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A201003 Triangular numbers, T(m), that are four-fifths of another triangular number: T(m) such that 5*T(m) = 4*T(k) for some k.

%I A201003 #40 Dec 02 2024 09:50:36
%S A201003 0,36,11628,3744216,1205625960,388207814940,125001710784756,
%T A201003 40250162664876528,12960427376379457296,4173217365031520372820,
%U A201003 1343763031112773180590780,432687522800947932629858376,139324038578874121533633806328,44861907734874666185897455779276
%N A201003 Triangular numbers, T(m), that are four-fifths of another triangular number: T(m) such that 5*T(m) = 4*T(k) for some k.
%C A201003 Also, numbers m such that 8*m+1 and 10*m+1 are squares. Example: 8*1205625960+1 = 98209^2 and 12056259601 = 109801^2. - _Bruno Berselli_, Mar 03 2016
%H A201003 Vincenzo Librandi, <a href="/A201003/b201003.txt">Table of n, a(n) for n = 0..200</a>
%H A201003 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (323,-323,1).
%F A201003 For n>1, a(n) = 322*a(n-1) - a(n-2) + 36. See A200993 for generalization.
%F A201003 G.f.: 36*x / ((1-x)*(x^2-322*x+1)). - _R. J. Mathar_, Aug 10 2014
%F A201003 From _Colin Barker_, Mar 02 2016: (Start)
%F A201003 a(n) = (-18+(9-4*sqrt(5))*(161+72*sqrt(5))^(-n)+(9+4*sqrt(5))*(161+72*sqrt(5))^n)/160.
%F A201003 a(n) = 323*a(n-1) - 323*a(n-2) + a(n-3) for n>2. (End)
%F A201003 a(n) = 36*A298271(n). - _Amiram Eldar_, Dec 01 2024
%e A201003 5*0 = 4*0;
%e A201003 5*36 = 4*45;
%e A201003 5*11628 = 4*14535;
%e A201003 5*3744216 = 4*4680270.
%t A201003 triNums = Table[(n^2 + n)/2, {n, 0, 4999}]; Select[triNums, MemberQ[triNums, (5/4)#] &] (* _Alonso del Arte_, Dec 20 2011 *)
%t A201003 CoefficientList[Series[-36 x/((x - 1) (x^2 - 322 x + 1)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 11 2014 *)
%t A201003 LinearRecurrence[{323,-323,1},{0,36,11628},20] (* _Harvey P. Dale_, Dec 21 2015 *)
%o A201003 (PARI) concat(0, Vec(36*x/((1-x)*(1-322*x+x^2)) + O(x^15))) \\ _Colin Barker_, Mar 02 2016
%o A201003 (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(36*x/((1-x)*(1-322*x+x^2)))); // _G. C. Greubel_, Jul 15 2018
%Y A201003 Cf. A001652, A029549, A053141, A075528, A200993-A201008, A298271.
%K A201003 nonn,easy
%O A201003 0,2
%A A201003 _Charlie Marion_, Dec 20 2011