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A133142
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Numbers which are both centered square and decagonal numbers.
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1
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1, 1201, 1731661, 2497053781, 3600749820361, 5192278743906601, 7487262347963498101, 10796627113484620354861, 15568728810382474588211281, 22450096147944414871580312161, 32373023076607035862344221924701, 46681876826371197769085496435106501
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OFFSET
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1,2
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COMMENTS
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We solve r^2+(r+1)^2=5*p^2-5*p+1 equivalent to 2*(2*r+1)^2=5*(2*p-1)^2-3. the Diophantine equation (2*X)^2=10*Y^2-6 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)^2-216)^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)^2-360*a(n)+36)^0.5 (new sequence it seems).
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LINKS
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Colin Barker, Table of n, a(n) for n = 1..317
Index entries for linear recurrences with constant coefficients, signature (1443,-1443,1).
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FORMULA
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a(n+2) = 1442*a(n+1)-a(n)-180.
a(n+1) = 721*a(n)-90+38*(360*a(n)^2-90*a(n)-45)^0.5.
G.f.: -x*(61*x^2-242*x+1) / ((x-1)*(x^2-1442*x+1)). - corrected by Colin Barker, Jan 02 2015
a(n) = (1/8)+(7/16)*[721-228*sqrt(10)]^n-(1/8)*[721-228*sqrt(10)]^n*sqrt(10)+(1/8)*[721+228 *sqrt(10)]^n*sqrt(10)+(7/16)*[721+228*sqrt(10)]^n, with n>=0. - Paolo P. Lava, Sep 26 2008
a(n) = 1443*a(n-1)-1443*a(n-2)+a(n-3).
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MATHEMATICA
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LinearRecurrence[{1443, -1443, 1}, {1, 1201, 1731661}, 20] (* Harvey P. Dale, Feb 13 2022 *)
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PROG
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(PARI) Vec(-x*(61*x^2-242*x+1)/((x-1)*(x^2-1442*x+1)) + O(x^100)) \\ Colin Barker, Jan 02 2015
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CROSSREFS
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Sequence in context: A156620 A214116 A221451 * A068534 A252340 A230838
Adjacent sequences: A133139 A133140 A133141 * A133143 A133144 A133145
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KEYWORD
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nonn,easy
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AUTHOR
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Richard Choulet, Sep 21 2007, corrected Sep 29 2007
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EXTENSIONS
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More terms from Paolo P. Lava, Sep 26 2008
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STATUS
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approved
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