A164608 -id:A164608 - cp's OEIS frontend

A164609 a(n) = 10a(n-1) - 17a(n-2) for n > 1; a(0) = 1, a(1) = 13.

1, 13, 113, 909, 7169, 56237, 440497, 3448941, 27000961, 211377613, 1654759793, 12954178509, 101410868609, 793887651437, 6214891748017, 48652827405741, 380875114341121, 2981653077513613, 23341653831337073, 182728435995639309

Offset: 0

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

Binomial transform of A164608. Fifth binomial transform of A164683.

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. A164608, A164683.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((2+4*r)*(5+2*r)^n+(2-4*r)*(5-2*r)^n)/4: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 22 2009

LinearRecurrence[{10,-17},{1,13},20] (* Harvey P. Dale, Nov 05 2014 *)

x='x+O('x^50); Vec((1+3*x)/(1-10*x+17*x^2)) \\ G. C. Greubel, Aug 10 2017

a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 1, a(1) = 13.
a(n) = ((2+4*sqrt(2))*(5+2*sqrt(2))^n + (2-4*sqrt(2))*(5-2*sqrt(2))^n)/4.
G.f.: (1+3*x)/(1-10*x+17*x^2).
E.g.f.: exp(5*x)*(cosh(2*sqrt(2)*x) + 2*sqrt(2)*sinh(2*sqrt(2)*x)). - G. C. Greubel, Aug 10 2017

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 22 2009

A154348 a(n) = 16a(n-1) - 56a(n-2) for n>1, with a(0)=1, a(1)=16.

Original entry on oeis.org

1, 16, 200, 2304, 25664, 281600, 3068416, 33325056, 361369600, 3915710464, 42414669824, 459354931200, 4974457389056, 53867442077696, 583309459456000, 6316374594945024, 68396663789584384, 740629643316428800

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jan 07 2009

Third binomial transform of A164609, fourth binomial transform of A164608, fifth binomial transform of A054490, sixth binomial transform of A164607, seventh binomial transform of A083100, eighth binomial transform of A164683.

lim_{n -> infinity} a(n)/a(n-1) = 8 + 2*sqrt(2) = 10.8284271247....

R. J. Mathar, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (16,-56).

Cf. A002193 (decimal expansion of sqrt(2)), A164609, A164608, A054490, A164607, A083100, A164683.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((8+2*r)^n-(8-2*r)^n)/(4*r): n in [1..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 12 2009

Join[{a=1,b=16},Table[c=16*b-56*a;a=b;b=c,{n,40}]] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2011*)
LinearRecurrence[{16,-56},{1,16},30] (* Harvey P. Dale, Aug 31 2016 *)

a(n) = 16*a(n-1) - 56*a(n-2) for n>1. - Philippe Deléham, Jan 12 2009
a(n) = ( (8 + 2*sqrt(2))^n - (8 - 2*sqrt(2))^n )/(4*sqrt(2)).
G.f.: 1/(1 - 16*x + 56*x^2). - Klaus Brockhaus, Jan 12 2009; corrected Oct 08 2009
E.g.f.: (1/(2*sqrt(2)))*exp(8*x)*sinh(2*sqrt(2)*x). - G. C. Greubel, Sep 13 2016

Extended beyond a(7) by Klaus Brockhaus, Jan 12 2009
Edited by Klaus Brockhaus, Oct 08 2009
Offset corrected. - R. J. Mathar, Jun 19 2021

cp's OEIS Frontend

A164609 a(n) = 10a(n-1) - 17a(n-2) for n > 1; a(0) = 1, a(1) = 13.

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A154348 a(n) = 16a(n-1) - 56a(n-2) for n>1, with a(0)=1, a(1)=16.

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cp's OEIS Frontend

A164609 a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 1, a(1) = 13.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A154348 a(n) = 16*a(n-1) - 56*a(n-2) for n>1, with a(0)=1, a(1)=16.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A164609 a(n) = 10a(n-1) - 17a(n-2) for n > 1; a(0) = 1, a(1) = 13.

A154348 a(n) = 16a(n-1) - 56a(n-2) for n>1, with a(0)=1, a(1)=16.