A163063 -id:A163063 - cp's OEIS frontend

A307268 Partial sums of the Lucas numbers of the form L(3n+2) (A163063).

3, 14, 61, 260, 1103, 4674, 19801, 83880, 355323, 1505174, 6376021, 27009260, 114413063, 484661514, 2053059121, 8696898000, 36840651123, 156059502494, 661078661101, 2800374146900, 11862575248703, 50250675141714, 212865275815561, 901711778403960, 3819712389431403

Offset: 0

Rigoberto Florez, Apr 01 2019

			L(2) + L(5) = 14;
L(2) + L(5) + L(8) = 61;
L(2) + L(5) + L(8) + L(11) = 260.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-3,-1).

Cf. A001076, A014448, A049651, A099919, A163063.

Table[(LucasL[3*n + 4] - 1)/2, {n, 0, 20}]
LinearRecurrence[{5,-3,-1},{3,14,61},30] (* Harvey P. Dale, Aug 10 2022 *)

L(n) = fibonacci(n+1)+fibonacci(n-1);
a(n) = (L(3*n+4)-1)/2; \\ Michel Marcus, Apr 01 2019

Vec((3 - x) / ((1 - x)*(1 - 4*x - x^2)) + O(x^25)) \\ Colin Barker, Apr 02 2019

a(n) = A001076(n+1) + A099919(n+1).
a(n) = Sum_{i=0..n} L(3i+2), L(i) = A000032(i).
a(n) = (L(3*n+4)-1)/2.
From Colin Barker, Apr 02 2019: (Start)
G.f.: (3 - x) / ((1 - x)*(1 - 4*x - x^2)).
a(n) = (-2 + (7-3*sqrt(5))*(2-sqrt(5))^n + (2+sqrt(5))^n*(7+3*sqrt(5))) / 4.
a(n) = 5*a(n-1) - 3*a(n-2) - a(n-3) for n > 2.
(End)

A163114 a(n) = 5*a(n-2) for n > 2; a(1) = 3, a(2) = 5.

Original entry on oeis.org

3, 5, 15, 25, 75, 125, 375, 625, 1875, 3125, 9375, 15625, 46875, 78125, 234375, 390625, 1171875, 1953125, 5859375, 9765625, 29296875, 48828125, 146484375, 244140625, 732421875, 1220703125, 3662109375, 6103515625, 18310546875

Offset: 1

Klaus Brockhaus, Jul 21 2009

Binomial transform is A163062, second binomial transform is A163063, third binomial transform is A098648 without initial 1, fourth binomial transform is A163064, fifth binomial transform is A163065.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,5).

Cf. A056487, A098648, A163062, A163063, A163064, A163065, A111386.

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

CoefficientList[Series[x*(3 + 5*x)/(1 - 5*x^2), {x, 0, 50}], x] (* G. C. Greubel, Dec 21 2017 *)
LinearRecurrence[{0,5},{3,5},30] (* Harvey P. Dale, Aug 01 2021 *)

x='x+O('x^30); Vec(x*(3+5*x)/(1-5*x^2)) \\ G. C. Greubel, Dec 21 2017

a(n) = (2-(-1)^n)*5^(1/4*(2*n-1+(-1)^n)).
G.f.: x*(3+5*x)/(1-5*x^2).
a(n) = A056487(n), n>=1.
E.g.f.: cosh(sqrt(5)*x) + 3*sinh(sqrt(5)*x)/sqrt(5) - 1. - Stefano Spezia, Nov 19 2023

A163062 a(n) = ((3+sqrt(5))(1+sqrt(5))^n + (3-sqrt(5))(1-sqrt(5))^n)/2.

Original entry on oeis.org

3, 8, 28, 88, 288, 928, 3008, 9728, 31488, 101888, 329728, 1067008, 3452928, 11173888, 36159488, 117014528, 378667008, 1225392128, 3965452288, 12832473088, 41526755328, 134383403008, 434873827328, 1407281266688

Offset: 0

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

nonn

Binomial transform of A163114. Inverse binomial transform of A163063.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,4).

Cf. A000032, A163063, A163114.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((3+r)*(1+r)^n+(3-r)*(1-r)^n)/2: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 21 2009

I:=[3,8]; [n le 2 select I[n] else 2*Self(n-1) + 4*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 22 2017

CoefficientList[Series[(3+2*x)/(1-2*x-4*x^2), {x,0,50}], x] (* or *) LinearRecurrence[{2,4}, {3,8}, 30] (* G. C. Greubel, Dec 22 2017 *)

x='x+O('x^30); Vec((3+2*x)/(1-2*x-4*x^2)) \\ G. C. Greubel, Dec 22 2017

a(n) = 2*a(n-1) + 4*a(n-2) for n > 1; a(0) = 3, a(1) = 8.
G.f.: (3+2*x)/(1-2*x-4*x^2).
a(n) = 2^n * A000032(n+2). - Diego Rattaggi, Jun 17 2020

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

A332936 Number of blue nodes in n-th power graph W exponentiation of a cycle graph with 7 blue nodes and 1 green node.

