A164682 -id:A164682 - cp's OEIS frontend

A164537 a(n) = 8a(n-1) - 14a(n-2) for n > 1; a(0) = 5, a(1) = 28.

5, 28, 154, 840, 4564, 24752, 134120, 726432, 3933776, 21300160, 115328416, 624425088, 3380802880, 18304471808, 99104534144, 536573667840, 2905125864704, 15728975567872, 85160042437120, 461074681546752

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009

Binomial transform of A102285 without initial term 1. Fourth binomial transform of A164682. Inverse binomial transform of A164538.

Vincenzo Librandi, Table of n, a(n) for n = 0..158
Index entries for linear recurrences with constant coefficients, signature (8, -14).

Cf. A102285, A164682, A164538.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+4*r)*(4+r)^n+(5-4*r)*(4-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 21 2009

a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 5, a(1) = 28.
G.f.: (5-12*x)/(1-8*x+14*x^2).
a(n) = ((5+4*sqrt(2))*(4+sqrt(2))^n + (5-4*sqrt(2))*(4-sqrt(2))^n)/2.

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 21 2009

A164538 a(n) = 10a(n-1) - 23a(n-2) for n > 1; a(0) = 5, a(1) = 33.

Original entry on oeis.org

5, 33, 215, 1391, 8965, 57657, 370375, 2377639, 15257765, 97891953, 627990935, 4028394431, 25840152805, 165748456137, 1063161046855, 6819395977399, 43741255696325, 280566449483073, 1799615613815255, 11543127800041871

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009

Binomial transform of A164537. Fifth binomial transform of A164682.

Vincenzo Librandi, Table of n, a(n) for n = 0..151
Index entries for linear recurrences with constant coefficients, signature (10,-23).

Cf. A164537, A164682.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+4*r)*(5+r)^n+(5-4*r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 21 2009

LinearRecurrence[{10,-23},{5,33},20] (* Harvey P. Dale, May 29 2019 *)

Vec((5-17*x)/(1-10*x+23*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jun 14 2011

a(n) = 10*a(n-1) - 23*a(n-2) for n > 1; a(0) = 5, a(1) = 33.
G.f.: (5-17*x)/(1-10*x+23*x^2).
a(n) = ((5+4*sqrt(2))*(5+sqrt(2))^n + (5-4*sqrt(2))*(5-sqrt(2))^n)/2.

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 21 2009

A228305 a(1) = 3; for n >= 1, a(2n) = 2^(n+1), a(2n+1) = 5*2^(n-1).

Original entry on oeis.org

3, 4, 5, 8, 10, 16, 20, 32, 40, 64, 80, 128, 160, 256, 320, 512, 640, 1024, 1280, 2048, 2560, 4096, 5120, 8192, 10240, 16384, 20480, 32768, 40960, 65536, 81920, 131072, 163840, 262144, 327680, 524288, 655360, 1048576, 1310720, 2097152, 2621440, 4194304, 5242880

Offset: 1

Arkadiusz Wesolowski, Aug 20 2013

Union of A020714 and A198633.
Essentially the same as A094958.
For every n, a(1)^3 + a(2)^3 + a(3)^3 + ... + a(2*n-1)^3 is a cube.

			a(9) = 40 because it is equal to 5*2^(4-1).

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

Cf. A020714, A094958, A121451, A164682, A198633.

[n le 3 select n+2 else 2*Self(n-2) : n in [1..43]];

CoefficientList[Series[(3 + 4 x - x^2)/(1 - 2 x^2), {x, 0, 50}], x] (* Bruno Berselli, Aug 20 2013 *)

r=43; print1(3); print1(", "); for(n=2, r, if(bitand(n, 1), print1(5*2^((n-3)/2)), print1(2^(n/2+1))); print1(", "));

a(n) = ceiling((9 - (- 1)^n)*2^(floor(n/2) - 2)).
a(n) = n + 2 for n <= 3; a(n) = 2*a(n-2) for n > 3.
From Bruno Berselli, Aug 20 2013: (Start)
G.f.: x*(3+4*x-x^2)/(1-2*x^2).
a(n) = (16-(8-5*r)*(1-(-1)^n))*r^(n-6) for n>1, r=sqrt(2). (End)
E.g.f.: (8*cosh(sqrt(2)*x) + 5*sqrt(2)*sinh(sqrt(2)*x) + 2*x - 8)/4. - Stefano Spezia, Apr 09 2025

cp's OEIS Frontend

A164537 a(n) = 8a(n-1) - 14a(n-2) for n > 1; a(0) = 5, a(1) = 28.

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A164538 a(n) = 10a(n-1) - 23a(n-2) for n > 1; a(0) = 5, a(1) = 33.

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A228305 a(1) = 3; for n >= 1, a(2n) = 2^(n+1), a(2n+1) = 5*2^(n-1).

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cp's OEIS Frontend

A164537 a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 5, a(1) = 28.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Formula

Extensions

A164538 a(n) = 10*a(n-1) - 23*a(n-2) for n > 1; a(0) = 5, a(1) = 33.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A228305 a(1) = 3; for n >= 1, a(2*n) = 2^(n+1), a(2*n+1) = 5*2^(n-1).

Views

Author

Keywords

Comments

Examples

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Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A164537 a(n) = 8a(n-1) - 14a(n-2) for n > 1; a(0) = 5, a(1) = 28.

A164538 a(n) = 10a(n-1) - 23a(n-2) for n > 1; a(0) = 5, a(1) = 33.

A228305 a(1) = 3; for n >= 1, a(2n) = 2^(n+1), a(2n+1) = 5*2^(n-1).