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Decimal expansion of (611 + sqrt(73))/36.
+10
8
1, 7, 2, 0, 9, 5, 5, 5, 6, 5, 9, 5, 9, 2, 1, 5, 3, 6, 4, 3, 5, 5, 1, 9, 9, 0, 2, 3, 1, 2, 8, 4, 4, 3, 6, 2, 8, 9, 8, 4, 9, 8, 4, 5, 9, 8, 1, 3, 7, 5, 9, 2, 4, 5, 0, 6, 7, 1, 9, 6, 8, 4, 7, 5, 7, 0, 4, 9, 2, 1, 2, 4, 6, 7, 2, 0, 3, 5, 3, 6, 0, 6, 6, 1, 4, 1, 1, 3, 8, 1
FORMULA
Equals lim_{n->infinity} A082640(2, n)^(1/n). Equals 288*x_2, where x_2 is the largest root of 5184*x^2 - 611*x + 18.
EXAMPLE
17.2095556595921536435519902312844...
MATHEMATICA
First[RealDigits[N[(611+Sqrt[73])/36, 90]]]
Decimal expansion of log_2((611 + sqrt(73))/36)/2.
+10
8
2, 0, 5, 2, 5, 6, 8, 9, 7, 1, 6, 1, 2, 7, 3, 5, 6, 6, 5, 1, 0, 7, 8, 7, 1, 5, 4, 0, 4, 7, 8, 6, 5, 5, 8, 7, 1, 0, 5, 3, 8, 4, 8, 7, 6, 2, 3, 7, 1, 2, 2, 1, 4, 3, 8, 8, 9, 2, 9, 8, 0, 3, 2, 7, 7, 4, 1, 7, 9, 0, 8, 2, 0, 0, 4, 1, 2, 0, 7, 1, 0, 4, 6, 5, 9, 3, 2, 3, 6, 3
FORMULA
Equals log_2(alpha)/2, where alpha = lim_{n->oo} A082640(2, n)^(1/n).
EXAMPLE
2.052568971612735665107871540478655871...
MATHEMATICA
First[RealDigits[N[Log[2, (611+Sqrt[73])/36]/2, 90]]]
Decimal expansion of lim_{n->infinity} f(3, n)^(1/(3*n)), where f(m, n) is the number of primitive lattice triangulations of m X n rectangle.
+10
8
4, 2, 3, 9, 3, 6, 9, 4, 8, 1, 5, 4, 8, 0, 2, 5, 6, 7, 1, 8, 7, 7, 6, 2, 5, 7, 4, 2, 0, 4, 5, 2, 3, 5, 7, 7, 2, 1, 0, 0, 6, 9, 5, 7, 1, 1, 2, 5, 1, 7, 9, 5, 4, 9, 9, 8, 3, 0, 8, 0, 1, 6, 8, 7, 8, 3, 3, 3, 5, 8, 2, 3, 8, 2, 7, 6, 7, 2, 8, 9, 8, 7, 8, 3, 7, 0, 5, 4, 8
EXAMPLE
4.2393694815480256718776257420452357721...
Decimal expansion of log_2(lim_{n->infinity} f(3, n)^(1/(3*n))), where f(m, n) is the number of primitive lattice triangulations of m X n rectangle.
+10
8
2, 0, 8, 3, 8, 4, 9, 7, 0, 9, 7, 2, 1, 0, 2, 3, 2, 0, 8, 2, 2, 4, 2, 1, 9, 2, 8, 9, 4, 9, 6, 1, 7, 0, 1, 3, 9, 6, 4, 8, 5, 1, 3, 4, 2, 3, 2, 4, 9, 5, 5, 2, 1, 3, 0, 7, 9, 9, 0, 5, 9, 9, 1, 8, 5, 5, 1, 7, 2, 9, 0, 6, 9, 2, 8, 1, 8, 0, 5, 2, 5, 1, 8, 6, 6, 5, 0, 8, 8
EXAMPLE
2.0838497097210232082242192894961701...
a(n) is the number of primitive triangulations of a 5 X n rectangle.
+10
8
252, 182132, 182881520, 208902766788, 260420548144996, 341816489625522032, 464476385680935656240, 645855159466371391947660, 913036902513499041820702784, 1306520849733616781789190513820, 1887591165891651253904039432371172, 2747848427721241461905176361078147168
a(n) is the number of primitive triangulations of a 6 X n rectangle.
+10
8
924, 2801708, 12244184472, 61756221742966, 341816489625522032, 1999206934751133055518, 12169409954141988707186052, 76083336332947513655554918994, 484772512167266688498399632918196, 3131521959869770128138491287826065904, 20443767611927599823217291769468449488548
a(n) is the number of primitive triangulations of a 7 X n rectangle.
+10
8
3432, 43936824, 839660660268, 18792896208387012, 464476385680935656240, 12169409954141988707186052, 332633840844113103751597995920, 9369363517501208819530429967280708, 269621109753732518252493257828413137272, 7880009979020501614060394747170100093057300, 233031642883906149386619647304562977586311372556
a(n) is the number of primitive triangulations of an 8 X n rectangle.
+10
8
12870, 698607816, 58591381296256, 5831528022482629710, 645855159466371391947660, 76083336332947513655554918994, 9369363517501208819530429967280708, 1191064812882685539785713745400934044308, 155023302820254133629368881178138076738462112, 20527337238769032315796332007167102984745417344046
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