748, 94450905714, 331837870408, 1165858298498, 4096053203771, 14390815650209, 50559786403254

Offset: 1

Gary W. Adamson, Jul 01 2006

The hendecagon is an 11-sided polygon. The preferred word in the OEIS is 11-gon.
The lengths of the diagonals of the regular 11-gon are r[k] = sin(k*Pi/11)/sin(Pi/11), 1 <= k <= 5, where r[1] = 1 is the length of the edge.
The value of limit(a(n)/a(n-1),n=infinity) equals the longest diagonal r[5].
The a(n) equal the matrix elements M^n[1,2], where M = Matrix([[1,1,1,1,1], [1,1,1,1,0], [1,1,1,0,0], [1,1,0,0,0], [1,0,0,0,0]]). The characteristic polynomial of M is (x^5 - 3x^4 - 3x^3 + 4x^2 + x - 1) with roots x1 = -r[4]/r[3], x2 = -r[2]/r[4], x3 = r[1]/r[2], x4 = r[3]/r[5] and x5 = r[5]/r[1].
Note that M^4*[1,0,0,0,0] = [55, 50, 41, 29, 15] which are all terms of the 5-wave sequence A038201. This is also the case for the terms of M^n*[1,0,0,0,0], n>=1.

			From _Johannes W. Meijer_, Aug 03 2011: (Start)
The lengths of the regular hendecagon edge and diagonals are:
  r[1] = 1.000000000, r[2] = 1.918985948, r[3] = 2.682507066,
  r[4] = 3.228707416, r[5] = 3.513337092.
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, ... (End)

G. C. Greubel, Table of n, a(n) for n = 1..1000
Jay Kappraff, Slavik Jablan, Gary W. Adamson and Radmila Sazdanovich, Golden Fields, Generalized Fibonacci Sequences and Chaotic Matrices, Forma, Vol. 19 No. 4, pp. 367-387, 2004.
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31, MR 1439165
Eric Weisstein's World of Mathematics, Hendecagon.
Index entries for linear recurrences with constant coefficients, signature (3,3,-4,-1,1).

Cf. A006358, A033304, A038201, A052963, A065941, A066170.
From Johannes W. Meijer, Aug 03 2011: (Start)
Cf. A006358 (T(n+2,1) and T(n+1,5)), A069006 (T(n+1,2)), A038342 (T(n+1,3)), this sequence (T(n,4)) (m=5: hendecagon or 11-gon).
Cf. A000045 (m=2; pentagon or 5-gon); A006356, A006054 and A038196 (m=3: heptagon or 7-gon); A006357, A076264, A091024 and A038197 (m=4: enneagon or 9-gon); A006359, A069007, A069008, A069009, A070778 (m=6; tridecagon or 13-gon); A025030 (m=7: pentadecagon or 15-gon); A030112 (m=8: heptadecagon or 17-gon). (End)

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x^2*(1+x-x^2)/(1-3*x-3*x^2+4*x^3+x^4-x^5) )); // G. C. Greubel, Nov 13 2022

nmax:=27: m:=5: 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 1 to nmax do a(n):=T(n,4) od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Aug 03 2011

LinearRecurrence[{3, 3, -4, -1, 1}, {0, 1, 4, 14, 50}, 41] (* G. C. Greubel, Nov 13 2022 *)

def A120747_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+x-x^2)/(1-3*x-3*x^2+4*x^3+x^4-x^5) ).list()
A120747_list(40) # G. C. Greubel, Nov 13 2022

