cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A060656 a(n) = 2*a(n-1)*a(n-2)/a(n-3), with a(0)=a(1)=1.

Original entry on oeis.org

1, 1, 2, 4, 16, 64, 512, 4096, 65536, 1048576, 33554432, 1073741824, 68719476736, 4398046511104, 562949953421312, 72057594037927936, 18446744073709551616, 4722366482869645213696, 2417851639229258349412352
Offset: 0

Views

Author

Henry Bottomley, Apr 18 2001

Keywords

Comments

a(n+1) is the Hankel transform of A135052. - Paul Barry, Nov 15 2007
a(n+1) is the Hankel transform of the aerated large Schroeder numbers. a(n) and a(n+1) both satisfy the trivial Somos-4 recurrence u(n)=4*u(n-2)^2/u(n-4). Associated with the elliptic curve y^2=1-6x^2+x^4 via Schroeder numbers. - Paul Barry, Dec 08 2009
Hankel transform of A089324. - Paul Barry, Mar 01 2010
a(n+1) is the number of n X n binary matrices that are symmetric about both diagonals (bisymmetric). For the derivation of this result, see the link below. - Dennis P. Walsh, Apr 03 2014
1 followed by {a(n-1)}A078495).%20-%20_Vladimir%20Shevelev">(n>=1) is the Somos-3 sequence: b(0)=b(1)=b(2)=1;for n>=3, b(n)=2*b(n-1)*b(n-2)/b(n-3) (cf. comment in A078495). - _Vladimir Shevelev, Apr 20 2016
If the Hankel transform is defined as in the link 'Sequence transformations' then a(n) is the Hankel transform of A151374. - Peter Luschny, Nov 30 2016

Examples

			a(6) = 2*64*16/4 = 512.
G.f. = 1 + x + 2*x^2 + 4*x^3 + 16*x^4 + 64*x^5 + 512*x^6 + 4096*x^7 + ...

Links

Crossrefs

Cf. A002416, A002620, A016116, A038754, A089324, A135052, A262666, A078495, A151374.

Programs

  • Maple
    A060656:=n->2^floor(n^2/4); seq(A060656(n), n=0..20); # Wesley Ivan Hurt, Apr 30 2014
  • Mathematica
    a[ n_] := 2^Quotient[n^2, 4]; (* Michael Somos, Jan 24 2014 *)
nxt[{a_,b_,c_}]:={b,c,(2c*b)/a}; NestList[nxt,{1,1,2},20][[All,1]] (* Harvey P. Dale, Nov 26 2017 *)
  • PARI
    { for (n=0, 100, write("b060656.txt", n, " ", 2^(n^2\4)); ) } \\ Harry J. Smith, Jul 09 2009
  • PARI
    {a(n) = 2^(n^2\4)}; /* Michael Somos, Jan 24 2014 */

Formula

a(n) = 2^floor( n^2/4 ) = a(n - 1) * 2^floor( n/2 ) = a(n - 2) * 2^(n - 1) = a(n - 1) * A016116(n) = 2^A002620(n).
0 = a(n) * a(n+3) + a(n+1) * ( -2*a(n+2) ) for all n in Z. - Michael Somos, Jan 24 2014
0 = a(n) * a(n+4) + a(n+2) * ( -4*a(n+2) ) for all n in Z. - Michael Somos, Jan 24 2014

A256938 Expansion of g.f.: (1-3*z-sqrt(1-6*z+5*z^2+8*z^3-4*z^4))/(2*z^2*(1-z)).

Original entry on oeis.org

1, 2, 7, 24, 86, 316, 1189, 4562, 17796, 70398, 281812, 1139658, 4649402, 19112962, 79096155, 329258424, 1377798890, 5792421108, 24454224310, 103631241912, 440674939192, 1879769835968, 8041447249926, 34490981798188, 148295899087660, 639036278210420
Offset: 0

Views

Author

José Luis Ramírez Ramírez, Apr 19 2015

Keywords

Comments

a(n) = number of lattice paths, never going below the x-axis, from (0,0) to (n,0) consisting of up steps U = (1,1), down steps D = (1,-1) and 2-colored horizontal steps H(k) = (k,0) for every positive integer k.

Links

Crossrefs

Cf. A135052.

Programs

  • Mathematica
    CoefficientList[Series[(1-3*x-Sqrt[1-6*x+5*x^2+8*x^3-4*x^4])/(2*x^2*(1-x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 20 2015 *)
  • PARI
    x='x+O('x^50); Vec((1-3*x-sqrt(1 -6*x +5*x^2 +8*x^3 -4*x^4) )/(2*x^2*(1-x))) \\ G. C. Greubel, Jun 03 2017

Formula

a(s) = Sum_{n=0..s} (Sum_{m=0..(s-2*n)} (C(n)*binomial(m+2*n,m) *binomial(s-2*n-1,m-1) * 2^m)), where C(n) = A000108(n).
G.f.: (1-3*z-sqrt(1-6*z+5*z^2+8*z^3-4*z^4))/(2*z^2*(1-z)).
a(n) ~ sqrt(221 + 53*sqrt(17)) * (5+sqrt(17))^n / (sqrt(Pi) * n^(3/2) * 2^(n+2)). - Vaclav Kotesovec, Apr 20 2015
Recurrence: (n+2)*a(n) = (7*n+5)*a(n-1) - (11*n-2)*a(n-2) - 3*(n-5)*a(n-3) + 12*(n-3)*a(n-4) - 4*(n-4)*a(n-5). - Vaclav Kotesovec, Apr 20 2015

A256939 Expansion of g.f.: (1-4*z-sqrt(1-8*z+12*z^2+8*z^3-4*z^4))/(2*z^2(1-z)).

Original entry on oeis.org

1, 3, 13, 57, 257, 1185, 5573, 26661, 129437, 636429, 3163725, 15877101, 80340813, 409495053, 2100558429, 10836262173, 56184433661, 292628726205, 1530338756093, 8032671187581, 42304703640701, 223484135199357, 1183921500416509, 6288098247289341
Offset: 0

Views

Author

José Luis Ramírez Ramírez, Apr 19 2015

Keywords

Comments

Number of lattice paths, never going below the x-axis, from (0,0) to (n,0) consisting of up steps U = (1,1), down steps D = (1,-1) and 3-colored horizontal steps H(k) = (k,0) for every positive integer k.

Links

Crossrefs

Cf. A135052.

Programs

  • Mathematica
    CoefficientList[Series[(1-4*x-Sqrt[1-8*x+12*x^2+8*x^3-4*x^4])/(2*x^2*(1-x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 20 2015 *)

Formula

a(s) = Sum_{n=0..s} (Sum_{m=0..s-2n} (C(n)binomial(m+2n,m)*binomial(s-2n-1,m-1)3^m)), where C(n)=A000108(n).
G.f.: (1-4z-sqrt(1-8z+12z^2+8z^3-4z^4))/(2z^2(1-z)).
a(n) ~ sqrt(77 + 29*sqrt(7)) * (3+sqrt(7))^n / (sqrt(3*Pi) * n^(3/2)). - Vaclav Kotesovec, Apr 20 2015
Recurrence: (n+2)*a(n) = 3*(3*n+2)*a(n-1) - 4*(5*n-2)*a(n-2) + 4*(n+2)*a(n-3) + 12*(n-3)*a(n-4) - 4*(n-4)*a(n-5). - Vaclav Kotesovec, Apr 20 2015
Showing 1-3 of 3 results.

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