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A076708
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Numbers n such that triangular numbers T(n) and T(n+1) sum to another triangular number (that is also a perfect square).
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5
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0, 5, 34, 203, 1188, 6929, 40390, 235415, 1372104, 7997213, 46611178, 271669859, 1583407980, 9228778025, 53789260174, 313506783023, 1827251437968, 10650001844789, 62072759630770, 361786555939835, 2108646576008244, 12290092900109633, 71631910824649558
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OFFSET
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1,2
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COMMENTS
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From T(k)+T(k+1) = (k*(k+1)+(k+1)*(k+2))/2 = (k+1)^2 any two consecutive triangular numbers sum to a square, the above sequence gives the sums that are also triangular. The units digit cycles through 0, 5, 4, 3, 8, 9, 0, 5, ...
Let P(b,e) be the polynomial 1+4*b+4*b^2+4*e+4*e^2. It appears that sequences A076708 and A076049 are special cases of the sequence of integers b such that P(b,b+n) is a perfect square. A076708 and A076049 for example are respectively the sequences of b's such that P(b,b+1) and P(b,b+2) are perfect squares. In fact it appears to be true that the sequence of integers b such that P(b,b+n) is a perfect square has the property that t(b)+t(b+n) is a triangular number. I have not had time to prove this but I do have evidence produced by Mathematica to support the assertion. - Robert Phillips (bobanne(AT)bellsouth.net), Sep 04 2009; corrected Sep 08 2009
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LINKS
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FORMULA
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Recursion: a(n+2) = 6*a(n+1)-a(n)+4, with a(0)=0 and a(1)=5.
G.f.: (5*x^2-x^3)/((1-x)*(1-6*x+x^2)).
Closed form: a(n)= ( sqrt(2)*( (3+2*sqrt(2))^(n+1) - (3-2*sqrt(2))^(n+1) )-8 )/8.
Also, if the entries in A001109 are denoted by b(n) then a(n) = b(n+1)-1.
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EXAMPLE
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a(1) = (sqrt(2)*((3+2*sqrt(2))^2-(3-2*sqrt(2))^2)-8)/8 = (sqrt(2)*(9+12*sqrt(2)+8-9+12*sqrt(2)-8)-8)/8 = (sqrt(2)*24*sqrt(2)-8)/8 = (48-8)/8 = 40/8 = 5.
T(5) + T(6) = 15 + 21 = 36 = T(8).
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MATHEMATICA
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Table[((3 + 2 Sqrt[2])^n - (3 - 2 Sqrt[2])^n)/(4 Sqrt[2]) - 1, {n, 1, 20}] (* Zerinvary Lajos, Jul 14 2009 *)
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PROG
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(PARI) concat(0, Vec(x^2*(x-5)/((x-1)*(x^2-6*x+1)) + O(x^100))) \\ Colin Barker, May 15 2015
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Bruce Corrigan (scentman(AT)myfamily.com), Oct 26 2002
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STATUS
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approved
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