A164110 -id:A164110

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A164095

a(n) = 2*a(n-2) for n > 2; a(1) = 5, a(2) = 6.

+10
7

5, 6, 10, 12, 20, 24, 40, 48, 80, 96, 160, 192, 320, 384, 640, 768, 1280, 1536, 2560, 3072, 5120, 6144, 10240, 12288, 20480, 24576, 40960, 49152, 81920, 98304, 163840, 196608, 327680, 393216, 655360, 786432, 1310720, 1572864, 2621440, 3145728

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Interleaving of A020714 and A007283 without initial term 3.

Partial sums are in A164096.

Binomial transform is A048655 without initial 1, second binomial transform is A161941 without initial 2, third binomial transform is A164037, fourth binomial transform is A161731 without initial 1, fifth binomial transform is A164038, sixth binomial transform is A164110.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (0, 2).

FORMULA

a(n) = A070876(n)/3.

a(n) = (4-(-1)^n)*2^(1/4*(2*n-1+(-1)^n)).

G.f.: x*(5+6*x)/(1-2*x^2).

MATHEMATICA

LinearRecurrence[{0, 2}, {5, 6}, 50] (* or *) With[{nn=20}, Riffle[NestList[ 2#&, 5, nn], NestList[2#&, 6, nn]]] (* Harvey P. Dale, Aug 15 2020 *)

PROG

(Magma) [ n le 2 select n+4 else 2*Self(n-2): n in [1..40] ];

CROSSREFS

Cf. A020714 (5*2^n), A007283 (3*2^n), A164096, A048655, A161941, A164037, A161731, A164038, A164110, A070876.

KEYWORD

nonn

AUTHOR

Klaus Brockhaus, Aug 10 2009

STATUS

approved

A164038

Expansion of (5-19*x)/(1-10*x+23*x^2).

+10
3

5, 31, 195, 1237, 7885, 50399, 322635, 2067173, 13251125, 84966271, 544886835, 3494644117, 22414043965, 143763624959, 922113238395, 5914569009893, 37937085615845, 243335768930911, 1560804720144675, 10011324516035797

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

Binomial transform of A161731 without initial 1. Fifth binomial transform of A164095. Inverse binomial transform of A164110.

LINKS

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 (10,-23).

FORMULA

a(n) = 10*a(n-1) - 23*a(n-2) for n > 1; a(0) = 5, a(1) = 31.

G.f.: (5-19*x)/(1-10*x+23*x^2).

a(n) = ((5+3*sqrt(2))*(5+sqrt(2))^n + (5-3*sqrt(2))*(5-sqrt(2))^n)/2.

E.g.f: (5*cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x))*exp(5*x). - G. C. Greubel, Sep 08 2017

MATHEMATICA

LinearRecurrence[{10, -23}, {5, 31}, 50] (* or *) CoefficientList[Series[(5 - 19*x)/(1 - 10*x + 23*x^2), {x, 0, 50}], x] (* G. C. Greubel, Sep 08 2017 *)

PROG

(Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((5+3*r)*(5+r)^n+(5-3*r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 10 2009

(PARI) Vec((5-19*x)/(1-10*x+23*x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

CROSSREFS

Cf. A161731, A164095, A164110.

KEYWORD

nonn,easy

AUTHOR

Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009

EXTENSIONS

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 10 2009

STATUS

approved

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