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A203135 Indices of hexagonal numbers that are also decagonal 2

%I #11 Aug 01 2015 10:00:37

%S 1,28,943,32026,1087933,36957688,1255473451,42649139638,1448815274233,

%T 49217070184276,1671931570991143,56796456343514578,

%U 1929407584108504501,65543061403345638448,2226534680129643202723,75636636063004523254126,2569419091462024147437553

%N Indices of hexagonal numbers that are also decagonal

%C As n increases, this sequence is approximately geometric with common ratio (1+sqrt(2))^4=17+12*sqrt(2).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (35, -35, 1).

%F G.f.: x(1-7*x-2*x^2) / ((1-x)*(1-34*x+x^2))

%F a(n) = 35*a(n-1)-35*a(n-2)+a(n-3)

%F a(n) = 34*a(n-1)-a(n-2)-8

%F a(n) = 1/16 *((3*sqrt(2)-2)*(1+sqrt(2))^(4*n-2)- (3*sqrt(2)+2)*(1-sqrt(2))^(4*n-2)+4)

%F a(n) = ceiling(1/16 *(3*sqrt(2)-2)*(1+sqrt(2))^(4*n-2))

%e The second hexagonal number which is also decagonal is A000384(28)=1540. Hence a(2) = 28.

%t LinearRecurrence[{35, -35, 1}, {1, 28, 943}, 17]

%Y Cf. A203134, A203136, A000384, A001107.

%K nonn,easy

%O 1,2

%A _Ant King_, Dec 30 2011

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Last modified July 18 21:02 EDT 2024. Contains 374388 sequences. (Running on oeis4.)