a(n) = 3*a(n-1) + 3*a(n-2) - 4*a(n-3) - a(n-4) + a(n-5).
G.f.: x^2*(1+x-x^2)/(1-3*x-3*x^2+4*x^3+x^4-x^5). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009
From Johannes W. Meijer, Aug 03 2011: (Start)
a(n) = T(n,4) with T(n,k) = Sum_{k1 = 6-k..6} T(n-1, k1), 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]
Sum_{k=1..5} T(n,k)*r[k] = r[5]^n, n>=1. [Steinbach]
r[k] = sin(k*Pi/11)/sin(Pi/11), 1 <= k <= 5. [Kappraff]
Sum_{k=1..5} T(n,k) = A006358(n-1).
Limit_{n -> 00} T(n,k)/T(n-1,k) = r[5], 1 <= k <= 5.
sequence(sequence( T(n,k), k=2..5), n>=1) = A038201(n-4).
G.f.: (x^2*(x - x1)*(x - x2))/((x - x3)*(x - x4)*(x - x5)*(x - x6)*(x - x7)) with x1 = phi, x2 = (1-phi), x3 = r[1] - r[3], x4 = r[3] - r[5], x5 = r[5] - r[4], x6 = r[4] - r[2], x7 = r[2], where phi = (1 + sqrt(5))/2 is the golden ratio A001622. (End)

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

A230080 Sequence needed for the nonpositive powers of rho(11) = 2*cos(Pi/11) in terms of the power basis of the degree 5 number field Q(rho(11)). Coefficients of the first power.

Original entry on oeis.org

0, 3, 5, 23, 73, 265, 920, 3245, 11385, 40018, 140574, 493911, 1735243, 6096533, 21419128, 75252674, 264387942, 928884046, 3263482673, 11465714843, 40282921096, 141527481021, 497233748352, 1746949771565, 6137623429414

Offset: 0

Wolfdieter Lang, Nov 04 2013

The formula for the nonpositive powers of rho(11) := 2*cos(Pi/11) (the length ratio (smallest diagonal/side) in the regular 11-gon), when written in the power basis of the degree 5 algebraic number field Q(rho(11)) is: 1/rho(11)^n = A038342(n)*1 + a(n)*rho(11) - A230081(n)*rho(11)^2 - A069006(n-1)*rho(11)^3 + A038342(n-1)*rho(11)^4, n >= 0, with A069006(-1) = 0 = A038342(-1).

			1/rho(11)^4 = 146*1 + 73*rho(11) - 173*rho(11)^2 - 29*rho(11)^3  + 41*rho(11)^4 (approximately 0.07374164519).

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

Cf. A038342, A069006, A230081.

LinearRecurrence[{3,3,-4,-1,1},{0,3,5,23,73},30] (* Harvey P. Dale, May 19 2017 *)

G.f.: x*(3 - 4*x - x^2 + x^3)/(1 - 3*x - 3*x^2 + 4*x^3 + x^4 - x^5).

a(n) = 3*a(n-1) +3*a(n-2) -4*a(n-3) -a(n-4) +a(n-5) for n >= 0, with a(-5)=-3, a(-4)=a(-3)=a(-2)=0, a(-1)=1.

A230081 Sequence needed for the nonpositive powers of rho(11) = 2*cos(Pi/11) in the power basis of the degree 5 number field Q(rho(11)). Negative of the coefficients of the second power.

Original entry on oeis.org

0, 4, 13, 50, 173, 613, 2149, 7557, 26543, 93264, 327657, 1151183, 4044478, 14209634, 49923211, 175397097, 616229093, 2165020570, 7606447024, 26724012524, 93890464368, 329868851103, 1158940469863, 4071748539926, 14305425173111

Offset: 0

Wolfdieter Lang, Nov 04 2013

The formula for the nonpositive powers of rho(11) := 2*cos(Pi/11) (the length ratio (smallest diagonal/side) in the regular 11-gon), when written in the power basis of the degree 5 algebraic number field Q(rho(11)) is: 1/rho(11)^n = A038342(n)*1 + A230080*rho(11) - a(n)*rho(11)^2 - A069006(n-1)*rho(11)^3 + A038342(n-1)*rho(11)^4, n >= 0, with A069006(-1) = 0 = A038342(-1).

			1/rho(11)^4 = 146*1 + 73*rho(11) - 173*rho(11)^2 - 29*rho(11)^3  + 41*rho(11)^4 (approximately 0.07374164519).

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

Cf. A038342, A069006, A230080.

G.f.: x*(4 + x - x^2) / (1 - 3*x - 3*x^2 + 4*x^3 + x^4 - x^5).

a(n) = 3*a(n-1) +3*a(n-2) -4*a(n-3) -a(n-4) +a(n-5) for n >= 0, with a(-5)=3, a(-4)=0, a(-3)=0, a(-2)=-1, a(-1)=0.

