cp's OEIS Frontend

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

%I A280181 #47 Oct 23 2022 05:27:38
%S A280181 1,17,561,19041,646817,21972721,746425681,25356500417,861374588481,
%T A280181 29261379507921,994025528680817,33767606595639841,1147104598723073761,
%U A280181 38967788749988868017,1323757712900898438801,44968794449880558051201,1527615253583038075302017
%N A280181 Indices of centered 9-gonal numbers (A060544) that are also squares (A000290).
%C A280181 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.
%C A280181 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
%H A280181 Colin Barker, <a href="/A280181/b280181.txt">Table of n, a(n) for n = 1..650</a>
%H A280181 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (35,-35,1).
%F A280181 a(n) = (6 + (3-2*sqrt(2))*(17+12*sqrt(2))^(-n) + (3+2*sqrt(2))*(17+12*sqrt(2))^n) / 12.
%F A280181 a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3) for n>3.
%F A280181 G.f.: x*(1 - 18*x + x^2) / ((1 - x)*(1 - 34*x + x^2)).
%F A280181 a(n) = (A002315(n-1)^2 + 2)/3 = (2*A001653(n)^2 + 1)/3. - _Jon E. Schoenfield_, Sep 06 2019
%F A280181 a(n) = A077420(floor((n-1)/2)) * A056771(floor(n/2)). - _Jon E. Schoenfield_, Sep 08 2019
%F A280181 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
%F A280181 Limit_{n->oo} a(n+1)/a(n) = 17 + 12*sqrt(2) = A156164. - _Andrea Pinos_, Oct 07 2022
%e A280181 17 is in the sequence because the 17th centered 9-gonal number is 1225, which is also the 35th square.
%e A280181 From _Jon E. Schoenfield_, Sep 06 2019: (Start)
%e A280181 The following table illustrates the relationship between the NSW numbers (A002315), the odd Pell numbers (A001653), and the terms of this sequence:
%e A280181 .
%e A280181   |  A002315(n-1)^2  |   2*A001653(n)^2  |
%e A280181 n |   = 3*a(n) - 2   |    = 3*a(n) - 1   |       3*a(n)
%e A280181 --+------------------+-------------------+-------------------
%e A280181 1 |    1^2 =       1 |   1^2*2 =       2 |      1*3 =       3
%e A280181 2 |    7^2 =      49 |   5^2*2 =      50 |     17*3 =      51
%e A280181 3 |   41^2 =    1681 |  29^2*2 =    1682 |    561*3 =    1683
%e A280181 4 |  239^2 =   57121 | 169^2*2 =   57122 |  19041*3 =   57123
%e A280181 5 | 1393^2 = 1940449 | 985^2*2 = 1940450 | 646817*3 = 1940451
%e A280181 (End)
%t A280181 LinearRecurrence[{35, -35, 1}, {1, 17, 561}, 50] (* _G. C. Greubel_, Dec 28 2016 *)
%o A280181 (PARI) Vec(x*(1 - 18*x + x^2) / ((1 - x)*(1 - 34*x + x^2)) + O(x^20))
%Y A280181 Cf. A000290, A001653, A002315, A046176, A046177, A056771, A060544, A077420.
%Y A280181 Cf. A156164.
%K A280181 nonn,easy
%O A280181 1,2
%A A280181 _Colin Barker_, Dec 28 2016