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Second order recursive series having the property that the product of any two adjacent terms equals 4 times a triangular number. That is a(n)*a(n+1)= 4*T(c) = 2c(c+1), where c = the term a(n+1) of related series A108261.
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%I #35 Jul 03 2023 08:03:11

%S 3,8,39,220,1275,7424,43263,252148,1469619,8565560,49923735,290976844,

%T 1695937323,9884647088,57611945199,335787024100,1957110199395,

%U 11406874172264,66484134834183,387497934832828,2258503474162779,13163522910143840,76722633986700255

%N Second order recursive series having the property that the product of any two adjacent terms equals 4 times a triangular number. That is a(n)*a(n+1)= 4*T(c) = 2c(c+1), where c = the term a(n+1) of related series A108261.

%H K. J. Ramsey, <a href="http://groups.yahoo.com/group/Triangular_and_Fibonacci_Numbers/message/16">Recursive Series Problem</a> [Edited by Kenneth J. Ramsey, May 14 2011]

%H Kenneth J. Ramsey, <a href="/A108261/a108261.txt">Recursive Series Problem</a>, digest of 4 messages in Triangular_and_Fibonacci_Numbers Yahoo group, May 28, 2005 - Mar 9, 2006. [Cached copy]

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7, -7, 1).

%F a(n) = 6*a(n-1) - a(n-2) - 6.

%F G.f.: (-4*x^2+13*x-3)/(x^3-7*x^2+7*x-1). [_Harvey P. Dale_, Mar 15 2011]

%e a(4) = 1275 as 220*1275 = 280500 is a term of A046092.

%t Transpose[NestList[{Last[#],6Last[#]-First[#]-6}&, {3,8}, 20]][[1]] (* _Harvey P. Dale_, Mar 15 2011 *)

%Y Cf. A046092, A108261.

%K nonn

%O 0,1

%A _Kenneth J Ramsey_, May 29 2005

%E More terms from _Harvey P. Dale_, Mar 15 2011