Search: id:A076139
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%I A076139
%S A076139 0,1,15,210,2926,40755,567645,7906276,110120220,1533776805,21362755051,
%T A076139 297544793910,4144264359690,57722156241751,803965923024825,
%U A076139 11197800766105800,155965244802456376,2172315626468283465
%N A076139 Triangular numbers that are one-third of another triangular number: T(m) 
               such that 3T(m)=T(k) for some k.
%C A076139 Both triangular and generalized pentagonal numbers: intersection of A000217 
               and A001318. - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 29 2004
%C A076139 Partial sums of Chebyshev polynomials S(n,14).
%H A076139 <a href="Sindx_Ch.html#Cheby">Index entries for sequences relate d to 
               Chebyshev polynomials.</a>
%F A076139 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)
%F A076139 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
%F A076139 a(n+1)= sum(S(k, 14), k=0..n), n>=0, with S(k, 14)=U(k, 7)=A007655(k+2).
%F A076139 a(n)= 14*a(n-1)-a(n-2)+1, n>=2, a(0)=0, a(1)=1.
%F A076139 a(n+1)= (S(n+1, 14)-S(n, 14) -1)/12, n>=0.
%F A076139 G.f.: x/(1-15*x+15*x^2-x^3) = x/((1-x)*(1-14*x+x^2)).
%e A076139 a(3)=210=T(20) and 3*210=630=T(35)
%o A076139 (PARI) a(n)=if(n<1,0,subst((-8+15*poltchebi(n)-poltchebi(n-1))/96,x, 
               7))
%Y A076139 The m values are in A061278, the k values are in A001571
%Y A076139 Cf. A076140.
%Y A076139 Sequence in context: A112496 A000483 A162785 this_sequence A163091 A163440 
               A163962
%Y A076139 Adjacent sequences: A076136 A076137 A076138 this_sequence A076140 A076141 
               A076142
%K A076139 easy,nonn
%O A076139 0,3
%A A076139 Bruce Corrigan (scentman(AT)myfamily.com), Oct 31 2002
%E A076139 More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), 
               Nov 01 2002
%E A076139 Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Aug 31 2004

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