A057031 -id:A057031 - cp's OEIS frontend

A090310 a(n) = 21*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 21.

2, 21, 443, 9324, 196247, 4130511, 86936978, 1829807049, 38512885007, 810600392196, 17061121121123, 359094143935779, 7558038143772482, 159077895163157901, 3348193836570088403, 70471148463135014364

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 25 2004

Lim_{n-> infinity} a(n)/a(n+1) = 0.0475115... = 2/(21+sqrt(445)) = (sqrt(445)-21)/2.
Lim_{n-> infinity} a(n+1)/a(n) = 21.0475115... = (21+sqrt(445))/2 = 2/(sqrt(445)-21).
a(2) = 443 divides a(14) = 3348193836570088403. Does this relate to the sequence being the (21,1)-weighted Fibonacci sequence with seed (2,21) and both 14 and 21 being multiples of 7? Primes in this sequence include: a(0) = 2, a(2) = 443, a(4) = 196247 Semiprimes in this sequence include: a(8) = 38512885007 = 97967 * 393121, a(14) = 3348193836570088403 = 443 * 7557999631083721. - Jonathan Vos Post, Feb 10 2005

			a(4) = 21*a(3) + a(2) = 21*9324 + 443 = ((21+sqrt(445))/2)^4 + ((21-sqrt(445))/2)^4 = 196246.9999949043 + 0.0000050956 = 196247.

G. C. Greubel, Table of n, a(n) for n = 0..500
Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for linear recurrences with constant coefficients, signature (21,1).

Cf. A014842, A057031.

Lucas polynomials Lucas(n,m): A000032 (m=1), A002203 (m=2), A006497 (m=3), A014448 (m=4), A087130 (m=5), A085447 (m=6), A086902 (m=7), A086594 (m=8), A087798 (m=9), A086927 (m=10), A001946 (m=11), A086928 (m=12), A088316 (m=13), A090300 (m=14), A090301 (m=15), A090305 (m=16), A090306 (m=17), A090307 (m=18), A090308 (m=19), A090309 (m=20), this sequence (m=21), A090313 (m=22), A090314 (m=23), A090316 (m=24), A330767 (m=25).

m:=21;; a:=[2,m];; for n in [3..20] do a[n]:=m*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Dec 30 2019

m:=21; I:=[2,m]; [n le 2 select I[n] else m*Self(n-1) +Self(n-2): n in [1..20]]; // G. C. Greubel, Dec 30 2019

seq(simplify(2*(-I)^n*ChebyshevT(n, 21*I/2)), n = 0..20); # G. C. Greubel, Dec 30 2019

LinearRecurrence[{21,1},{2,21},40] (* or *) CoefficientList[ Series[ (2-21x)/(1-21x-x^2),{x,0,40}],x]  (* Harvey P. Dale, Apr 24 2011 *)
LucasL[Range[20]-1,21] (* G. C. Greubel, Dec 30 2019 *)

vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 21*I/2) ) \\ G. C. Greubel, Dec 30 2019

[2*(-I)^n*chebyshev_T(n, 21*I/2) for n in (0..20)] # G. C. Greubel, Dec 30 2019

a(n) = 21*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 21.
a(n) = ((21+sqrt(445))/2)^n + ((21-sqrt(445))/2)^n.
(a(n))^2 = a(2n) - 2 if n=1, 3, 5... .
(a(n))^2 = a(2n) + 2 if n=2, 4, 6... .
G.f.: (2-21*x)/(1-21*x-x^2). - Philippe Deléham, Nov 02 2008
a(n) = Lucas(n, 21) = 2*(-i)^n * ChebyshevT(n, 21*i/2). - G. C. Greubel, Dec 30 2019

More terms from Ray Chandler, Feb 14 2004

A249991 Start with the natural numbers, reverse the order in each pair, skip one number, reverse the order in each triple, skip one number, and so on.

Original entry on oeis.org

2, 3, 5, 10, 12, 13, 21, 26, 28, 39, 41, 46, 54, 65, 67, 82, 84, 85, 109, 114, 122, 137, 139, 160, 178, 179, 181, 222, 230, 235, 269, 274, 276, 313, 331, 336, 370, 381, 383, 424, 426, 437, 471, 476, 536, 541, 549, 554, 618, 629, 647, 704, 706, 707, 761, 818

Offset: 1

Alex Ratushnyak, Nov 27 2014

nonn

Start with the natural numbers. Reverse the order of numbers in each pair. Skip one number. In the remainder (that is, "1, 4, 3, 6, 5, 8, 7, 10, 9, 12, 11,...") reverse the order in each triple. Skip one number. In the remainder (it starts with "4, 1, 8, 5, 6, 9, 10, 7") reverse the order in each tetrad. Skip one number. And so on.

Cf. A000960, A026239, A249990.

Partial sums of A057031.

TOP = 100000
a = list(range(TOP))
for step in range(2,TOP):
  numBlocks = (len(a)-1) // step
  if numBlocks==0:  break
  a = a[:(1+numBlocks*step)]
  for pos in range(1,len(a),step):
    a[pos:pos+step] = a[pos+step-1:pos-1:-1]
  print(a[1], end=', ')
  a[1:] = a[2:]

cp's OEIS Frontend

A090310 a(n) = 21*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 21.

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