A099206 -id:A099206 - cp's OEIS frontend

A103748 Triangle read by rows, based on the morphism f: 1->{2}, 2->{3}, 3->{3,3,2,1}. First row is 1. If current row is a,b,c,..., then the next row is a,b,c,...,f(a),f(b),f(c),...

1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 3, 3, 2, 1, 1, 2, 2, 3, 2, 3, 3, 3, 3, 2, 1, 2, 3, 3, 3, 3, 2, 1, 3, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 2, 1, 2, 2, 3, 2, 3, 3, 3, 3, 2, 1, 2, 3, 3, 3, 3, 2, 1, 3, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 2, 2, 3, 3, 3, 3, 2, 1, 3, 3, 3, 2, 1, 3

Offset: 0

Roger L. Bagula, Mar 28 2005

Kenyon tile substitution sequence.

Richard Kenyon, The Construction of Self-Similar Tilings

Cf. A073058, A099206.

s[n_] := n /. {1 -> 2, 2 -> 3, 3 -> {3, 3, 2, 1}}; t[a_] := Join[a, Flatten[s /@ a]]; Flatten[ NestList[t, {1}, 5]]

A103749 Expansion of x(1+2x)/(1+x+x^2-2*x^3).

Original entry on oeis.org

0, 1, 1, -2, 3, 1, -8, 13, -3, -26, 55, -35, -72, 217, -215, -146, 795, -1079, -8, 2677, -4827, 2134, 8047, -19835, 16056, 19873, -75599, 87838, 27507, -266543, 414712, -93155, -854643, 1777222, -1108889, -2377619, 7040952, -6881111, -4915079, 25878094, -34725237, -983015

Offset: 0

Roger L. Bagula, Mar 28 2005

Insert n=3, p=2, q=-1 and r=1 in Kenyon's characteristic polynomial x^n-p*x^(n-1)+q*x+r=0 .

Richard Kenyon, The Construction of Self-Similar Tilings, arXiv:math.MG/9405210
Index entries for linear recurrences with constant coefficients, signature (-1,-1,2).

Cf. A099206.

LinearRecurrence[{-1,-1,2},{0,1,1},50] (* Harvey P. Dale, Jul 23 2013 *)

a(n) = -a(n-1)-a(n-2)+2*a(n-3) = A077975(n-1)+2*A077975(n-2).

Edited, replaced by signed variant by the Assoc. Eds. of the OEIS - Jul 31 2010

A103750 Expansion of (1+2x^3)/(1-x+x^3-2x^4).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 4, 5, 7, 11, 14, 17, 20, 28, 39, 53, 65, 82, 107, 148, 196, 253, 319, 419, 558, 745, 964, 1244, 1615, 2141, 2825, 3698, 4787, 6244, 8196, 10805, 14135, 18427, 24014, 31489, 41332, 54172, 70711, 92357, 120849, 158482, 207547, 271412, 354628, 464045

Offset: 0

Roger L. Bagula, Mar 28 2005

Set n=4, p=2, q=-1 and r=-1 in the characteristic polynomial x^n-p*x^(n-1)+q*x+r=0.

Richard Kenyon, The Construction of Self-Similar Tilings, arXiv:math.MG/9505210
Index entries for linear recurrences with constant coefficients, signature (1, 0, -1, 2).

Cf. A099206.

CoefficientList[Series[(1+2x^3)/(1-x+x^3-2x^4),{x,0,50}],x] (* or *) LinearRecurrence[{1,0,-1,2},{1,1,1,2},50] (* Harvey P. Dale, Mar 30 2012 *)

a(n) = a(n-1) -a(n-3) +2*a(n-4).

All values replaced consistent with the recurrence - the Assoc. Eds. of the OEIS - Jul 31 2010

cp's OEIS Frontend

A103748 Triangle read by rows, based on the morphism f: 1->{2}, 2->{3}, 3->{3,3,2,1}. First row is 1. If current row is a,b,c,..., then the next row is a,b,c,...,f(a),f(b),f(c),...

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A103749 Expansion of x(1+2x)/(1+x+x^2-2*x^3).

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A103750 Expansion of (1+2x^3)/(1-x+x^3-2x^4).

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cp's OEIS Frontend

A103748 Triangle read by rows, based on the morphism f: 1->{2}, 2->{3}, 3->{3,3,2,1}. First row is 1. If current row is a,b,c,..., then the next row is a,b,c,...,f(a),f(b),f(c),...

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A103749 Expansion of x*(1+2*x)/(1+x+x^2-2*x^3).

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A103750 Expansion of (1+2*x^3)/(1-x+x^3-2*x^4).

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A103749 Expansion of x(1+2x)/(1+x+x^2-2*x^3).

A103750 Expansion of (1+2x^3)/(1-x+x^3-2x^4).