A164737 -id:A164737 - cp's OEIS frontend

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

Creighton Dement, Jan 23 2005

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)

T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, 1964, Theorem 109, pp. 207-208 with Theorem 104, pp. 197-198.

Colin Barker, Table of n, a(n) for n = 0..1000
Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 55, 56.
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).

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

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

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!((5 - 3*x)/(1-6*x+x^2))); // G. C. Greubel, Jul 26 2018

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

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 *)

Vec((5-3*x)/(1-6*x+x^2) + O(x^30)) \\ Colin Barker, Feb 05 2015

[5*chebyshev_U(n,3) -3*chebyshev_U(n-1,3) for n in (0..30)] # G. C. Greubel, Mar 17 2020

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

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

A164593 a(n) = ((5 + sqrt(18))(2 + sqrt(8))^n + (5 - sqrt(18))(2 - sqrt(8))^n)/2.

Original entry on oeis.org

5, 22, 108, 520, 2512, 12128, 58560, 282752, 1365248, 6592000, 31828992, 153683968, 742051840, 3582943232, 17299980288, 83531694080, 403326697472, 1947433566208, 9403041054720, 45401898483712, 219219758153728, 1058486626549760, 5110825538813952, 24677248661454848

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009

Binomial transform of A096980 without initial 1. Second binomial transform of A164737. Inverse binomial transform of A101386.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Index entries for linear recurrences with constant coefficients, signature (4,4).

Cf. A000129, A096980, A101386, A164737.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+3*r)*(2+2*r)^n+(5-3*r)*(2-2*r)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 24 2009

seq(coeff(series( (5+2*x)/(1-4*x-4*x^2) , x, n+1), x, n), n = 0..25); # G. C. Greubel, Apr 16 2020

LinearRecurrence[{4, 4}, {5, 22}, 25] (* G. C. Greubel, Aug 12 2017 *)
Table[2^n*(Fibonacci[n+2, 2] + 3*Fibonacci[n+1, 2]), {n,0,25}] (* G. C. Greubel, Apr 16 2020 *)

my(x='x+O('x^25)); Vec((5+2*x)/(1-4*x-4*x^2)) \\ G. C. Greubel, Aug 12 2017

[2^n*(lucas_number1(n+2, 2, -1) + 3*lucas_number1(n+1, 2, -1)) for n in range(25)] # G. C. Greubel, Apr 16 2020

a(n) = 4*a(n-1) + 4*a(n-2) for n > 1; a(0) = 5, a(1) = 22.
G.f.: (5 + 2*x)/(1-4*x-4*x^2).
E.g.f.: exp(2*x)*(5*cosh(2*sqrt(2)*x) + 3*sqrt(2)*sinh(2*sqrt(2)*x)). - G. C. Greubel, Aug 12 2017
a(n) = 2^n*(Pell(n+2) + 3*Pell(n+1)), where Pell(n) = A000129(n). - G. C. Greubel, Apr 16 2020

A164594 a(n) = ((5 + sqrt(18))(4 + sqrt(8))^n + (5 - sqrt(18))(4 - sqrt(8))^n)/2.

Original entry on oeis.org

5, 32, 216, 1472, 10048, 68608, 468480, 3198976, 21843968, 149159936, 1018527744, 6954942464, 47491317760, 324291002368, 2214397476864, 15120851795968, 103251634552832, 705046262054912, 4814357020016640, 32874486063693824, 224481032349417472

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009

nonn

Binomial transform of A101386. Fourth binomial transform of A164737. Inverse binomial transform of A164595.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (8,-8).

Cf. A101386, A164737, A164595.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+3*r)*(4+2*r)^n+(5-3*r)*(4-2*r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 24 2009

A164594:= (n) -> simplify( (2*sqrt(2))^n*(5*ChebyshevU(n, sqrt(2)) - 2*sqrt(2)*ChebyshevU(n-1, sqrt(2))) ); seq( A164594(n), n = 0..25); # G. C. Greubel, Apr 21 2020

