A147958 -id:A147958 - cp's OEIS frontend

A164599 a(n) = 14a(n-1) - 47a(n-2), for n > 1, with a(0) = 1, a(1) = 15.

1, 15, 163, 1577, 14417, 127719, 1110467, 9543745, 81420481, 691330719, 5851867459, 49433600633, 417032638289, 3515077706295, 29610553888547, 249339102243793, 2099051398651393, 17667781775661231, 148693529122641763

Offset: 0

Klaus Brockhaus, Aug 17 2009

nonn

Binomial transform of A164598. Seventh binomial transform of A164587. Inverse binomial transform of A081185 without initial term 0.

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 have 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 11 2021

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 (14,-47).

Sequences in the class a(n, m): A164298 (m=1), A164299 (m=2), A164300 (m=3), A164301 (m=4), A164598 (m=5), this sequence (m=6), A081185 (m=7), A164600 (m=8).

Cf. A002203, A081184, A081185, A147958, A164587.

[ n le 2 select 14*n-13 else 14*Self(n-1)-47*Self(n-2): n in [1..30] ];

m:=30; S:=series( (1+x)/(1-14*x+47*x^2), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 11 2021

LinearRecurrence[{14,-47}, {1,15}, 30] (* G. C. Greubel, Aug 11 2017 *)

my(x='x+O('x^30)); Vec((1+x)/(1-14*x+47*x^2)) \\ G. C. Greubel, Aug 11 2017

def A164599_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)/(1-14*x+47*x^2) ).list()
A164599_list(30) # G. C. Greubel, Mar 11 2021

a(n) = ((1+4*sqrt(2))*(7+sqrt(2))^n + (1-4*sqrt(2))*(7-sqrt(2))^n)/2.
G.f.: (1+x)/(1-14*x+47*x^2).
E.g.f.: exp(7*x)*(cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x)). - G. C. Greubel, Aug 11 2017
From G. C. Greubel, Mar 11 2021: (Start)
a(n) = A147958(n) + 8*A081184(n).
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*6^(n-k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)

A147957 a(n) = ((6 + sqrt(2))^n + (6 - sqrt(2))^n)/2.

Original entry on oeis.org

1, 6, 38, 252, 1732, 12216, 87704, 637104, 4663312, 34298208, 253025888, 1870171584, 13839178816, 102484311936, 759279663488, 5626889356032, 41707163713792, 309171726460416, 2292017151256064, 16992367115418624, 125979822242317312, 934017384983574528, 6924894663564105728

Offset: 0

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

6th binomial transform of A077957. Binomial transform of A083880. Inverse binomial transform of A147958. - Philippe Deléham, Nov 30 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-34).

Cf. A077957, A083880, A098158, A147958.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((6+r2)^n+(6-r2)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 19 2008

LinearRecurrence[{12, -34}, {1, 6}, 50] (* G. C. Greubel, Aug 17 2018 *)

my(x='x+O('x^50)); Vec((1-6*x)/(1-12*x+34*x^2)) \\ G. C. Greubel, Aug 17 2018

From Philippe Deléham, Nov 19 2008: (Start)
a(n) = 12*a(n-1) - 34*a(n-2), n > 1; a(0)=1, a(1)=6.
G.f.: (1 - 6*x)/(1 - 12*x + 34*x^2).
a(n) = (Sum_{k=0..n} A098158(n,k)*6^(2k)*2^(n-k))/6^n. (End)
E.g.f.: exp(6*x)*cosh(sqrt(2)*x). - Ilya Gutkovskiy, Aug 11 2017

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

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

cp's OEIS Frontend

A164599 a(n) = 14*a(n-1) - 47*a(n-2), for n > 1, with a(0) = 1, a(1) = 15.

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A147957 a(n) = ((6 + sqrt(2))^n + (6 - sqrt(2))^n)/2.

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Author

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Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A164599 a(n) = 14a(n-1) - 47a(n-2), for n > 1, with a(0) = 1, a(1) = 15.