Original entry on oeis.org

7, 51, 387, 2943, 22383, 170235, 1294731, 9847143, 74892951, 569602179, 4332138579, 32948302095, 250590001023, 1905875101899, 14495230812123, 110244221191287, 838468077093927, 6377011953177555, 48500691394138659, 368874495293576607, 2805493888166196879, 21337327619448845211

Offset: 0

George Strand Vajagich, Mar 02 2020

The series of green nodes in n-th power W exponentiation for all n<6 n blue 1 green, 2 edge per node graphs already corresponds with an existing OEIS sequence (empirical). For example the number of blue nodes in n-th power W exponentiation of a square containing 3 blue nodes and 1 green node corresponds to A163063.

			For n = 2 take g(1)=15 and b(1)=51. Multiply b(1) by 7 to get 357 add 30 to get 387.
For n = 3 take g(2)=117 and b(2)=387. Multiply b(2) by 7 to get 774 add 234 to get 2943.

George Strand Vajagich, Youtube video explaining graph W multiplication, YouTube video.
Index entries for linear recurrences with constant coefficients, signature (8,-3).

Cf. A331211.

Vec((1 + 43*x - 18*x^2) / (1 - 8*x + 3*x^2) + O(x^40)) \\ Colin Barker, Mar 03 2020

g=1
b=7
sg=0
sb=0
bl=[]
gl=[]
for int in range(1,20):
  sg=g*1+b*2
  sb=b*7+g*2
  g=sg
  b=sb
  gl.append(g)
  bl.append(b)
print(bl)

g(n) = g(n-1) + 2*a(n-1), a(n) = 2*g(n-1) + 7*a(n-1) with g(0) = 1 and b(0) = 7, where g(n) = A332211(n).
From Colin Barker, Mar 03 2020: (Start)
G.f.: (1 + 43*x - 18*x^2) / (1 - 8*x + 3*x^2).
a(n) = 8*a(n-1) - 3*a(n-2) for n > 1.
(End)
From Stefano Spezia, Mar 03 2020: (Start)
a(n) = ((4 - sqrt(13))^n*(-23 + 7*sqrt(13)) + (4 + sqrt(13))^n*(23 + 7*sqrt(13)))/(2*sqrt(13)).
E.g.f.: exp(4*x)*(91*cosh(sqrt(13)*x) + 23*sqrt(13)*sinh(sqrt(13)*x))/13.
(End)
a(n) = 7*A190976(n+1) -5*A190976(n). - R. J. Mathar, Apr 30 2020

A122909 a(n) = F(n+1)F(2n+2) + F(n)F(2n).

Original entry on oeis.org

1, 4, 19, 79, 338, 1427, 6053, 25628, 108583, 459931, 1948354, 8253271, 34961561, 148099316, 627359147, 2657535383, 11257501522, 47687540107, 202007664157, 855718193164, 3624880442591, 15355239954179, 65045840274434

Offset: 0

Paul Barry, Sep 18 2006

Let M be the matrix M(n,k)=F(k+1)*sum{j=0..n, (-1)^(n-j)C(n,j)C(j+1,k+1)}. a(n) gives the row sums of M^3.

Index entries for linear recurrences with constant coefficients, signature (3,6,-3,-1).

Cf. A000045, A037451, A061084, A163063.

Table[Fibonacci[n+1]Fibonacci[2n+2]+Fibonacci[n]Fibonacci[2n],{n,0,30}] (* or *) LinearRecurrence[{3,6,-3,-1},{1,4,19,79},30] (* Harvey P. Dale, Dec 11 2016 *)

G.f.: (1+x)*(1+x^2) / ( (x^2+4*x-1)*(x^2-x-1) ).
a(n) = (sqrt(5)+2)^n(sqrt(5)/5+3/5)-2^(-n-1)(sqrt(5)-1)^n(sqrt(5)/5+1/5)+ 2^(-n-1)(sqrt(5)+1)^n(sqrt(5)/5-1/5)(-1)^n+(sqrt(5)-2)^n(3/5-sqrt(5)/5)(-1)^n;
a(n) = (2*A163063(n) -A061084(n))/5. - R. J. Mathar, Jun 08 2016

cp's OEIS Frontend

A307268 Partial sums of the Lucas numbers of the form L(3n+2) (A163063).

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A163114 a(n) = 5*a(n-2) for n > 2; a(1) = 3, a(2) = 5.

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A163062 a(n) = ((3+sqrt(5))*(1+sqrt(5))^n + (3-sqrt(5))*(1-sqrt(5))^n)/2.

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A332936 Number of blue nodes in n-th power graph W exponentiation of a cycle graph with 7 blue nodes and 1 green node.

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A122909 a(n) = F(n+1)*F(2n+2) + F(n)*F(2n).

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Formula

A163062 a(n) = ((3+sqrt(5))(1+sqrt(5))^n + (3-sqrt(5))(1-sqrt(5))^n)/2.

A122909 a(n) = F(n+1)F(2n+2) + F(n)F(2n).