

A133142


Numbers which are both centered square and decagonal numbers.


1



1, 1201, 1731661, 2497053781, 3600749820361, 5192278743906601, 7487262347963498101, 10796627113484620354861, 15568728810382474588211281, 22450096147944414871580312161, 32373023076607035862344221924701, 46681876826371197769085496435106501
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OFFSET

1,2


COMMENTS

We solve r^2+(r+1)^2=5*p^25*p+1 equivalent to 2*(2*r+1)^2=5*(2*p1)^23. the Diophantine equation (2*X)^2=10*Y^26 is such that
X is given by 1, 49,1861,70669,... with a(n+2) = 38*a(n+1)a(n) and also a(n+1) = 19*a(n)+(360*a(n)^2+540)^0.5
Y is given by 1, 31,1177,44695,... with a(n+2) = 38*a(n+1)a(n) and also a(n+1) = 19*a(n)+(360*a(n)^2216)^0.5
r is given by 0, 24,930,35334,... with a(n+2) = 38*a(n+1)a(n)+18 and also a(n+1) = 19*a(n)+9+(360*a(n)^2+360*a(n)+225)^0.5 (new sequence it seems)
p is given by 1, 16,589, 22345,... with a(n+2) = 38*a(n+1)a(n)18 and also a(n+1) = 19*a(n)9+(360*a(n)^2360*a(n)+36)^0.5 (new sequence it seems).


LINKS



FORMULA

a(n+2) = 1442*a(n+1)a(n)180.
a(n+1) = 721*a(n)90+38*(360*a(n)^290*a(n)45)^0.5.
G.f.: x*(61*x^2242*x+1) / ((x1)*(x^21442*x+1)).  corrected by Colin Barker, Jan 02 2015
a(n) = 1443*a(n1)1443*a(n2)+a(n3).


MATHEMATICA

LinearRecurrence[{1443, 1443, 1}, {1, 1201, 1731661}, 20] (* Harvey P. Dale, Feb 13 2022 *)


PROG

(PARI) Vec(x*(61*x^2242*x+1)/((x1)*(x^21442*x+1)) + O(x^100)) \\ Colin Barker, Jan 02 2015


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS



STATUS

approved



