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Search: id:A076139
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| A076139 |
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Triangular numbers that are one-third of another triangular number: T(m) such that 3T(m)=T(k) for some k. |
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+0
9
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| 0, 1, 15, 210, 2926, 40755, 567645, 7906276, 110120220, 1533776805, 21362755051, 297544793910, 4144264359690, 57722156241751, 803965923024825, 11197800766105800, 155965244802456376, 2172315626468283465
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Both triangular and generalized pentagonal numbers: intersection of A000217 and A001318. - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 29 2004
Partial sums of Chebyshev polynomials S(n,14).
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LINKS
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Index entries for sequences relate d to Chebyshev polynomials.
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FORMULA
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a(n)=(A061278(n))*(A061278(n)+1)/2 closed form: a(n)=(1/288)*(-24+(12-6*sqrt(3))*(7-4*sqrt(3))^n+(12+6*sqrt(3))*(7+4*sqrt(3))^n)
Recurrence: a(0)=0, a(1)=1, a(2)=15; a(n) = 15*(a(n-1)-a(n-2))+a(n-3) for n>=3. G.f.: x/(1-15*x+15*x^2-x^3). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002
a(n+1)= sum(S(k, 14), k=0..n), n>=0, with S(k, 14)=U(k, 7)=A007655(k+2).
a(n)= 14*a(n-1)-a(n-2)+1, n>=2, a(0)=0, a(1)=1.
a(n+1)= (S(n+1, 14)-S(n, 14) -1)/12, n>=0.
G.f.: x/(1-15*x+15*x^2-x^3) = x/((1-x)*(1-14*x+x^2)).
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EXAMPLE
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a(3)=210=T(20) and 3*210=630=T(35)
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PROGRAM
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(PARI) a(n)=if(n<1, 0, subst((-8+15*poltchebi(n)-poltchebi(n-1))/96, x, 7))
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CROSSREFS
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The m values are in A061278, the k values are in A001571
Cf. A076140.
Sequence in context: A019553 A112496 A000483 this_sequence A001880 A113362 A135519
Adjacent sequences: A076136 A076137 A076138 this_sequence A076140 A076141 A076142
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KEYWORD
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easy,nonn
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AUTHOR
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Bruce Corrigan (scentman(AT)myfamily.com), Oct 31 2002
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EXTENSIONS
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More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002
Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 31 2004
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