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