A048911 -id:A048911 - cp's OEIS frontend

A036411 9-gonal square numbers.

1, 9, 1089, 8281, 978121, 7436529, 878351769, 6677994961, 788758910641, 5996832038649, 708304623404049, 5385148492712041, 636056763057925561, 4835857349623374369, 571178264921393749929, 4342594514813297471521

Offset: 1

Jean-Francois Chariot (jeanfrancois.chariot(AT)afoc.alcatel.fr)

From Ant King, Nov 17 2011: (Start)
lim_{n -> infinity} a(2n+1)/a(2n) = 1/625 * (36913 + 9864 * sqrt(14));
lim_{n -> infinity} a(2n)/a(2n-1) = 1/625 * (2417 + 624 * sqrt(14)).
(End)

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Nonagonal Square Number.
Index entries for linear recurrences with constant coefficients, signature (1,898,-898,-1,1)

Cf. A001106, A048911, A048919.

I:=[1, 9, 1089, 8281]; [n le 4 select I[n] else 898*Self(n-2)-Self(n-4)+200: n in [1..20]]; // Vincenzo Librandi, Nov 18 2011

a(0):=1:a(1):=9:a(2):=1089:a(3):=8281: a(4):=978121:for n from 0 to 20 do a(n+5):=a(n+4)+898*a(n+3)-898*a(n+2)-a(n+1)+a(n):od:seq(a(n),n=0..20); # Richard Choulet, May 08 2009

LinearRecurrence[ {1, 898, - 898, - 1, 1 }, { 1, 9, 1089, 8281, 978121 }, 16] (* Ant King, Nov 17 2011 *)

O.g.f. f(z) = 1 + 9*z + ... = ((1 + 8*z + 182*z^2 + 8*z^3 + z^4)/((1-z)*(1 - 898*z^2 + z^4))). With the first values, for n > 0: a(n+5) = a(n+4) + 898*a(n+3) - 898*a(n+2) - a(n+1) + a(n). On every bisection modulo 2: a(n+2) = 30*a(n+1) - a(n) + 200. On every bisection modulo 2: a(n+1) = 449*a(n) + 100 + 60*sqrt(56*a(n)^2 + 25*a(n)). a(n) = -25/112 + (11/28 + (11/112)*sqrt(14))*(15 + 4*sqrt(14))^n + (11/28 - (11/112)*sqrt(14))*(15 - 4*sqrt(14))^n + (7/32 - (1/16)*sqrt(14))*(-15 + 4*sqrt(14))^n + (7/32 + (1/16)*sqrt(14))*(-15 - 4*sqrt(14))^n. - Richard Choulet, May 08 2009

a(n) = 898 * a(n-2) - a(n-4) + 200. - Ant King, Nov 17 2011

More terms from Eric W. Weisstein

More terms from Richard Choulet, May 08 2009

A048910 Indices of 9-gonal numbers that are also square.

Original entry on oeis.org

1, 2, 18, 49, 529, 1458, 15842, 43681, 474721, 1308962, 14225778, 39225169, 426298609, 1175446098, 12774732482, 35224157761, 382815675841, 1055549286722, 11471695542738, 31631254443889, 343768050606289, 947882084029938, 10301569822645922, 28404831266454241

Offset: 1

Eric W. Weisstein

From Ant King, Nov 18 2011: (Start)
lim( n -> Infinity, a(2n+1)/a(2n)) = 1/25 * (137 + 36 * sqrt(14)) = 1/25 * (9 + 2 * sqrt(14))^2.
lim( n -> Infinity, a(2n)/a(2n-1)) = 1/25 * (39 + 8 * sqrt(14)).
(14 * a(n) - 5)^2 - 56 *  A048911(n) ^ 2 = 25.
(End)

Colin Barker, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Nonagonal Square Number
Index entries for linear recurrences with constant coefficients, signature (1,30,-30,-1,1).

Cf. A048911, A036411.

LinearRecurrence[ {1, 30, - 30, -1, 1 }, {1, 2, 18, 49, 529}, 21 ] (* Ant King, Nov 18 2011 *)

Vec(-x*(x^4+x^3-14*x^2+x+1)/((x-1)*(x^4-30*x^2+1)) + O(x^50)) \\ Colin Barker, Jun 22 2015

From Ant King, Nov 18 2011: (Start)
a(n) = 30 * a(n - 2) - a(n-4) - 10.
a(n) = a(n - 1) + 30 * a(n - 2) - 30 * a(n - 3) - a(n - 4) + a(n - 5).
Let p = 9 + 4 * sqrt(2) + sqrt(7) + 2 * sqrt(14) and q = 9 - 4 * sqrt(2) - sqrt(7) + 2 * sqrt(14). Then
a(n) = 1/56 * ( ( p - q * (-1) ^ n) * ( 2 * sqrt(2) + sqrt(7))^(n - 1) + ( p + q * (-1)^n) * ( 2 * sqrt(2) - sqrt(7))^n + 20 ).
a(n) = ceiling (1/56 * ( p - q * (-1) ^ n) * ( 2 * sqrt(2) + sqrt(7))^(n - 1) ).
G.f.: x * (1 + x - 14 * x^2 + x^3 + x^4) / ((1 - x) * (1 - 30 * x^2 + x^4)).
(End)

cp's OEIS Frontend

A036411 9-gonal square numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

Extensions