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

5, 27, 157, 915, 5333, 31083, 181165, 1055907, 6154277, 35869755, 209064253, 1218515763, 7102030325, 41393666187, 241259966797, 1406166134595, 8195736840773, 47768254910043, 278413792619485, 1622714500806867, 9457873212221717, 55124524772523435, 321289275422918893

Offset

0,1

Comments

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.

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

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. 207-208 with Theorem 104, pp. 197-198.

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), 25-28.

Tanya Khovanova, Recursive Sequences

Morris Newman, Daniel Shanks, and H. C. Williams, Simple groups of square order and an interesting sequence of primes, Acta Arith., 38 (1980/1981) 129-140. 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(n-1) - a(n-2), 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 + (5-sqrt(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*A001109(n+1) - 3*A001109(n). - 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

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

Maple

A101386:= (n) -> simplify(5*ChebyshevU(n, 3) - 3*ChebyshevU(n-1, 3)); seq( A101386(n), n = 0..30); # G. C. Greubel, Mar 17 2020

Mathematica

CoefficientList[ Series[(5-3x)/(1-6x+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

(PARI) Vec((5-3*x)/(1-6*x+x^2) + O(x^30)) \\ Colin Barker, Feb 05 2015
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!((5 - 3*x)/(1-6*x+x^2))); // G. C. Greubel, Jul 26 2018
(SageMath) [5*chebyshev_U(n, 3) -3*chebyshev_U(n-1, 3) for n in (0..30)] # G. C. Greubel, Mar 17 2020

Crossrefs

Cf. A000004, A000129, A001109, A002315, A038761, A038762, A046184, A054490, A164737.

Cf. A077442, A077443(n+1), A077445, A253811.

Sequence in context: A351015 A052227 A357227 * 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