A276175 -id:A276175 - cp's OEIS frontend

A276123 a(0) = a(1) = a(2) = 1; for n > 2, a(n) = (a(n-1) + 1)*(a(n-2) + 1) / a(n-3).

1, 1, 1, 4, 10, 55, 154, 868, 2449, 13825, 39025, 220324, 621946, 3511351, 9912106, 55961284, 157971745, 891869185, 2517635809, 14213945668, 40124201194, 226531261495, 639469583290, 3610286238244, 10191389131441, 57538048550401, 162422756519761

Offset: 0

Bruno Langlois, Aug 21 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Matthew Christopher Russell, Using experimental mathematics to conjecture and prove theorems in the theory of partitions and commutative and non-commutative recurrences", PhD Dissertation, Mathematics Department, Rutgers University, May 2016.
Index entries for linear recurrences with constant coefficients, signature (0,17,0,-17,0,1).

Cf. A072881, A076839, A276175.

I:=[1,1,1,4,10,55]; [n le 6 select I[n] else 17*Self(n-2)-17*Self(n-4)+Self(n-6): n in [1..30]]; // Vincenzo Librandi, Aug 27 2016

LinearRecurrence[{0, 17, 0, -17, 0, 1}, {1, 1, 1, 4, 10, 55}, 40] (* Vincenzo Librandi, Aug 27 2016 *)
nxt[{a_,b_,c_}]:={b,c,((c+1)(b+1))/a}; NestList[nxt,{1,1,1},30][[All,1]] (* Harvey P. Dale, Oct 01 2021 *)

Vec((1+x-16*x^2-13*x^3+10*x^4+4*x^5)/((1-x)*(1+x)*(1-16*x^2+x^4)) + O(x^30)) \\ Colin Barker, Aug 21 2016

a(n) = (9-3*(-1)^n)/2*a(n-1) - a(n-2) - 1.
From Colin Barker, Aug 21 2016: (Start)
a(n) = 17*a(n-2) - 17*a(n-4) + a(n-6) for n > 5.
G.f.: (1 + x - 16*x^2 - 13*x^3 + 10*x^4 + 4*x^5) / ((1-x)*(1+x)*(1 - 16*x^2 + x^4)). (End)
a(2n+1) = A073352(n). a(2n) = A048907(n). - R. J. Mathar, Jul 04 2024

More terms from Colin Barker, Aug 21 2016

A276453 a(n) = (a(n-1)+1)(a(n-2)+1)(a(n-3)+1)/a(n-4) with a(0) = a(1) = 1, a(2) = 2, a(3) = 6.

Original entry on oeis.org

1, 1, 2, 6, 42, 903, 136052, 881442036, 2581196224947732, 342795531574625708871288171, 5732512385084161208637718426682572229606557631, 5754497648510061274107897581706624823818534711463525598519384262130236399970112

Offset: 0

Seiichi Manyama, Sep 03 2016

nonn

From Antoine de Saint Germain, Dec 30 2024: (Start)
Sequence consists of integers, see Math StackExchange link.
Values of a unitary Y-frieze pattern associated to the linearly oriented quiver K4 (i.e., the quiver whose underlying graph is the complete graph on the vertices {1,2,3,4}, oriented such that i -> j whenever i < j). (End)

Seiichi Manyama, Table of n, a(n) for n = 0..15
Math StackExchange, Is A276175 integer-only?

Cf. A051786, A101879, A276175.

I:=[1, 1, 2, 6]; [n le 4 select I[n] else (Self(n-1)+1)*(Self(n-2)+1)*(Self(n-3)+1)/Self(n-4): n in [1..13]]; // Vincenzo Librandi, Dec 30 2024

RecurrenceTable[{a[n] == (a[n - 1] + 1) (a[n - 2] + 1) (a[n - 3] + 1)/a[n - 4], a[0] == 1, a[1] == 1, a[2] == 2, a[3] == 6}, a, {n, 0, 11}] (* Michael De Vlieger, Sep 03 2016 *)

def A276453(n)
  a = [1, 1, 2, 6]
  ary = [1]
  while ary.size < n + 1
    i = a[1..-1].inject(1){|s, i| s * (i + 1)}
    break if i % a[0] > 0
    a = *a[1..-1], i / a[0]
    ary << a[0]
  end
  ary
end

a(n) = A051786(n)*A051786(n+1)*A051786(n+2).

