A029826 -id:A029826 - cp's OEIS frontend

A177738 a(n) = floor( (x^n - x^(-n)) / (x - x^(-1)) ) where x = Pi-2.

0, 1, 2, 3, 4, 5, 6, 8, 9, 11, 13, 15, 17, 20, 23, 26, 30, 35, 40, 46, 52, 60, 69, 78, 90, 103, 117, 134, 153, 175, 199, 228, 260, 297, 339, 387, 442, 505, 576, 658, 751, 858, 979, 1118, 1277, 1457, 1664, 1900, 2169, 2476, 2826

Offset: 0

Roger L. Bagula, May 12 2010

nonn

The ratio a(n+1)/a(n) approaches Pi-2 as n approaches infinity, and is lower than even Salem polynomial expansions based on A073011.

The idea is the emulation of quadratic beta integer domains using a transcendental number base with a ratio below A073011.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A073011, A029826, A000796.

Clear[a, n, b]; b = Pi - 2; a[n_] = (b^n - b^(-n))/(b - b^(-1));
Table[Floor[a[n]], {n, 0, 50}]

Undefined terminology removed from the definition - The Assoc. Eds. of the OEIS, May 14 2010

A225396 Expansion of 1/(1 - x - x^2 + x^10 - x^12).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 88, 142, 229, 369, 595, 959, 1546, 2492, 4017, 6475, 10438, 16826, 27123, 43722, 70479, 113611, 183139, 295217, 475885, 767119, 1236583, 1993351, 3213249, 5179704, 8349597, 13459412, 21696349, 34974155, 56377758, 90880011

Offset: 0

Roger L. Bagula, May 06 2013

Limiting ratio is 1.61198..., the largest real root of -1 + x^2 - x^10 - x^11 + x^12 = 0.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Roger L. Bagula, Demo file
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1).

Cf. A117791, A107293, A204631, A225393, A225394, A029826, A147660.

CoefficientList[Series[1/(1 - x - x^2 + x^10 - x^12), {x, 0, 50}], x]
LinearRecurrence[{1,1,0,0,0,0,0,0,0,-1,0,1},{1,1,2,3,5,8,13,21,34,55,88,142},50] (* Harvey P. Dale, Apr 12 2014 *)

a(0)=1, a(1)=1, a(2)=2, a(3)=3, a(4)=5, a(5)=8, a(6)=13, a(7)=21, a(8)=34, a(9)=55, a(10)=88, a(11)=142, a(n)=a(n-1)+a(n-2)- a(n-10)+ a(n-12). - Harvey P. Dale, Apr 12 2014

A173244 G.f. (x^10 +x^9 +x^8 +.... +x+1) / (x^10 +x^9 -x^7 -x^6 -x^5 -x^4 -x^3 +x +1).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 7, 8, 9, 11, 13, 15, 18, 21, 25, 29, 34, 41, 47, 56, 66, 77, 91, 107, 126, 148, 174, 205, 241, 283, 334, 392, 461, 543, 638, 751, 883, 1039, 1222, 1437, 1691, 1989, 2339, 2752, 3237, 3807, 4479, 5268, 6197, 7289, 8574, 10086, 11863

Offset: 0

Roger L. Bagula, Feb 13 2010

nonn

Limiting ratio is: a(n+1)/a(n)->1.1762808182599176...

Cf. A029826

p[x_] = Sum[x^i, {i, 0, 10}]/(x^10 + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1);;
a = Table[SeriesCoefficient[ Series[p[x], {x, 0, 50}], n], {n, 0, 50}]

G.f.: (1-x^11)/(1-x^2-x^3+x^8+x^9-x^11).

Removed unused variables - The Assoc. Editors of the OEIS, Feb 24 2010

cp's OEIS Frontend

A177738 a(n) = floor( (x^n - x^(-n)) / (x - x^(-1)) ) where x = Pi-2.

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A225396 Expansion of 1/(1 - x - x^2 + x^10 - x^12).

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A173244 G.f. (x^10 +x^9 +x^8 +.... +x+1) / (x^10 +x^9 -x^7 -x^6 -x^5 -x^4 -x^3 +x +1).

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