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A182334
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Triangular numbers that differ from a square by 1.
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4
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0, 1, 3, 10, 15, 120, 325, 528, 4095, 11026, 17955, 139128, 374545, 609960, 4726275, 12723490, 20720703, 160554240, 432224101, 703893960, 5454117903, 14682895930, 23911673955, 185279454480, 498786237505, 812293020528, 6294047334435, 16944049179226
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OFFSET
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1,3
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COMMENTS
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Actually this sequence is the union of two subsequences; the triangular numbers that are less than a square by 1 and those that are greater than a square by 1.
The first sequence by index of the triangular numbers is A072221: b(n) = 6b(n-1) - b(n-2) + 2, with b(0)=1, b(1)=4.
And obviously the second sequence by index of the triangular numbers is A006451: c(n) = 6c(n-2) - c(n-4) + 2 with c(0)=0, c(1)=2, c(2)=5, c(3)=15.
(End)
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REFERENCES
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Edward J. Barbeau, Pell's Equation (Springer 2003) at 17.
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LINKS
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FORMULA
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G.f.: x^2*(1+3*x+10*x^2-20*x^3+15*x^4-25*x^5+38*x^6+x^8-x^9) / ((1-x)*(1+x+x^2)*(1-34*x^3+x^6)). - Colin Barker, Sep 17 2016
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EXAMPLE
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T(2) = 3 = 2^2 - 1, T(4) = 10 = 3^2 + 1, T(5) = 15 = 4^2 - 1, and T(15) = 120 = 11^2 - 1.
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MATHEMATICA
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lst = {}; Do[t = n*(n + 1)/2; If[IntegerQ[(t - 1)^(1/2)] || IntegerQ[(t + 1)^(1/2)], AppendTo[lst, t]], {n, 0, 10^4}]; lst (* Arkadiusz Wesolowski, Aug 06 2012 *)
b[n_] := b[n] = 6 b[n - 1] - b[n - 2] + 2; b[0] = 1; b[1] = 4; c[n_] := c[n] = 6 c[n - 2] - c[n - 4] + 2; c[0] = 0; c[1] = 2; c[2] = 5; c[3] = 15; #(# + 1)/2 & /@ Union@ Join[ Array[b, 9, 0], Array[c, 18, 0]] (* or *)
#(# + 1)/2 & /@ Join[{0, 1}, LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {2, 4, 5, 15, 25, 32, 90}, 35]] (* or *)
#(# + 1)/2 & /@ CoefficientList[ Series[x + x^2 (1 + x) (2 + x^2 - 3 x^3 + x^4)/((1 - x) (1 - 6 x^3 + x^6)), {x, 0, 36}], x] (* Robert G. Wilson v, Jun 20 2015 *)
a[n_] := a[n] = 35 a[n - 3] - 35 a[n - 6] + a[n - 9]; a[1] = 0; a[2] = 1; a[3] = 3; a[4] = 10; a[5] = 15; a[6] = 120; a[7] = 325; a[8] = 528; a[9] = 4095; a[10] = 11026; a[11] = 17955; Array[a, 36] (* Robert G. Wilson v after Charles R Greathouse IV, Apr 25 2012 *)
Select[Accumulate[Range[0, 6*10^6]], AnyTrue[Sqrt[#+{1, -1}], IntegerQ]&] (* or *) LinearRecurrence[{0, 0, 35, 0, 0, -35, 0, 0, 1}, {0, 1, 3, 10, 15, 120, 325, 528, 4095, 11026, 17955}, 40] (* The first program uses the AnyTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 24 2015 *)
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PROG
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(Magma) I:=[0, 1, 3, 10, 15, 120, 325, 528, 4095, 11026, 17955]; [n le 11 select I[n] else 35*Self(n-3)-35*Self(n-6)+Self(n-9): n in [1..30]]; // Vincenzo Librandi, Jun 21 2015
(PARI) concat(0, Vec(x^2*(1+3*x+10*x^2-20*x^3+15*x^4-25*x^5+38*x^6+x^8-x^9)/((1-x)*(1+x+x^2)*(1-34*x^3+x^6)) + O(x^30))) \\ Colin Barker, Sep 17 2016
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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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STATUS
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approved
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