A108404 -id:A108404 - cp's OEIS frontend

A163141 a(n) = 5*a(n-2) for n > 2; a(1) = 4, a(2) = 5.

4, 5, 20, 25, 100, 125, 500, 625, 2500, 3125, 12500, 15625, 62500, 78125, 312500, 390625, 1562500, 1953125, 7812500, 9765625, 39062500, 48828125, 195312500, 244140625, 976562500, 1220703125, 4882812500, 6103515625, 24414062500

Offset: 1

Klaus Brockhaus, Jul 21 2009

nonn

Apparently the same as A133632 without initial 1.

Binomial transform is A163069, second binomial transform is A163070, third binomial transform is A163071, fourth binomial transform is A108404 without initial 1, fifth binomial transform is A163072.

Index entries for linear recurrences with constant coefficients, signature (0,5).

Cf. A133632, A108404, A163069, A163070, A163071, A163072.

[ n le 2 select n+3 else 5*Self(n-2): n in [1..29] ];

LinearRecurrence[{0,5},{4,5},30] (* Harvey P. Dale, Dec 20 2021 *)

Vec(x*(4+5*x)/(1-5*x^2) + O(x^30)) \\ Jinyuan Wang, Mar 23 2020

a(n) = (5-3*(-1)^n)*5^(1/4*(2*n-1+(-1)^n))/2.

G.f.: x*(4+5*x)/(1-5*x^2).

A163071 a(n) = ((4+sqrt(5))(3+sqrt(5))^n + (4-sqrt(5))(3-sqrt(5))^n)/2.

Original entry on oeis.org

4, 17, 86, 448, 2344, 12272, 64256, 336448, 1761664, 9224192, 48298496, 252894208, 1324171264, 6933450752, 36304019456, 190090313728, 995325804544, 5211593572352, 27288258215936, 142883175006208, 748146017173504

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009

nonn

Binomial transform of A163070. Third binomial transform of A163141. Inverse binomial transform of A108404 without initial 1.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6, -4).

Cf. A163070, A163141, A108404.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((4+r)*(3+r)^n+(4-r)*(3-r)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 21 2009

LinearRecurrence[{6,-4},{4,17},40] (* Harvey P. Dale, Feb 12 2013 *)

a(n) = 6*a(n-1) - 4*a(n-2) for n > 1; a(0) = 4, a(1) = 17.
G.f.: (4-7*x)/(1-6*x+4*x^2).
a(n) = 2^(n+1) * A000032(2*n) + 5 * 2^(n-1) * A000045(2*n) = 2^(n+1) * A005248(n) + 5 * 2^(n-1) * A001906(n). - Diego Rattaggi, Aug 02 2020

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

A228842 Binomial transform of A014448.

Original entry on oeis.org

2, 6, 28, 144, 752, 3936, 20608, 107904, 564992, 2958336, 15490048, 81106944, 424681472, 2223661056, 11643240448, 60964798464, 319215828992, 1671435780096, 8751751364608, 45824765067264, 239941584945152, 1256350449401856, 6578336356630528, 34444616342175744

Offset: 0

R. J. Mathar, Nov 10 2013

The binomial transform of this sequence is 2, 8, 42, 248,... = 2*A108404(n).

C. Smith, A Treatise on Algebra, Macmillan, London, 5th ed., 1950, p. 360, Example 44.

Colin Barker, Table of n, a(n) for n = 0..1000
P. Bhadouria, D. Jhala, and B. Singh, Binomial Transforms of the k-Lucas Sequences and its Properties, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 81-92; sequence B_4.
Takao Komatsu, Asymmetric Circular Graph with Hosoya Index and Negative Continued Fractions, arXiv:2105.08277 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (6,-4).

Cf. A014448, A108404, A098648.

When divided by 2^n this becomes(essentially) A005248.

CoefficientList[Series[2*(1 - 3 x)/(1 - 6 x + 4 x^2), {x, 0, 23}], x] (* Michael De Vlieger, Aug 26 2021 *)
LinearRecurrence[{6,-4},{2,6},30] (* Harvey P. Dale, Jun 30 2024 *)

Vec(2*(1 - 3*x) / (1 - 6*x + 4*x^2) + O(x^30)) \\ Colin Barker, Sep 21 2017

G.f.: 2*( 1-3*x ) / ( 1-6*x+4*x^2 ).
a(n) = 2*A098648(n).
From Colin Barker, Sep 21 2017: (Start)
a(n) = (3-sqrt(5))^n + (3+sqrt(5))^n.
a(n) = 6*a(n-1) - 4*a(n-2) for n>1.
(End)

A292466 Triangle read by rows: T(n,k) = 4T(n-1,k-1) + T(n,k-1) with T(2m,0) = 0 and T(2*m+1,0) = 5^m.

