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
Cf.
A006359,
A069007,
A069008,
A069009,
A070778,
A006359(offset), for x(n), y(n), z(n), t(n), u(n), v(n).
-
LinearRecurrence[{3,3,-4,-1,1},{1,2,9,29,105},30] (* Harvey P. Dale, Apr 16 2015 *)
A069007
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) = y(n).
Original entry on oeis.org
1, 5, 20, 85, 350, 1456, 6034, 25038, 103849, 430794, 1786960, 7412548, 30748055, 127546530, 529077571, 2194674687, 9103762600, 37763453591, 156647144355, 649790354877, 2695405055655, 11180849888139, 46379450073255
Offset: 0
Cf.
A006359,
A069007,
A069008,
A069009,
A070778,
A006359 (offset), for x(n), y(n), z(n), t(n), u(n), v(n).
-
a:= n->(Matrix(6, (i, j)->`if`(i+j>7, 0, 1))^n.<<[1$6][]>>)[2, 1]:
seq(a(n), n=0..30); # Alois P. Heinz, Jun 18 2013
-
m = Table[ If[i + j <= 7, 1, 0], {i, 1, 6}, {j, 1, 6}]; mp[n_] := MatrixPower[m, n].m[[1]]; a[n_] := mp[n][[2]]; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Jun 18 2013 *)
A069008
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) = z(n).
Original entry on oeis.org
1, 4, 18, 74, 309, 1280, 5313, 22035, 91410, 379171, 1572857, 6524375, 27063881, 112264055, 465684247, 1931711700, 8012962189, 33238687760, 137877896315, 571933356551, 2372445281505, 9841175633650, 40822327332150, 169335704473650, 702423959724591
Offset: 0
Cf.
A006359,
A069007,
A069008,
A069009,
A070778,
A006359 (offset), for x(n), y(n), z(n), t(n), u(n), v(n).
-
a:= n->(Matrix(6, (i, j)->`if`(i+j>7, 0, 1))^n.<<[1$6][]>>)[3, 1]:
seq(a(n), n=0..30); # Alois P. Heinz, Jun 14 2013
-
m = Table[ If[i + j <= 7, 1, 0], {i, 1, 6}, {j, 1, 6}]; mp[n_] := MatrixPower[m, n].m[[1]]; a[n_] := mp[n][[3]]; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Jun 18 2013 *)
A070778
Let M denote the 6 X 6 matrix = row by row /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) = u(n).
Original entry on oeis.org
1, 2, 11, 41, 176, 721, 3003, 12439, 51623, 214103, 888173, 3684174, 15282475, 63393324, 262962987, 1090800411, 4524765831, 18769248040, 77856998326, 322959774150, 1339674254489, 5557122741105, 23051583675890, 95620617831960, 396645310086831, 1645330322871807
Offset: 0
Cf.
A006359,
A069007,
A069008,
A069009,
A070778,
A006359 (offset), for x(n), y(n), z(n), t(n), u(n), v(n).
-
I:=[1,2,11,41,176,721]; [n le 6 select I[n] else 3*Self(n-1)+6*Self(n-2)-4*Self(n-3)-5*Self(n-4)+Self(n-5)+Self(n-6): n in [1..30]]; // Vincenzo Librandi, Oct 10 2017
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a:= n-> (Matrix(6, (i, j)->`if`(i+j>7, 0, 1))^n.<<[1$6][]>>)[5, 1]:
seq(a(n), n=0..30); # Alois P. Heinz, Jun 14 2013
-
CoefficientList[Series[(x^2 + x - 1)/(x^6 + x^5 - 5*x^4 - 4*x^3 + 6*x^2 + 3*x - 1), {x, 0, 30}], x] (* Wesley Ivan Hurt, Oct 09 2017 *)
LinearRecurrence[{3, 6, -4, -5, 1, 1}, {1, 2, 11, 41, 176, 721}, 30] (* Vincenzo Librandi, Oct 10 2017 *)
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
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 (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).
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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
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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
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LinearRecurrence[{3, 3, -4, -1, 1}, {0, 1, 4, 14, 50}, 41] (* G. C. Greubel, Nov 13 2022 *)
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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
Showing 1-5 of 5 results.
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