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A055792 a(n) and floor(a(n)/2) are both squares; i.e., squares which remain squares when written in base 2 and last digit is removed.

%I A055792 #48 Jul 22 2024 05:32:32
%S A055792 0,1,9,289,9801,332929,11309769,384199201,13051463049,443365544449,
%T A055792 15061377048201,511643454094369,17380816062160329,590436102659356801,
%U A055792 20057446674355970889,681362750825443653409,23146276081390728245001,786292024016459316676609
%N A055792 a(n) and floor(a(n)/2) are both squares; i.e., squares which remain squares when written in base 2 and last digit is removed.
%C A055792 a(n) > 0 is a square such that a(n) - 1 is a product of powers. - _Michel Lagneau_, Feb 16 2012
%H A055792 Charles R Greathouse IV, <a href="/A055792/b055792.txt">Table of n, a(n) for n = 0..654</a>
%H A055792 M. F. Hasler, <a href="/wiki/M._F._Hasler/Truncated_squares">Truncated squares</a>, OEIS wiki, Jan 16 2012
%H A055792 Giovanni Lucca, <a href="http://forumgeom.fau.edu/FG2018volume18/FG201808index.html">Integer Sequences and Circle Chains Inside a Circular Segment</a>, Forum Geometricorum, Vol. 18 (2018), 47-55.
%H A055792 Giovanni Lucca, <a href="https://ijgeometry.com/product/giovanni-lucca-circle-chains-inside-the-arbelos-and-integer-sequences/">Circle chains inside the arbelos and integer sequences</a>, Int'l J. Geom. (2023) Vol. 12, No. 1, 71-82.
%H A055792 <a href="/index/Sq#sqtrunc">Index to sequences related to truncating digits of squares</a>.
%H A055792 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (35,-35,1).
%F A055792 a(n) = 34*a(n-1) - a(n-2) - 16 = A001541(n-1)^2 = 2*A001542(n-1)^2 + 1 = 8*A001110(n-1) + 1.
%F A055792 From _Colin Barker_, Sep 15 2014: (Start)
%F A055792 a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3) for n > 3.
%F A055792 G.f.: -x*(9*x^2 - 26*x + 1) / ((x-1)*(x^2 - 34*x + 1)). (End)
%F A055792 a(n) = c*k^n + 1/2 + o(1) with k = 17+sqrt(288) = 33.97... and c = 17/4 - sqrt(18). - _Charles R Greathouse IV_, May 07 2015
%F A055792 a(n) = (4 + 2*(17 + 12*sqrt(2))^(1-n) + (34 - 24*sqrt(2))*(17 + 12*sqrt(2))^n)/8 for n > 0. - _Colin Barker_, Mar 02 2016
%e A055792 a(2) = 9 because 9 = 3^2 = 1001_2 and 100_2 = 4 = 2^2.
%t A055792 LinearRecurrence[{35, -35, 1}, {0, 1, 9, 289}, 25] (* _Paolo Xausa_, Jul 22 2024 *)
%o A055792 (PARI) concat(0, Vec(-x*(9*x^2-26*x+1)/((x-1)*(x^2-34*x+1)) + O(x^100))) \\ _Colin Barker_, Sep 15 2014
%o A055792 (PARI) is(n)=issquare(n) && issquare(n\2) \\ _Charles R Greathouse IV_, May 07 2015
%Y A055792 Cf. A023110, A247375.
%K A055792 nonn,base,easy
%O A055792 0,3
%A A055792 _Henry Bottomley_, Jul 14 2000