A276308 a(n) = (a(n-1)+1)*(a(n-3)+1)/a(n-4) for n > 3, a(0) = a(1) = a(2) = a(3) = 1.

Original entry on oeis.org

1, 1, 1, 1, 4, 10, 22, 115, 319, 736, 3886, 10816, 24991, 131989, 367405, 848947, 4483720, 12480934, 28839196, 152314471, 423984331, 979683706, 5174208274, 14402986300, 33280406797, 175770766825, 489277549849, 1130554147381, 5971031863756, 16621033708546

Offset: 0

Seiichi Manyama, Aug 29 2016

Colin Barker, Table of n, a(n) for n = 0..1000
S. Fomin and A. Zelevinsky, The Laurent Phenomenon, Advances in Applied Mathematics, 28 (2002), 119-144.
Matthew Christopher Russell, Using experimental mathematics to conjecture and prove theorems in the theory of partitions and commutative and non-commutative recurrences, PhD Dissertation, Mathematics Department, Rutgers University, May 2016. See Eq. (6.137).
Index entries for linear recurrences with constant coefficients, signature (0,0,35,0,0,-35,0,0,1).

Cf. A276123, A276175.

Vec((1+x+x^2-34*x^3-31*x^4-25*x^5+22*x^6+10*x^7+4*x^8)/((1-x)*(1+x+x^2)*(1-34*x^3+x^6)) + O(x^35)) \\ Colin Barker, Aug 29 2016

a276308(maxn) = {a=vector(maxn); a[1]=a[2]=a[3]=a[4]=1; for(n=5, maxn, a[n]=(a[n-1]+1)*(a[n-3]+1)/a[n-4]); a} \\ Colin Barker, Aug 30 2016

def A(m, n)
  a = Array.new(m, 1)
  ary = [1]
  while ary.size < n + 1
    i = (a[1] + 1) * (a[-1] + 1)
    break if i % a[0] > 0
    a = *a[1..-1], i / a[0]
    ary << a[0]
  end
  ary
end
def A276308(n)
  A(4, n)
end

From Colin Barker, Aug 29 2016: (Start)
a(n) = 35*a(n-3)-35*a(n-6)+a(n-9) for n>8.
G.f.: (1+x+x^2-34*x^3-31*x^4-25*x^5+22*x^6+10*x^7+4*x^8) / ((1-x)*(1+x+x^2)*(1-34*x^3+x^6)).
(End)

A362884 a(n) = (a(n-1)a(n-2)a(n-3)+64)/(4*a(n-4)) with a(0) = a(2) = a(3) = 2 and a(1) = 16.

Original entry on oeis.org

2, 16, 2, 2, 16, 2, 16, 72, 37, 5336, 222112, 152263946, 1219335473828432, 1932041718420459629645062, 403742785702569426305018937234491996105486, 1561663327784579146423924791055619905538560428937482474084426146608982032

Offset: 0

Max Alekseyev, May 07 2023

nonn

Conjecture 1: all terms are integer.

Conjecture 2: a(n) is never divisible by 64.

Cf. A276175.

For n >= 2, a(n) = 4 * A276175(n) * A276175(n-2) / (A276175(n-1) * (A276175(n-1) + 1)).

cp's OEIS Frontend

A276123 a(0) = a(1) = a(2) = 1; for n > 2, a(n) = (a(n-1) + 1)*(a(n-2) + 1) / a(n-3).

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A276453 a(n) = (a(n-1)+1)*(a(n-2)+1)*(a(n-3)+1)/a(n-4) with a(0) = a(1) = 1, a(2) = 2, a(3) = 6.

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Programs

Magma

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Formula

A276308 a(n) = (a(n-1)+1)*(a(n-3)+1)/a(n-4) for n > 3, a(0) = a(1) = a(2) = a(3) = 1.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Ruby

Formula

A362884 a(n) = (a(n-1)*a(n-2)*a(n-3)+64)/(4*a(n-4)) with a(0) = a(2) = a(3) = 2 and a(1) = 16.

Views

Author

Keywords

Comments

Crossrefs

Formula

A276453 a(n) = (a(n-1)+1)(a(n-2)+1)(a(n-3)+1)/a(n-4) with a(0) = a(1) = 1, a(2) = 2, a(3) = 6.

A362884 a(n) = (a(n-1)a(n-2)a(n-3)+64)/(4*a(n-4)) with a(0) = a(2) = a(3) = 2 and a(1) = 16.