Original entry on oeis.org
0, 64, 4096, 438976, 32768000, 2537716544, 192699928576, 14662949322176, 1114512556032000, 84722519068761664, 6440005257691303936, 489526700523945017536, 37210462578386173952000, 2828485190904971429628224
Offset: 0
A152152
a(n) = Product_{k=1..n} (1 + 4*sin(2*Pi*k/n)^2).
Original entry on oeis.org
0, 1, 1, 16, 25, 121, 256, 841, 2025, 5776, 14641, 39601, 102400, 271441, 707281, 1860496, 4862025, 12752041, 33362176, 87403801, 228765625, 599074576, 1568239201, 4106118241, 10749542400, 28143753121, 73680216481, 192900153616
Offset: 0
- Seiichi Manyama, Table of n, a(n) for n = 0..2000
- M. Baake, J. Hermisson, and P. Pleasants, The torus parametrization of quasiperiodic LI-classes, J. Phys. A 30 (1997), no. 9, 3029-3056. See Table 4.
- Kh. Bibak and M. H. Shirdareh Haghighi, Some Trigonometric Identities Involving Fibonacci and Lucas Numbers , Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.4
- N. Garnier and O. Ramaré, Fibonacci numbers and trigonometric identities, April 2006.
- N. Garnier and O. Ramaré, Fibonacci numbers and trigonometric identities, Fibonacci Quart. 46/47 (2008/2009), no. 1, 56-61.
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[(1-Lucas(n)+(-1)^n)^2: n in [0..30]]; // G. C. Greubel, Mar 13 2019
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Table[(1 + Fibonacci[n] - 2*Fibonacci[n+1] + (-1)^n)^2, {n, 0, 30}]
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{a(n) = (1-fibonacci(n-1)-fibonacci(n+1)+(-1)^n)^2}; \\ G. C. Greubel, Mar 13 2019
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[(1-lucas_number2(n,1,-1)+(-1)^n)^2 for n in (0..30)] # G. C. Greubel, Mar 13 2019
Original entry on oeis.org
1, 0, 21, 31, 266, 672, 3484, 11375, 48768, 177023, 716418, 2730315, 10878520, 42485638, 169181010, 670042125, 2678678730, 10705526976, 43007270292, 173003915322, 698235680844, 2822901487191, 11439823946306, 46438021798875, 188856966693230, 769224288476860, 3137871076604544, 12817404260955810
Offset: 1
-
a001350(n) = fibonacci(n+1)+fibonacci(n-1)-1-(-1)^n;
a(n) = sumdiv(n, d, moebius(n/d)*a001350(d)^3)/n; \\ Seiichi Manyama, Apr 29 2021
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f(x) = ((1-3*x+x^2)*(1+3*x+x^2))^3*(1-x^2)^10/((1-4*x-x^2)*(1-x-x^2)^6*(1+x-x^2)^9);
my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, moebius(k)*log(f(x^k))/k)) \\ Seiichi Manyama, Apr 29 2021
A324488
Inflation orbit counts b^{(3)}_n for Danzer's F-type tiling and other 3D cut and project patterns with tau-inflation.
Original entry on oeis.org
1, 0, 63, 124, 1330, 4032, 24388, 91000, 438912, 1770230, 7880598, 32763780, 141420760, 594798932, 2537715150, 10720674000, 45537538410, 192699485568, 817138135548, 3460078306440, 14662949297724, 62103832718202, 263115950765038, 1114512523173000, 4721424167330750
Offset: 1
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a001350(n) = fibonacci(n+1)+fibonacci(n-1)-1-(-1)^n;
a(n) = sumdiv(n, d, moebius(n/d)*a001350(d)^3); \\ Seiichi Manyama, Apr 29 2021
Showing 1-4 of 4 results.