cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A164035 a(n) = ((4+3*sqrt(2))*(5+sqrt(2))^n + (4-3*sqrt(2))*(5-sqrt(2))^n)/4.

Original entry on oeis.org

2, 13, 84, 541, 3478, 22337, 143376, 920009, 5902442, 37864213, 242885964, 1557982741, 9993450238, 64100899337, 411159637896, 2637275694209, 16916085270482, 108503511738013, 695965156159044, 4464070791616141
Offset: 0

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Author

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

Keywords

Comments

Binomial transform of A164034. Fifth binomial transform of A164090.

Links

Crossrefs

Cf. A164034, A164090.

Programs

  • Magma
    Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((4+3*r)*(5+r)^n+(4-3*r)*(5-r)^n)/4: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 09 2009
  • Mathematica
    LinearRecurrence[{10,-23}, {2,13}, 50] (* or *) CoefficientList[Series[(2 - 7*x)/(1 - 10*x + 23*x^2), {x,0,50}], x] (* _G. C. Greubel, Sep 08 2017 *)
  • PARI
    x='x+O('x^50); Vec((2-7*x)/(1-10*x+23*x^2)) \\ G. C. Greubel, Sep 08 2017

Formula

a(n) = 10*a(n-1) - 23*a(n-2) for n > 1; a(0) = 2, a(1) = 13.
G.f.: (2-7*x)/(1-10*x+23*x^2).
E.g.f.: (2*cosh(sqrt(2)*x) + (3*sqrt(2)/2)*sinh(sqrt(2)*x))*exp(5*x). - G. C. Greubel, Sep 08 2017

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 09 2009

A265207 Draw a square and follow these steps: Take a square and place at its edges isosceles right triangles with the edge as hypotenuse. Draw a square at every new edge of the triangles. Repeat for all the new squares of the same size. New figures are only placed on empty space. The structure is symmetric about the first square. The sequence gives the numbers of squares of equal size in successive rings around the center.

Original entry on oeis.org

1, 8, 20, 36, 60, 92, 140, 204, 300, 428, 620, 876, 1260, 1772, 2540, 3564, 5100, 7148, 10220, 14316, 20460, 28652, 40940, 57324, 81900, 114668, 163820, 229356, 327660, 458732, 655340, 917484, 1310700, 1834988, 2621420, 3669996, 5242860, 7340012, 10485740, 14680044, 20971500, 29360108, 41943020, 58720236
Offset: 1

Views

Author

Marian Kraus, Dec 04 2015

Keywords

Examples

			By recursion:
a(3)=2*a(1)+20=2*8+20=36
a(4)=2*a(2)+20=2*20+20=60
By function:
a(3)=4*sum_{k=1}^{[(3+1)/2]}(2^k)+6*sum_{k=1}^{[3/2]}(2^k)
=4*sum_{k=1}^{[2]}(2^k)+6*sum_{k=1}^{[1.5]}(2^k)
=4*sum_{k=1}^{2}(2^k)+6*sum_{k=1}^{1}(2^k)
=4*(2^1+2^2)+6*(2^1)
=4*(2+4)+6*(2)=24+12=36
a(4)=4*sum_{k=1}^{[(4+1)/2]}(2^k)+6*sum_{k=1}^{[4/2]}(2^k)
=4*sum_{k=1}^{[2.5]}(2^k)+6*sum_{k=1}^{[2]}(2^k)
=4*sum_{k=1}^{2}(2^k)+6*sum_{k=1}^{2}(2^k)
=4*(2^1+2^2)+6*(2^1+2^2)
=4*(2+4)+6*(2+4)=24+36=60

Links

Crossrefs

For the differences (a(n)-a(n-1))/4, n>2, see A163978.
Cf. A029744, A063759, A164090.

Programs

  • R
    rm(a)
a <- vector() powerof2 <- vector()
x <- 300
n <- x/2
for (i in 1:x){
   powerof2[i] <- 2^(i-1)}
powerof2 for (i in 1:n){
   a[2*i]   <- 8*(sum(powerof2[1:i]))+12*(sum(powerof2[1:i]))}
for (i in 1:(n+1)){
   a[2*i+1] <- 8*(sum(powerof2[1:(i+1)]))+12*(sum(powerof2[1:i]))}
a[1]<-8
a

Formula

Conjectured recurrence:
a(0)=1,
a(1)=8,
a(2)=20, and thereafter
a(n)=2*a(n-2)+20.
Conjectured formula: ("[]" is the floor function)
a(n)=4*sum_{k=1}^{[(n+1)/2]}(2^k)+6*sum_{k=1}^{[n/2]}(2^k).
Conjectures from Colin Barker, Dec 07 2015: (Start)
a(n) = (-20+2^(1/2*(-1+n))*(10-10*(-1)^n+7*sqrt(2)+7*(-1)^n*sqrt(2))) for n>1.
a(n) = 5*2^(n/2+1/2)-5*(-1)^n*2^(n/2+1/2)+7*2^(n/2)+7*(-1)^n*2^(n/2)-20 for n>1.
a(n) = a(n-1)+2*a(n-2)-2*a(n-3) for n>4.
G.f.: x*(1+7*x+10*x^2+2*x^3) / ((1-x)*(1-2*x^2)).
(End)
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