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A076139
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Triangular numbers that are one-third of another triangular number: T(m) such that 3*T(m)=T(k) for some k.
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13
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0, 1, 15, 210, 2926, 40755, 567645, 7906276, 110120220, 1533776805, 21362755051, 297544793910, 4144264359690, 57722156241751, 803965923024825, 11197800766105800, 155965244802456376, 2172315626468283465
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OFFSET
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0,3
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COMMENTS
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Both triangular and generalized pentagonal numbers: intersection of A000217 and A001318. - Vladeta Jovovic, Aug 29 2004
Partial sums of Chebyshev polynomials S(n,14).
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LINKS
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Table of n, a(n) for n=0..17.
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
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a(n) = (A061278(n))*(A061278(n)+1)/2.
a(n) = (1/288)*(-24+(12-6*sqrt(3))*(7-4*sqrt(3))^n+(12+6*sqrt(3))*(7+4*sqrt(3))^n)
a(0)=0, a(1)=1, a(2)=15; a(n) = 15*a(n-1)-15*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+1)= (S(n+1, 14)-S(n, 14) -1)/12, n>=0.
a(n) = 14 * a(n-1) - a(n-2) + 1. a(0)=0, a(1)=1.
a(-n) = a(n-1).
G.f.: x / ((1 - x) * (1 - 14*x +x^2)).
a(2*n) = A108281(n + 1). a(2*n + 1) = A014979(n + 2). - Michael Somos Jun 16 2011
a(n) = 1/2*A217855(n) = 1/3*A076140(n) = 1/4*A123480(n) = 1/8*A045899(n). - Peter Bala, Dec 31 2012
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EXAMPLE
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x + 15*x^2 + 210*x^3 + 2926*x^4 + 40755*x^5 + 567645*x^6 + ...
a(3)=210=T(20) and 3*210=630=T(35)
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MATHEMATICA
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a[n_] := a[n] = 14*a[n-1] - a[n-2] + 1; a[0] = 0; a[1] = 1; Table[ a[n], {n, 0, 17}] (* From Jean-François Alcover, Dec 15 2011, after given formula *)
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PROG
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(PARI) {a(n) = polchebyshev( n, 2, 7) / 14 + polchebyshev( n, 1, 7)/ 84 - 1 / 12} /* Michael Somos Jun 16 2011 */
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CROSSREFS
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The m values are in A061278, the k values are in A001571
Cf. A014979, A076140, A108281. A045899, A123480, A217855.
Sequence in context: A112496 A000483 A162785 * A163091 A163440 A163962
Adjacent sequences: A076136 A076137 A076138 * A076140 A076141 A076142
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KEYWORD
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easy,nonn,changed
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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 Wolfdieter Lang, Aug 31 2004
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STATUS
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approved
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