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A055793 Numbers k such that k and floor[k/3] are both squares; i.e., squares which remain squares when written in base 3 and last digit is removed.

%I A055793 #59 Jul 11 2025 16:18:22
%S A055793 0,1,4,49,676,9409,131044,1825201,25421764,354079489,4931691076,
%T A055793 68689595569,956722646884,13325427460801,185599261804324,
%U A055793 2585064237799729,36005300067391876,501489136705686529,6984842613812219524,97286307456665386801,1355023461779503195684,18873042157456379352769
%N A055793 Numbers k such that k and floor[k/3] are both squares; i.e., squares which remain squares when written in base 3 and last digit is removed.
%C A055793 Or, squares of the form 3k^2+1.
%C A055793 See A023110, A204503, A204512, A204517, A204519, A055812, A055808 and A055792 for the analog in other bases.
%H A055793 Vincenzo Librandi, <a href="/A055793/b055793.txt">Table of n, a(n) for n = 1..800</a>
%H A055793 Tom C. Brown and Peter J Shiue, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/33-4/brown.pdf">Squares of second-order linear recurrence sequences</a>, Fib. Quart., 33 (1994), 352-356.
%H A055793 Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, <a href="http://dx.doi.org/10.13140/RG.2.2.16511.73129">Lune and Lens Sequences</a>, ResearchGate preprint, 2024. See pp. 48, 56.
%H A055793 M. F. Hasler, <a href="/wiki/M. F. Hasler/Truncated_squares">Truncated squares</a>, OEIS wiki, Jan 16 2012
%H A055793 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 A055793 <a href="/index/Sq#sqtrunc">Index to sequences related to truncating digits of squares</a>.
%H A055793 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (15,-15,1).
%F A055793 a(n) = 3*A098301(n-2)+1. - _R. J. Mathar_, Jun 11 2009
%F A055793 a(n) = 14*a(n-1)-a(n-2)-6, with a(0)=1, a(1)=4. (See Brown and Shiue)
%F A055793 a(n) = (A001075(n-2))^2. - _Johannes Boot_ Dec 16 2011, corrected by _M. F. Hasler_, Jan 15 2012
%F A055793 G.f.: x*(1 - 11*x + 4*x^2)/((1 - x)*(1 - 14*x + x^2)). - _M. F. Hasler_, Jan 15 2012
%e A055793 a(3) = 49 because 49 = 7^2 = 1211 base 3 and 121 base 3 = 16 = 4^2.
%p A055793 A055793 := proc(n) coeftayl(x*(1-11*x+4*x^2)/((1-x)*(1-14*x+x^2)), x=0, n); end proc: seq(A055793(n), n=0..20); # _Wesley Ivan Hurt_, Sep 28 2014
%t A055793 CoefficientList[Series[x*(1 - 11*x + 4*x^2)/((1 - x)*(1 - 14*x + x^2)), {x, 0, 20}], x] (* _Wesley Ivan Hurt_, Sep 28 2014 *)
%t A055793 LinearRecurrence[{15,-15,1},{0,1,4,49},40] (* _Harvey P. Dale_, Jun 19 2021 *)
%o A055793 (PARI) sq3nsqplus1(n) = { for(x=1,n, y = 3*x*x+1; \ print1(y" ") if(issquare(y),print1(y" ")) ) }
%o A055793 (Magma) I:=[0, 1, 4]; [n le 3 select I[n] else 14*Self(n-1) - Self(n-2) - 6: n in [1..30]]; // _Vincenzo Librandi_, Jan 27 2013
%Y A055793 Cf. A001075, A023110, A098301.
%Y A055793 Cf. also A023110, A204503, A204512, A204517, A204519, A055812, A055808 and A055792 for the analog in other bases.
%K A055793 base,nonn,easy
%O A055793 1,3
%A A055793 _Henry Bottomley_, Jul 14 2000
%E A055793 More terms from _Cino Hilliard_, Mar 01 2003