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A101386 Expansion of g.f.: (5 - 3*x)/(1 - 6*x + x^2).

%I A101386 #73 Jul 10 2025 11:03:43
%S A101386 5,27,157,915,5333,31083,181165,1055907,6154277,35869755,209064253,
%T A101386 1218515763,7102030325,41393666187,241259966797,1406166134595,
%U A101386 8195736840773,47768254910043,278413792619485,1622714500806867,9457873212221717,55124524772523435,321289275422918893
%N A101386 Expansion of g.f.: (5 - 3*x)/(1 - 6*x + x^2).
%C A101386 A floretion-generated sequence relating to NSW numbers and numbers n such that (n^2 - 8)/2 is a square. It is also possible to label this sequence as the "tesfor-transform of the zero-sequence" under the floretion given in the program code, below. This is because the sequence "vesseq" would normally have been A046184 (indices of octagonal numbers which are also a square) using the floretion given. This floretion, however, was purposely "altered" in such a way that the sequence "vesseq" would turn into A000004. As (a(n)) would not have occurred under "natural" circumstances, one could speak of it as the transform of A000004.
%C A101386 Floretion Algebra Multiplication Program FAMP code: - tesforseq[ + 3'i - 2'j + 'k + 3i' - 2j' + k' - 4'ii' - 3'jj' + 4'kk' - 'ij' - 'ji' + 3'jk' + 3'kj' + 4e], Note: vesforseq = A000004, lesforseq = A002315, jesforseq = A077445
%C A101386 From _Wolfdieter Lang_, Feb 05 2015: (Start)
%C A101386 All positive solutions x = a(n) of the (generalized) Pell equation x^2 - 2*y^2 = +7 based on the  fundamental solution (x2,y2) = (5,3) of the second class of (proper) solutions. The corresponding y solutions are given by y(n) = A253811(n).
%C A101386 All other positive solutions come from the first class of (proper) solutions based on the fundamental solution (x1,y1) = (3,1). These are given in A038762 and A038761.
%C A101386 All solutions of this Pell equation are found in A077443(n+1) and A077442(n), for n >= 0. See, e.g., the Nagell reference on how to find all solutions.
%C A101386 (End)
%D A101386 T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, 1964, Theorem 109, pp. 207-208 with Theorem 104, pp. 197-198.
%H A101386 Colin Barker, <a href="/A101386/b101386.txt">Table of n, a(n) for n = 0..1000</a>
%H A101386 Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, <a href="http://dx.doi.org/10.13140/RG.2.2.16511.73129">Lune and Lens Sequences</a>, ResearchGate preprint, 2024. See pp. 55, 56.
%H A101386 M. A. Gruber, Artemas Martin, A. H. Bell, J. H. Drummond, A. H. Holmes and H. C. Wilkes, <a href="http://www.jstor.org/stable/2968551">Problem 47</a>, Amer. Math. Monthly, 4 (1897), 25-28.
%H A101386 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A101386 Morris Newman, Daniel Shanks, and H. C. Williams, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa38/aa3826.pdf">Simple groups of square order and an interesting sequence of primes</a>, Acta Arith., 38 (1980/1981) 129-140. MR82b:20022.
%H A101386 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NSWNumber.html">NSW Number.</a>
%H A101386 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-1).
%F A101386 a(n) = A002315(n) + A077445(n+1). Note: the offset of A077445 is 1.
%F A101386 a(n+1) - a(n) = 2*A054490(n+1).
%F A101386 a(n) = 6*a(n-1) - a(n-2), a(0)=5, a(1)=27. - _Philippe Deléham_, Nov 17 2008
%F A101386 From Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009: (Start)
%F A101386 a(n) = ((5+sqrt(18))*(3 + sqrt(8))^n + (5-sqrt(18))*(3 - sqrt(8))^n)/2.
%F A101386 Third binomial transform of A164737. (End)
%F A101386 a(n) = rational part of z(n), with z(n) = (5+3*sqrt(2))*(3+2*sqrt(2))^n, n >= 0, the general positive solutions of the second class of proper solutions. See the preceding formula. - _Wolfdieter Lang_, Feb 05 2015
%F A101386 a(n) = 5*A001109(n+1) - 3*A001109(n). - _G. C. Greubel_, Mar 17 2020
%F A101386 a(n) = Pell(2*n+2) + 3*Pell(2*n+1), where Pell(n) = A000129(n). - _G. C. Greubel_, Apr 17 2020
%F A101386 E.g.f.: exp(3*x)*(5*cosh(2*sqrt(2)*x) + 3*sqrt(2)*sinh(2*sqrt(2)*x)). - _Stefano Spezia_, Mar 16 2024
%p A101386 A101386:= (n) -> simplify(5*ChebyshevU(n, 3) - 3*ChebyshevU(n-1, 3)); seq( A101386(n), n = 0..30); # _G. C. Greubel_, Mar 17 2020
%t A101386 CoefficientList[ Series[(5-3x)/(1-6x+x^2), {x,0,30}], x] (* _Robert G. Wilson v_, Jan 29 2005 *)
%t A101386 LinearRecurrence[{6,-1},{5,27},30] (* _Harvey P. Dale_, Apr 23 2016 *)
%o A101386 (PARI) Vec((5-3*x)/(1-6*x+x^2) + O(x^30)) \\ _Colin Barker_, Feb 05 2015
%o A101386 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!((5 - 3*x)/(1-6*x+x^2))); // _G. C. Greubel_, Jul 26 2018
%o A101386 (SageMath) [5*chebyshev_U(n,3) -3*chebyshev_U(n-1,3) for n in (0..30)] # _G. C. Greubel_, Mar 17 2020
%Y A101386 Cf. A000004, A000129, A001109, A002315, A038761, A038762, A046184, A054490, A164737.
%Y A101386 Cf. A077442, A077443(n+1), A077445, A253811.
%K A101386 nonn,easy
%O A101386 0,1
%A A101386 _Creighton Dement_, Jan 23 2005
%E A101386 More terms from _Robert G. Wilson v_, Jan 29 2005