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A072261
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a(n) = 4*a(n-1) + 1, a(1)=7.
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8
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7, 29, 117, 469, 1877, 7509, 30037, 120149, 480597, 1922389, 7689557, 30758229, 123032917, 492131669, 1968526677, 7874106709, 31496426837, 125985707349, 503942829397, 2015771317589, 8063085270357, 32252341081429, 129009364325717, 516037457302869
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OFFSET
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1,1
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COMMENTS
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These are the integers N which on application of the Collatz function yield the number 11. The Collatz function: if N is an odd number then (3N+1)/2^r yields a positive odd integer for some value of r (which in this case is 11).
These numbers reach 11 in Collatz function iteration after 2(n+1) steps and so end in 1 after exactly 2n+18 steps. - Lambert Klasen (lambert.klasen(AT)gmx.de), Nov 08 2004
Numbers whose binary representation is 111 together with n - 1 times 01. For example, a(4) = 469 = 111010101 (2). - Omar E. Pol, Nov 24 2012
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LINKS
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FORMULA
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a(n) = (11*4^n - 2)/6 = 22*A002450(n-1) + 7. - Lambert Klasen (lambert.klasen(AT)gmx.de), Nov 08 2004
G.f.: x*(7 - 6*x) / ((1 - x)*(1 - 4*x)).
a(n) = 5*a(n-1) - 4*a(n-2) for n>2. (End)
E.g.f.: (-9 - 2*exp(x) + 11*exp(4*x))/6. - G. C. Greubel, Jan 14 2020
a(n) = a(n-1) + 11*2^(2*n-3), for n >= 2, with a(1) = 7. - Wolfdieter Lang, Aug 16 2021
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MAPLE
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seq(coeff(series(x*(7-6*x)/((1-x)*(1-4*x)), x, n+1), x, n), n = 1..25); # G. C. Greubel, Jan 14 2020
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MATHEMATICA
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a[n_]:= 4a[n-1] +1; a[1]=7; Table[a[n], {n, 25}]
NestList[4#+1&, 7, 30] (* or *) LinearRecurrence[{5, -4}, {7, 29}, 30] (* Harvey P. Dale, Sep 04 2023 *)
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PROG
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(PARI) Vec(x*(7-6*x)/((1-x)*(1-4*x)) + O(x^25)) \\ Colin Barker, Oct 27 2019
(Magma) [(11*4^n -2)/6: n in [1..25]]; // G. C. Greubel, Jan 14 2020
(Sage) [(11*4^n -2)/6 for n in (1..25)] # G. C. Greubel, Jan 14 2020
(GAP) List([1..25], n-> (11*4^n -2)/6); # G. C. Greubel, Jan 14 2020
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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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N. Rathankar (rathankar(AT)yahoo.com), Jul 08 2002
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EXTENSIONS
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STATUS
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approved
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