CoefficientList[Series[(5-8*x)/(1-8*x+8*x^2), {x,0,25}], x] (* G. C. Greubel, Aug 12 2017 *)
Table[(2*Sqrt[2])^n*(3*ChebyshevU[n, Sqrt[2]] + 2*ChebyshevT[n, Sqrt[2]]), {n, 0, 25}] (* G. C. Greubel, Apr 21 2020 *)
LinearRecurrence[{8,-8},{5,32},30] (* Harvey P. Dale, Jul 09 2022 *)

my(x='x+O('x^25)); Vec((5-8*x)/(1-8*x+8*x^2)) \\ G. C. Greubel, Aug 12 2017

[(2*sqrt(2))^n*(5*chebyshev_U(n, sqrt(2)) - 2*sqrt(2)*chebyshev_U(n-1, sqrt(2))) for n in (0..25)] # G. C. Greubel, Apr 21 2020

a(n) = 8*a(n-1) - 8*a(n-2) for n > 1; a(0) = 5, a(1) = 32.
G.f.: (5-8*x)/(1-8*x+8*x^2).
E.g.f.: exp(4*x)*(5*cosh(2*sqrt(2)*x) + 3*sqrt(2)*sinh(2*sqrt(2)*x)). - G. C. Greubel, Aug 12 2017
a(n) = (2*sqrt(2))^n * (3*ChebyshevU(n, sqrt(2)) + 2*ChebyshevT(n, sqrt(2))). - G. C. Greubel, Apr 21 2020

Extended by Klaus Brockhaus and R. J. Mathar Aug 24 2009

A164595 a(n) = 10a(n-1) - 17a(n-2) for n > 1; a(0) = 5, a(1) = 37.

Original entry on oeis.org

5, 37, 285, 2221, 17365, 135893, 1063725, 8327069, 65187365, 510313477, 3994949565, 31274166541, 244827522805, 1916614396853, 15004076080845, 117458316061949, 919513867245125, 7198347299398117, 56351737250814045, 441145468418372461, 3453475150919885845

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009

Binomial transform of A164594. Fifth binomial transform of A164737.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..144 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (10,-17).

Cf. A002203, A164594, A164737.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+3*r)*(5+2*r)^n+(5-3*r)*(5-2*r)^n)/2: n in [0..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 24 2009

seq(coeff(series( (5-13*x)/(1-10*x+17*x^2) , x, n+1), x, n), n = 0..25); # G. C. Greubel, Apr 21 2020

CoefficientList[Series[(5 -13z)/(1 -10z +17z^2), {z,0,25}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 12 2011 *)
LinearRecurrence[{10,-17}, {5,37}, 25] (* G. C. Greubel, Aug 11 2017 *)

my(x='x+O('x^25)); Vec((5-13*x)/(1-10*x+17*x^2)) \\ G. C. Greubel, Aug 11 2017

def A164595_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (5-13*x)/(1-10*x+17*x^2) ).list()
A164595_list(25) # G. C. Greubel, Apr 21 2020

a(n) = ((5 + sqrt(18))*(5 + sqrt(8))^n + (5 - sqrt(18))*(5 - sqrt(8))^n)/2.
G.f.: (5-13*x)/(1-10*x+17*x^2).
E.g.f.: exp(5*x)*(5*cosh(2*sqrt(2)*x) + 3*sqrt(2)*sinh(2*sqrt(2)*x)). - G. C. Greubel, Aug 11 2017
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*2^(n-k)*(3*Q(2*k+1) + 2*Q(2*k)), where Q(n) are the Pell-Lucas numbers (A002203). - G. C. Greubel, Apr 21 2020

Extended by Klaus Brockhaus and R. J. Mathar, Aug 24 2009

cp's OEIS Frontend

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

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A164593 a(n) = ((5 + sqrt(18))*(2 + sqrt(8))^n + (5 - sqrt(18))*(2 - sqrt(8))^n)/2.

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A164594 a(n) = ((5 + sqrt(18))*(4 + sqrt(8))^n + (5 - sqrt(18))*(4 - sqrt(8))^n)/2.

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A164595 a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 5, a(1) = 37.

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Magma

Maple

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Extensions

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

A164593 a(n) = ((5 + sqrt(18))(2 + sqrt(8))^n + (5 - sqrt(18))(2 - sqrt(8))^n)/2.

A164594 a(n) = ((5 + sqrt(18))(4 + sqrt(8))^n + (5 - sqrt(18))(4 - sqrt(8))^n)/2.

A164595 a(n) = 10a(n-1) - 17a(n-2) for n > 1; a(0) = 5, a(1) = 37.