Original entry on oeis.org

0, 1, 1, 0, 4, 8, 5, 5, 21, 53, 0, 20, 40, 124, 336, 25, 25, 105, 265, 761, 2105, 0, 100, 200, 620, 1680, 4724, 13144, 125, 125, 525, 1325, 3805, 10525, 29421, 81997, 0, 500, 1000, 3100, 8400, 23620, 65720, 183404, 511392, 625, 625, 2625, 6625, 19025, 52625

Offset: 0

Seiichi Manyama, Sep 22 2017

			First few rows are:
    0;
    1,   1;
    0,   4,   8;
    5,   5,  21,   53;
    0,  20,  40,  124,  336;
   25,  25, 105,  265,  761,  2105;
    0, 100, 200,  620, 1680,  4724, 13144;
  125, 125, 525, 1325, 3805, 10525, 29421, 81997.
--------------------------------------------------------------
The diagonal is      {0, 1,  8,  53, 336, 2105, ...} and
the next diagonal is {1, 4, 21, 124, 761, 4724, ...}.
Two sequences have the following property:
     1^2 - 5*   0^2 = 1      (= 11^0),
     4^2 - 5*   1^2 = 11     (= 11^1),
    21^2 - 5*   8^2 = 121    (= 11^2),
   124^2 - 5*  53^2 = 1331   (= 11^3),
   761^2 - 5* 336^2 = 14641  (= 11^4),
  4724^2 - 5*2105^2 = 161051 (= 11^5),
  ...

Seiichi Manyama, Rows n = 0..139, flattened

The diagonal of the triangle is A091870.
The next diagonal of the triangle is A108404.
T(n,k) = b*T(n-1,k-1) + T(n,k-1): A292789 (b=-3), A292495 (b=-2), A117918 and A228405 (b=1), A227418 (b=2), this sequence (b=4).

T(n+1,n)^2 - 5*T(n,n)^2 = 11^n.

A163072 a(n) = ((4+sqrt(5))(5+sqrt(5))^n + (4-sqrt(5))(5-sqrt(5))^n)/2.

Original entry on oeis.org

4, 25, 170, 1200, 8600, 62000, 448000, 3240000, 23440000, 169600000, 1227200000, 8880000000, 64256000000, 464960000000, 3364480000000, 24345600000000, 176166400000000, 1274752000000000, 9224192000000000, 66746880000000000

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009

nonn

Binomial transform of A108404 without initial 1. Fifth binomial transform of A163141.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-20).

Cf. A108404, A163141.

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

LinearRecurrence[{10, -20}, {4, 25}, 30] (* G. C. Greubel, Jan 08 2018 *)

x='x+O('x^30); Vec((4-15*x)/(1-10*x+20*x^2)) \\ G. C. Greubel, Jan 08 2018

a(n) = 10*a(n-1) - 20*a(n-2) for n > 1; a(0) = 4, a(1) = 25.

G.f.: (4-15*x)/(1-10*x+20*x^2).

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

cp's OEIS Frontend

A163141 a(n) = 5*a(n-2) for n > 2; a(1) = 4, a(2) = 5.

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A163071 a(n) = ((4+sqrt(5))*(3+sqrt(5))^n + (4-sqrt(5))*(3-sqrt(5))^n)/2.

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A228842 Binomial transform of A014448.

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A292466 Triangle read by rows: T(n,k) = 4*T(n-1,k-1) + T(n,k-1) with T(2*m,0) = 0 and T(2*m+1,0) = 5^m.

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A163072 a(n) = ((4+sqrt(5))*(5+sqrt(5))^n + (4-sqrt(5))*(5-sqrt(5))^n)/2.

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Extensions

A163071 a(n) = ((4+sqrt(5))(3+sqrt(5))^n + (4-sqrt(5))(3-sqrt(5))^n)/2.

A292466 Triangle read by rows: T(n,k) = 4T(n-1,k-1) + T(n,k-1) with T(2m,0) = 0 and T(2*m+1,0) = 5^m.

A163072 a(n) = ((4+sqrt(5))(5+sqrt(5))^n + (4-sqrt(5))(5-sqrt(5))^n)/2.