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%I #66 Nov 05 2020 05:04:39
%S 0,2,20,182,1640,14762,132860,1195742,10761680,96855122,871696100,
%T 7845264902,70607384120,635466457082,5719198113740,51472783023662,
%U 463255047212960,4169295424916642,37523658824249780,337712929418248022
%N Numbers whose base-9 representation is 22222222.......2.
%C If f(1) := 1/x and f(n+1) = (f(n) + 2/f(n))/3, then f(n) = 3^(1-n) * (1/x + a(n)*x + O(x^3)). - _Michael Somos_, Jul 28 2020
%H G. Benkart, D. Moon, <a href="http://arxiv.org/abs/1409.8154">A Schur-Weyl Duality Approach to Walking on Cubes</a>, arXiv preprint arXiv:1409.8154 [math.RT], 2014 and <a href="http://dx.doi.org/10.1007/s00026-016-0311-3">Ann. Combin. 20 (3) (2016) 397-417</a>
%H E. Estrada and J. A. de la Pena, <a href="http://arxiv.org/abs/1302.1176">From Integer Sequences to Block Designs via Counting Walks in Graphs</a>, arXiv preprint arXiv:1302.1176 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 28 2013
%H E. Estrada and J. A. de la Pena, <a href="http://www.nntdm.net/papers/nntdm-19/NNTDM-19-3-78-84.pdf">Integer sequences from walks in graphs</a>, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, No. 3, 78-84.
%H R. J. Mathar, <a href="/A102518/a102518.pdf">Counting Walks on Finite Graphs</a>, Nov 2020, Section 5.
%H Vladimir Pletser, <a href="http://arxiv.org/abs/1409.7969">Congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers</a>, arXiv:1409.7969 [math.NT], 2014.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-9).
%F a(n) = (9^(n-1) - 1)*2/8.
%F a(n) = 9*a(n-1) + 2 (with a(1)=0). - _Vincenzo Librandi_, Sep 30 2010
%F a(n) = 2 * A002452(n). - _Vladimir Pletser_, Mar 29 2014
%F From _Colin Barker_, Sep 30 2014: (Start)
%F a(n) = 10*a(n-1) - 9*a(n-2).
%F G.f.: 2*x^2 / ((x-1)*(9*x-1)). (End)
%F a(n) = -a(2-n) * 9^(n-1) for all n in Z. - _Michael Somos_, Jul 02 2017
%F a(n) = A191681(n-1)/2. - _Klaus Purath_, Jul 03 2020
%e G.f. = 2*x^2 + 20*x^3 + 182*x^4 + 1640*x^5 + 14762*x^6 + 132860*x^7 + ... - _Michael Somos_, Jul 28 2020
%p seq((9^n-1)*2/8, n=0..19);
%t FromDigits[#, 9]&/@Table[PadRight[{2}, n, 2], {n, 0, 20}] (* _Harvey P. Dale_, Feb 02 2011 *)
%t Table[(9^(n - 1) - 1)*2/8, {n, 20}] (* _Wesley Ivan Hurt_, Mar 29 2014 *)
%o (PARI) Vec(2*x^2/((x-1)*(9*x-1)) + O(x^100)) \\ _Colin Barker_, Sep 30 2014
%o (PARI) {a(n) = (9^(n-1) - 1)/4}; /* _Michael Somos_, Jul 02 2017 */
%Y Cf. A002452.
%K easy,nonn,base
%O 1,2
%A _Zerinvary Lajos_, Feb 03 2007
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