

A101386


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


14



5, 27, 157, 915, 5333, 31083, 181165, 1055907, 6154277, 35869755, 209064253, 1218515763, 7102030325, 41393666187, 241259966797, 1406166134595, 8195736840773, 47768254910043, 278413792619485, 1622714500806867, 9457873212221717, 55124524772523435
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OFFSET

0,1


COMMENTS

A floretiongenerated 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 "tesfortransform of the zerosequence" 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.
From Wolfdieter Lang, Feb 05 2015: (Start)
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).
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.
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.
(End)


REFERENCES

T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, 1964, Theorem 109, pp. 207208 with Theorem 104, pp. 197198.


LINKS

Colin Barker, Table of n, a(n) for n = 0..1000
M. A. Gruber, Artemas Martin, A. H. Bell, J. H. Drummond, A. H. Holmes and H. C. Wilkes, Problem 47, Amer. Math. Monthly, 4 (1897), 2528.
Tanya Khovanova, Recursive Sequences
Morris Newman, Daniel Shanks, H. C. Williams, Simple groups of square order and an interesting sequence of primes, Acta Arith., 38 (1980/1981) 129140. MR82b:20022.
Eric Weisstein's World of Mathematics, NSW Number.
Index entries for linear recurrences with constant coefficients, signature (6,1).


FORMULA

a(n) = A002315(n) + A077445(n+1). Note: the offset of A077445 is 1.
a(n+1)  a(n) = 2*A054490(n+1).
a(n) = 6*a(n1)  a(n2), a(0)=5, a(1)=27.  Philippe Deléham, Nov 17 2008
From Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009: (Start)
a(n) = ((5+sqrt(18))*(3 + sqrt(8))^n + (5sqrt(18))*(3  sqrt(8))^n)/2.
Third binomial transform of A164737. (End)
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
a(n) = 5*ChebyshevU(n, 3)  3*ChebyshevU(n1, 3).  G. C. Greubel, Mar 17 2020
a(n) = Pell(2*n+2) + 3*Pell(2*n+1), where Pell(n) = A000129(n).  G. C. Greubel, Apr 17 2020


MAPLE

A101386:= (n) > simplify(5*ChebyshevU(n, 3)  3*ChebyshevU(n1, 3)); seq( A101386(n), n = 0..30); # G. C. Greubel, Mar 17 2020


MATHEMATICA

CoefficientList[ Series[(53x)/(16x+x^2), {x, 0, 30}], x] (* Robert G. Wilson v, Jan 29 2005 *)
LinearRecurrence[{6, 1}, {5, 27}, 30] (* Harvey P. Dale, Apr 23 2016 *)


PROG

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
(PARI) Vec((53*x)/(16*x+x^2) + O(x^30)) \\ Colin Barker, Feb 05 2015
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!((5  3*x)/(16*x+x^2))); // G. C. Greubel, Jul 26 2018
(Sage) [5*chebyshev_U(n, 3) 3*chebyshev_U(n1, 3) for n in (0..30)] # G. C. Greubel, Mar 17 2020


CROSSREFS

Cf. A000004, A000129, A002315, A038761, A038762, A054490, A164737.
Cf. A077442, A077443(n+1), A077445, A253811.
Sequence in context: A098409 A351015 A052227 * A153233 A084076 A355252
Adjacent sequences: A101383 A101384 A101385 * A101387 A101388 A101389


KEYWORD

nonn,easy


AUTHOR

Creighton Dement, Jan 23 2005


EXTENSIONS

More terms from Robert G. Wilson v, Jan 29 2005


STATUS

approved



