A147960 -id:A147960 - cp's OEIS frontend

A164600 a(n) = 18a(n-1) - 79a(n-2) for n > 1; a(0) = 1, a(1) = 17.

1, 17, 227, 2743, 31441, 349241, 3802499, 40854943, 434991553, 4602307457, 48477201539, 509007338599, 5332433173201, 55772217368297, 582637691946467, 6081473282940943, 63438141429166081, 661450156372654961

Offset: 0

Klaus Brockhaus, Aug 17 2009

nonn

Binomial transform of A081185 without initial term 0. Ninth binomial transform of A164587.

This sequence is part of a class of sequences defined by the recurrence a(n,m) = 2*(m+1)*a(n-1,m) - ((m+1)^2 - 2)*a(n-2,m) with a(0) = 1 and a(1) = m+9. The generating function is Sum_{n>=0} a(n,m)*x^n = (1 - (m-7)*x)/(1 - 2*(m+1)*x + ((m+1)^2 - 2)*x^2) and has a series expansion in terms of Pell-Lucas numbers defined by a(n, m) = (1/2)*Sum_{k=0..n} binomial(n,k)*m^(n-k)*(5*Q(k) + 4*Q(k-1)). - G. C. Greubel, Mar 12 2021

G. C. Greubel, Table of n, a(n) for n = 0..975 (terms 0..100 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (18, -79).

Sequences in the class a(n, m): A164298 (m=1), A164299 (m=2), A164300 (m=3), A164301 (m=4), A164598 (m=5), A164599 (m=6), A081185 (m=7), this sequence (m=8).

Cf. A147960, A153593, A164587.

[ n le 2 select 16*n-15 else 18*Self(n-1)-79*Self(n-2): n in [1..18] ];

m:=30; S:=series( (1-x)/(1-18*x+79*x^2), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 12 2021

LinearRecurrence[{18,-79},{1,17},30] (* Harvey P. Dale, Oct 30 2013 *)

my(x='x+O('x^50)); Vec((1-x)/(1-18*x+79*x^2)) \\ G. C. Greubel, Aug 11 2017

[( (1-x)/(1-18*x+79*x^2) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Mar 12 2021

a(n) = ((1+4*sqrt(2))*(9+sqrt(2))^n + (1-4*sqrt(2))*(9-sqrt(2))^n)/2.
G.f.: (1-x)/(1-18*x+79*x^2).
E.g.f.: exp(9*x)*(cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x)). - G. C. Greubel, Aug 11 2017
From G. C. Greubel, Mar 12 2021: (Start)
a(n) = 2*A147960(n) + 8*A153593(n).
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*8^(n-k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)

A147959 a(n) = ((8 + sqrt(2))^n + (8 - sqrt(2))^n)/2.

Original entry on oeis.org

1, 8, 66, 560, 4868, 43168, 388872, 3545536, 32618512, 302072960, 2810819616, 26244590336, 245642629184, 2303117466112, 21620036448384, 203127300275200, 1909594544603392, 17959620096591872, 168959059780059648

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Nov 17 2008

Binomial transform of A147958. Inverse binomial transform of A147960. 8th binomial transform of A077957. - Philippe Deléham, Nov 30 2008

			a(3) = ((8 + sqrt(2))^3 + (8 - sqrt(2))^3)/2
     = (8^3 + 3*8^2*sqrt(2) + 3*8*2 + 2*sqrt(2)
      + 8^3 - 3*8^2*sqrt(2) + 3*8*2 - 2*sqrt(2))/2
     =  8^3                 + 3*8*2
     =  560.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-62).

Cf. A077957, A147958, A147960.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((8+r2)^n+(8-r2)^n)/2: n in [0..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 19 2008

LinearRecurrence[{16, -62}, {1, 8}, 50] (* G. C. Greubel, Aug 17 2018 *)

x='x+O('x^30); Vec((1-8*x)/(1-16*x+62*x^2)) \\ G. C. Greubel, Aug 17 2018

From Philippe Deléham, Nov 19 2008: (Start)
a(n) = 16*a(n-1) - 62*a(n-2), n > 1; a(0)=1, a(1)=8.
G.f.: (1 - 8*x)/(1 - 16*x + 62*x^2).
a(n) = (Sum_{k=0..n} A098158(n,k)*8^(2k)*2^(n-k))/8^n. (End)
E.g.f.: exp(8*x)*cosh(sqrt(2)*x). - Ilya Gutkovskiy, Aug 11 2017

Extended beyond a(6) by Klaus Brockhaus, Nov 19 2008

A162666 a(n) = 20a(n-1) - 98a(n-2) for n > 1; a(0) = 1, a(1) = 10.

Original entry on oeis.org

1, 10, 102, 1060, 11204, 120200, 1306008, 14340560, 158822416, 1771073440, 19856872032, 223572243520, 2525471411264, 28599348360320, 324490768902528, 3687079238739200, 41941489422336256, 477496023050283520

Offset: 0

Klaus Brockhaus, Jul 20 2009

Binomial transform of A147960. Tenth binomial transform of A077957.

G. C. Greubel, Table of n, a(n) for n = 0..940
Index entries for linear recurrences with constant coefficients, signature (20,-98).

Cf. A147960, A077957.

a:=[1,10];; for n in [3..20] do a[n]:=20*a[n-1]-98*a[n-2]; od; a; # G. C. Greubel, Aug 27 2019

[ n le 2 select 9*n-8 else 20*Self(n-1)-98*Self(n-2): n in [1..18] ];

seq(coeff(series((1-10*x)/(1-20*x+98*x^2), x, n+1), x, n), n = 0..20); # G. C. Greubel, Aug 27 2019

Union[Flatten[NestList[{#[[2]],20#[[2]]-98#[[1]]}&,{1,10},20]]]  (* Harvey P. Dale, Feb 25 2011 *)
LinearRecurrence[{20,-98}, {1,10}, 20] (* G. C. Greubel, Aug 27 2019 *)

my(x='x+O('x^20)); Vec((1-10*x)/(1-20*x+98*x^2)) \\ G. C. Greubel, Aug 27 2019

def A162666_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-10*x)/(1-20*x+98*x^2)).list()
A162666_list(20) # G. C. Greubel, Aug 27 2019

a(n) = ((10+sqrt(2))^n + (10-sqrt(2))^n)/2.
G.f.: (1-10*x)/(1-20*x+98*x^2).
E.g.f.: exp(10*x)*cosh(sqrt(2)*x). - Ilya Gutkovskiy, Aug 11 2017

cp's OEIS Frontend

A164600 a(n) = 18*a(n-1) - 79*a(n-2) for n > 1; a(0) = 1, a(1) = 17.

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A147959 a(n) = ((8 + sqrt(2))^n + (8 - sqrt(2))^n)/2.

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Magma

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A162666 a(n) = 20*a(n-1) - 98*a(n-2) for n > 1; a(0) = 1, a(1) = 10.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A164600 a(n) = 18a(n-1) - 79a(n-2) for n > 1; a(0) = 1, a(1) = 17.

A162666 a(n) = 20a(n-1) - 98a(n-2) for n > 1; a(0) = 1, a(1) = 10.