A231186 Decimal expansion of the ratio (longest diagonal)/side in a regular 11-gon (or hendecagon).

Original entry on oeis.org

3, 5, 1, 3, 3, 3, 7, 0, 9, 1, 6, 6, 6, 1, 3, 5, 1, 8, 8, 7, 8, 2, 1, 7, 1, 5, 9, 6, 2, 9, 7, 9, 8, 1, 8, 4, 2, 0, 7, 4, 5, 9, 4, 8, 1, 7, 7, 7, 0, 1, 4, 9, 4, 2, 2, 1, 3, 7, 7, 4, 6, 9, 0, 0, 1, 2, 0, 1, 8, 1, 7, 7, 5, 6, 9, 3, 0, 3, 0, 5, 2, 5, 9, 2, 8, 9, 1, 5, 3, 2, 9, 1, 7, 1, 4, 9, 9, 3, 7, 0, 0, 1, 6

Offset: 1

Wolfdieter Lang, Nov 20 2013

omega(11):= S(4, x) = 1 - 3*x^2 + x^4 with x = rho(11) := 2*cos(Pi/11). See A049310 for Chebyshev s-polynomials. rho(11) is the ratio (shortest diagonal)/side in a regular 11-gon. See the Q(2*cos(Pi/n)) link given in A187360. This is the power basis representation of omega(11) in the algebraic number field Q(2*cos(Pi/11)) of degree 5.
omega(11) = 1/(2*cos(Pi*5/11)) = 1/R(5, rho(11)) with the R-polynomial given in A127672. This follows from a computation of the power basis coefficients of the reciprocal of R(5, x) (mod C(11, x)) = 1+2*x-3*x^2-x^3+x^4, where C(11, x) is the minimal polynomial of rho(11) given in A187360. The result for this reciprocal (mod C(11, x)) is 1 - 3*x^2  + x^4 giving the power base coefficients [1,0,-3,0,1] for omega(11).
omega(11) is the analog in the regular 11-gon of the golden section in the regular 5-gon (pentagon), because it is the limit of a(n+1)/a(n) for n -> infinity of sequences like A038342, A069006, A230080 and A230081.
The ratio diagonal/side of the second and third shortest diagonals in a regular 11-gon are respectively x^2 - 1 and x^3 - 2*x, where x = 2*cos(Pi/11). - Mohammed Yaseen, Nov 03 2020

			3.51333709166613518878217159629798184207459481777014...

Cf. A038342, A049310, A069006, A127672, A187360, A230080, A230081.

RealDigits[Sin[5*Pi/11]/Sin[Pi/11], 10, 120][[1]] (* Amiram Eldar, Jun 01 2023 *)

omega(11) = 1 - 3*x^2 + x^4 with x = rho(11) := 2*cos(Pi/11) = 1/(2*cos(Pi*5/11)) = 3.5133370916661... See the comments above.

Equals sin(5*Pi/11)/sin(Pi/11). - Mohammed Yaseen, Nov 03 2020

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A038201 5-wave sequence.

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A069006 Let M denote the 5 X 5 matrix with rows /1,1,1,1,1/1,1,1,1,0/1,1,1,0,0/1,1,0,0,0/1,0,0,0,0/ and A(n) = vector (x(n),y(n),z(n),t(n),u(n)) = M^n*A where A is the vector (1,1,1,1,1); then a(n) = t(n).

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A120747 Sequence relating to the 11-gon (or hendecagon).

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A230080 Sequence needed for the nonpositive powers of rho(11) = 2*cos(Pi/11) in terms of the power basis of the degree 5 number field Q(rho(11)). Coefficients of the first power.

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A230081 Sequence needed for the nonpositive powers of rho(11) = 2*cos(Pi/11) in the power basis of the degree 5 number field Q(rho(11)). Negative of the coefficients of the second power.

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A231186 Decimal expansion of the ratio (longest diagonal)/side in a regular 11-gon (or hendecagon).

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