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A226500
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Triangular numbers representable as 3 * x^2.
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1
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0, 3, 300, 29403, 2881200, 282328203, 27665282700, 2710915376403, 265642041604800, 26030209161894003, 2550694855824007500, 249942065661590841003, 24491771739980078410800, 2399943688452386093417403, 235169989696593857076494700, 23044259046577745607403063203
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = 99*a(n-1) - 99*a(n-2) + a(n-3), for n > 3. a(n) = floor((49 + 20*sqrt(6))^(n-1)/32). - Giovanni Resta, Jun 09 2013
a(n) = (49+20*sqrt(6))^(-n)*(49+20*sqrt(6)-2*(49+20*sqrt(6))^n+(49-20*sqrt(6))*(49+20*sqrt(6))^(2*n))/32. - Colin Barker, Mar 03 2016
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MATHEMATICA
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a[1]=0; a[2]=3; a[3]=300; a[n_] := a[n] = 99*(a[n-1] - a[n-2]) + a[n-3]; Array[a, 10] (* Giovanni Resta, Jun 09 2013 *)
Rest@ CoefficientList[Series[3 x^2 (1 + x)/((1 - x) (1 - 98 x + x^2)), {x, 0, 16}], x] (* or *)
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PROG
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(C)
#include <stdio.h>
#include <math.h>
typedef unsigned long long U64;
U64 isTriangular(U64 a) { // input must be < 1ULL<<63
U64 r = sqrt(a*2);
return (r*(r+1) == a*2);
}
int main() {
for (U64 j, i = 0; (j=i*i*3) < (1ULL<<63); i++)
if (isTriangular(j)) printf("%llu, ", j);
return 0;
}
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CROSSREFS
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Cf. A029549 (triangular numbers representable as x^2 + x).
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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