A147957 -id:A147957 - cp's OEIS frontend

A164598 a(n) = 12a(n-1) - 34a(n-2), for n > 1, with a(0) = 1, a(1) = 14.

1, 14, 134, 1132, 9028, 69848, 531224, 3999856, 29936656, 223244768, 1661090912, 12342768832, 91636134976, 679979479424, 5044125163904, 37410199666432, 277422140424448, 2057118896434688, 15253073982785024, 113094845314640896

Offset: 0

Klaus Brockhaus, Aug 17 2009

Binomial transform of A164301. Sixth binomial transform of A164587. Inverse binomial transform of A164599.

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 (12,-34).

Sequences in the class a(n, m): A164298 (m=1), A164299 (m=2), A164300 (m=3), A164301 (m=4), this sequence (m=5), A164599 (m=6), A081185 (m=7), A164600 (m=8).

Cf. A081183, A147957, A164587.

[ n le 2 select 13*n-12 else 12*Self(n-1)-34*Self(n-2): n in [1..30] ];

m:=30; S:=series( (1+2*x)/(1-12*x+34*x^2), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 11 2021

LinearRecurrence[{12,-34}, {1,14}, 30] (* G. C. Greubel, Aug 11 2017 *)

my(x='x+O('x^30)); Vec((1+2*x)/(1-12*x+34*x^2)) \\ G. C. Greubel, Aug 11 2017

[( (1+2*x)/(1-12*x+34*x^2) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Mar 11 2021

a(n) = ((1+4*sqrt(2))*(6+sqrt(2))^n + (1-4*sqrt(2))*(6-sqrt(2))^n)/2.
G.f.: (1+2*x)/(1-12*x+34*x^2).
E.g.f.: exp(6*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) = A147957(n) + 8*A081183(n).
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*5^(n-k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)

A083880 a(0)=1, a(1)=5, a(n) = 10a(n-1) - 23a(n-2), n >= 2.

Original entry on oeis.org

1, 5, 27, 155, 929, 5725, 35883, 227155, 1446241, 9237845, 59114907, 378678635, 2427143489, 15561826285, 99793962603, 640017621475, 4104915074881, 26328745454885, 168874407826587, 1083182932803515, 6947717948023649

Offset: 0

Paul Barry, May 08 2003

Binomial transform of A083879.

Inverse binomial transform of A147957. 5th binomial transform of A077957. - Philippe Deléham, Nov 30 2008

Index entries for linear recurrences with constant coefficients, signature (10,-23).

Cf. A083878, A006012.

[ n eq 1 select 1 else n eq 2 select 5 else 10*Self(n-1)-23*Self(n-2): n in [1..21] ]; // Klaus Brockhaus, Dec 16 2008

LinearRecurrence[{10,-23},{1,5},30] (* Harvey P. Dale, May 14 2018 *)

a(n)=if(n<0,0,polsym(23-10*x+x^2,n)[n+1]/2)

G.f.: (1-5x)/(1-10x+23x^2).
E.g.f.: exp(5x)cosh(x*sqrt(2)).
a(n) = ((5-sqrt(2))^n + (5+sqrt(2))^n)/2;
a(n) = Sum_{k=0..n} C(n, 2k)*5^(n-2k)*2^k.
a(n) = (Sum_{k=0..n} A098158(n,k)*5^(2k)*2^(n-k))/5^n. - Philippe Deléham, Nov 30 2008

Typo in definition corrected by Klaus Brockhaus, Dec 16 2008

A147958 a(n) = ((7 + sqrt(2))^n + (7 - sqrt(2))^n)/2.

Original entry on oeis.org

1, 7, 51, 385, 2993, 23807, 192627, 1577849, 13036417, 108350935, 904201491, 7566326929, 63431106929, 532418131343, 4472591813139, 37592633210825, 316085049734017, 2658336935367463, 22360719757645683, 188108240644768801

Offset: 0

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

nonn

7th binomial transform of A077957. Binomial transform of A147957. Inverse binomial transform of A147959. - 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 (14, -47).

Cf. A077957, A098158, A147957, A147959.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((7+r2)^n+(7-r2)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 19 2008

LinearRecurrence[{14, -47}, {1, 7}, 50] (* G. C. Greubel, Aug 17 2018 *)

x='x+O('x^30); Vec((1-7*x)/(1-14*x+47*x^2)) \\ G. C. Greubel, Aug 17 2018

From Philippe Deléham, Nov 19 2008: (Start)
a(n) = 14*a(n-1) - 47*a(n-2), n > 1; a(0)=1, a(1)=7.
G.f.: (1 - 7*x)/(1 - 14*x + 47*x^2).
a(n) = (Sum_{k=0..n} A098158(n,k)*7^(2k)*2^(n-k))/7^n. (End)
E.g.f.: exp(7*x)*cosh(sqrt(2)*x). - Ilya Gutkovskiy, Aug 11 2017

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

cp's OEIS Frontend

A164598 a(n) = 12*a(n-1) - 34*a(n-2), for n > 1, with a(0) = 1, a(1) = 14.

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A083880 a(0)=1, a(1)=5, a(n) = 10*a(n-1) - 23*a(n-2), n >= 2.

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A147958 a(n) = ((7 + sqrt(2))^n + (7 - sqrt(2))^n)/2.

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Author

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Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A164598 a(n) = 12a(n-1) - 34a(n-2), for n > 1, with a(0) = 1, a(1) = 14.

A083880 a(0)=1, a(1)=5, a(n) = 10a(n-1) - 23a(n-2), n >= 2.