A011920
a(n) = b(n)*(b(n)+1) = b(n) + ... + c(n), where b(n) = A011916(n), c(n) = A011918(n).
Original entry on oeis.org
12, 1980, 378840, 73419192, 14241916260, 2762844014580, 535977297450672, 103976830083273840, 20170969020163148220, 3913064012542622257452, 759114247456742016195720, 147264250942490855924510760
Offset: 1
Mario Velucchi (mathchess(AT)velucchi.it)
- Mario Velucchi "Seeing couples" in Recreational and Educational Computing, to appear 1997.
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A011922 := proc(n) (2+sqrt(1+((((2+sqrt(3))^(2*n)-(2-sqrt(3))^(2*n))^2)/4)))/3 ; expand(%) ; simplify(%) ; end proc:
A011916 := proc(n) ((A011922(n)-1)+sqrt(3*A011922(n)^2-4*A011922(n)+1))/2 ; end proc:
A011920 := proc(n) A011916(n)*(A011916(n)+1) ; end proc:
seq(A011920(n),n=1..20) ; # R. J. Mathar, Apr 15 2010
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LinearRecurrence[{209,-2926,2926,-209,1},{12,1980,378840,73419192,14241916260},20] (* Harvey P. Dale, Jan 01 2021 *)
A011916
a(n) = ((b(n)-1)+sqrt(3*b(n)^2-4*b(n)+1))/2, where b(n) is A011922(n).
Original entry on oeis.org
0, 3, 44, 615, 8568, 119339, 1662180, 23151183, 322454384, 4491210195, 62554488348, 871271626679, 12135248285160, 169022204365563, 2354175612832724, 32789436375292575, 456697933641263328, 6360981634602394019
Offset: 0
Mario Velucchi (mathchess(AT)velucchi.it)
- Mario Velucchi, "Seeing couples" in Recreational and Educational Computing, to appear 1997. [apparently never materialized, Colin Barker, Dec 23 2014]
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RecurrenceTable[{a[n] == 15 a[n - 1] - 15 a[n - 2] + a[n - 3], a[0] == 0, a[1] == 3, a[2] == 44}, a, {n, 0, 17}] (* Michael De Vlieger, Jul 02 2015 *)
LinearRecurrence[{15,-15,1},{0,3,44},30] (* Harvey P. Dale, Jul 26 2018 *)
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{a(n) = if( n<0, n = -n; polcoeff( x*(1 - 3*x) / ((x-1) * (x^2 - 14*x + 1)) + x * O(x^n), n), polcoeff( x*(x - 3) / ((x-1) * (x^2 - 14*x + 1)) + x * O(x^n), n))} /* Michael Somos, Jul 27 2012 */
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concat(0, Vec(x*(-3+x)/((x-1)*(x^2-14*x+1)) + O(x^100))) \\ Colin Barker, Dec 20 2014
A011922
a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3) with a(0)=1, a(1)=3, and a(2)=33.
Original entry on oeis.org
1, 3, 33, 451, 6273, 87363, 1216801, 16947843, 236052993, 3287794051, 45793063713, 637815097923, 8883618307201, 123732841202883, 1723376158533153, 24003533378261251, 334326091137124353, 4656561742541479683, 64857538304443591201, 903348974519668797123, 12582028104970919568513
Offset: 0
Mario Velucchi (mathchess(AT)velucchi.it)
- Mario Velucchi, Seeing couples, in Recreational and Educational Computing, to appear 1997.
- Vincenzo Librandi, Table of n, a(n) for n = 0..100
- Christian Aebi and Grant Cairns, Lattice Equable Parallelograms, arXiv:2006.07566 [math.NT], 2020.
- Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
- Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 31, 56.
- Z. Franusic, On the Extension of the Diophantine Pair {1,3} in Z[surd d], J. Int. Seq. 13 (2010) # 10.9.6.
- Giovanni Lucca, Circle Chains Inscribed in Symmetrical Lenses and Integer Sequences, Forum Geometricorum, Volume 16 (2016) 419-427.
- Index entries for linear recurrences with constant coefficients, signature (15,-15,1).
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I:=[1,3,33]; [n le 3 select I[n] else 15*Self(n-1)-15*Self(n-2)+Self(n-3): n in [1..17]]; // Bruno Berselli, Jul 09 2011
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a:= gfun:-rectoproc({a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3), a(0)=1,a(1)=3,a(2)=33},a(n),remember):
map(a,[$0..100]); # Robert Israel, Jul 02 2015
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RecurrenceTable[{a[n] == 15 a[n - 1] - 15 a[n - 2] + a[n - 3], a[0] == 1, a[1] == 3, a[2] == 33}, a, {n, 0, 15}] (* Michael De Vlieger, Jul 02 2015 *)
LinearRecurrence[{15,-15,1},{1,3,33},30] (* Harvey P. Dale, Dec 04 2018 *)
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a[0]:1$ a[1]:3$ a[2]:33$ a[n]:=15*a[n-1]-15*a[n-2]+a[n-3]$ makelist(a[n], n, 0, 16); /* Bruno Berselli, Jul 09 2011 */
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a(n)=([0,1,0; 0,0,1; 1,-15,15]^n*[1;3;33])[1,1] \\ Charles R Greathouse IV, Jul 02 2015
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