A280181 Indices of centered 9-gonal numbers (A060544) that are also squares (A000290).

1, 17, 561, 19041, 646817, 21972721, 746425681, 25356500417, 861374588481, 29261379507921, 994025528680817, 33767606595639841, 1147104598723073761, 38967788749988868017, 1323757712900898438801, 44968794449880558051201, 1527615253583038075302017

Offset: 1

Author

Colin Barker, Dec 28 2016

Keywords

• nonn
• easy

Comments

Also positive integers y in the solutions to 2*x^2 - 9*y^2 + 9*y - 2 = 0, the corresponding values of x being A046176.

Consider all ordered triples of consecutive integers (k, k+1, k+2) such that k is a square and k+1 is twice a square; then the values of k are the squares of the NSW numbers (A002315), the values of k+1 are twice the squares of the odd Pell numbers (A001653), and the values of k+2 are thrice the terms of this sequence. (See the Example section.) - Jon E. Schoenfield, Sep 06 2019

Examples

			17 is in the sequence because the 17th centered 9-gonal number is 1225, which is also the 35th square.
From _Jon E. Schoenfield_, Sep 06 2019: (Start)
The following table illustrates the relationship between the NSW numbers (A002315), the odd Pell numbers (A001653), and the terms of this sequence:
.
  |  A002315(n-1)^2  |   2*A001653(n)^2  |
n |   = 3*a(n) - 2   |    = 3*a(n) - 1   |       3*a(n)
--+------------------+-------------------+-------------------
1 |    1^2 =       1 |   1^2*2 =       2 |      1*3 =       3
2 |    7^2 =      49 |   5^2*2 =      50 |     17*3 =      51
3 |   41^2 =    1681 |  29^2*2 =    1682 |    561*3 =    1683
4 |  239^2 =   57121 | 169^2*2 =   57122 |  19041*3 =   57123
5 | 1393^2 = 1940449 | 985^2*2 = 1940450 | 646817*3 = 1940451
(End)
		

Links

• Colin Barker, Table of n, a(n) for n = 1..650
• Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Crossrefs

Cf. A000290, A001653, A002315, A046176, A046177, A056771, A060544, A077420.

Cf. A156164.

Programs

Mathematica

LinearRecurrence[{35, -35, 1}, {1, 17, 561}, 50] (* G. C. Greubel, Dec 28 2016 *)

PARI

Vec(x*(1 - 18*x + x^2) / ((1 - x)*(1 - 34*x + x^2)) + O(x^20))

Formula

a(n) = (6 + (3-2*sqrt(2))*(17+12*sqrt(2))^(-n) + (3+2*sqrt(2))*(17+12*sqrt(2))^n) / 12.

a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3) for n>3.

G.f.: x*(1 - 18*x + x^2) / ((1 - x)*(1 - 34*x + x^2)).

a(n) = (A002315(n-1)^2 + 2)/3 = (2*A001653(n)^2 + 1)/3. - Jon E. Schoenfield, Sep 06 2019

a(n) = A077420(floor((n-1)/2)) * A056771(floor(n/2)). - Jon E. Schoenfield, Sep 08 2019

E.g.f.: -1+(1/12)*(6*exp(x)+(3-2*sqrt(2))*exp((17-12*sqrt(2))*x)+(3+2*sqrt(2))*exp((17+12*sqrt(2))*x)). - Stefano Spezia, Sep 08 2019

Limit_{n->oo} a(n+1)/a(n) = 17 + 12*sqrt(2) = A156164. - Andrea Pinos, Oct 07 2022