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.

Showing 1-6 of 6 results.

A204062 Expansion of g.f.: exp( Sum_{n>=1} A002203(n)^2 * x^n/n ) where A002203 are the companion Pell numbers.

Original entry on oeis.org

1, 4, 26, 148, 867, 5048, 29428, 171512, 999653, 5826396, 33958734, 197925996, 1153597255, 6723657520, 39188347880, 228406429744, 1331250230601, 7759094953844, 45223319492482, 263580822001028, 1536261612513707, 8953988853081192, 52187671505973468
Offset: 0

Views

Author

Paul D. Hanna, Jan 10 2012

Keywords

Examples

			G.f.: A(x) = 1 + 4*x + 26*x^2 + 148*x^3 + 867*x^4 + 5048*x^5 + ...
where
log(A(x)) = 2^2*x + 6^2*x^2/2 + 14^2*x^3/3 + 34^2*x^4/4 + 82^2*x^5/5 + 198^2*x^6/6 + 478^2*x^7/7 + ... + A002203(n)^2*x^n/n + ...

Links

Crossrefs

Cf. A000129, A002203, A026933, A204061, A212442.

Programs

  • Magma
    I:=[1,4,26,148]; [n le 4 select I[n] else 4*Self(n-1) +10*Self(n-2) +4*Self(n-3) -Self(n-4): n in [1..31]]; // G. C. Greubel, May 25 2021
  • Mathematica
    LinearRecurrence[{4,10,4,-1},{1,4,26,148},30] (* Vincenzo Librandi, Feb 12 2012 *)
Table[(Fibonacci[2*n+4, 2] + 2*(-1)^n*(n+2))/16, {n, 0, 30}] (* G. C. Greubel, May 25 2021 *)
  • PARI
    {A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, A002203(k)^2*x^k/k)+x*O(x^n)), n)}
  • Sage
    [(lucas_number1(2*n+4,2,-1) +2*(-1)^n*(n+2))/16 for n in (0..30)] # G. C. Greubel, May 25 2021

Formula

G.f.: 1/((1+x)^2*(1-6*x+x^2)).
Self-convolution of A026933.
Self-convolution 4th power of A204061.
a(n) = Pell(n-1)^2 + a(n-2) where Pell(n) = A000129(n).
a(n) = (1/8)*(A001109(n+2) + (-1)^n*(n+2)). - Bruno Berselli, Jan 10 2012
a(n) = (1/16)*(A000129(2*n+4) + 2*(-1)^n*(n+2)). - G. C. Greubel, May 25 2021

A026933 Self-convolution of array T given by A008288.

Original entry on oeis.org

1, 2, 11, 52, 269, 1414, 7575, 41064, 224665, 1237898, 6859555, 38187164, 213408805, 1196524814, 6727323439, 37915058384, 214140178225, 1211694546194, 6867622511675, 38981807403268, 221562006394173, 1260814207833750, 7182599953332423, 40958645048598840, 233779564099963081
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A002203, A008288, A204061, A204062.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/(1+x)/Sqrt[1-6*x+x^2],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 08 2012 *)
a[ n_]:= Sum[ SeriesCoefficient[ SeriesCoefficient[1/(1-x-y-x*y) , {x,0,n-k}] , {y, 0, k}]^2, {k, 0, n}]; (* Michael Somos, Jun 27 2017 *)
A026933[n_]:= Sum[(Binomial[n, k]*Hypergeometric2F1[-k,k-n,-n,-1])^2, {k,0,n}];
Table[A026933[n], {n, 0, 40}] (* G. C. Greubel, May 25 2021 *)
  • PARI
    /* Sum of squares of Delannoy numbers: */
{a(n)=sum(k=0,n,polcoeff(polcoeff(1/(1-x-y-x*y +x*O(x^n)+y*O(y^k)),n-k,x),k,y)^2)} \\ Paul D. Hanna, Jan 10 2012
  • PARI
    /* Involving squares of companion Pell numbers: */
{A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, A002203(k)^2/2*x^k/k)+x*O(x^n)), n)}
\\ Paul D. Hanna, Jan 10 2012
  • PARI
    my(x='x+O('x^66)); Vec( 1/(1+x)/sqrt(1-6*x+x^2) ) \\ Joerg Arndt, May 04 2013
  • Sage
    def A026933_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1+x)*sqrt(1-6*x+x^2)) ).list()
A026933_list(40) # G. C. Greubel, May 25 2021

Formula

a(n) = Sum_{k=0..n} D(n-k,k)^2 where D(n,k) = A008288(n,k) are the Delannoy numbers. - Paul D. Hanna, Jan 10 2012
G.f.: 1/((1+x)*sqrt(1-6*x+x^2)). - Vladeta Jovovic, May 13 2003
a(n) = (-1)^n*Sum_{k=0...n} (-1)^k*A001850(k). - Benoit Cloitre, Sep 28 2005
G.f.: exp( Sum_{n>=1} A002203(n)^2/2 * x^n/n ), where A002203 are the companion Pell numbers. - Paul D. Hanna, Jan 10 2012
Self-convolution yields A204062; self-convolution of A204061. - Paul D. Hanna, Jan 10 2012
From Vaclav Kotesovec, Oct 08 2012: (Start)
Recurrence: n*a(n) = (5*n-3)*a(n-1) + (5*n-2)*a(n-2) - (n-1)*a(n-3).
a(n) ~ sqrt(24+17*sqrt(2))*(3+2*sqrt(2))^n/(8*sqrt(Pi*n)). (End)
0 = +a(n)*(+a(n+1) -8*a(n+2) -7*a(n+3) +2*a(n+4)) +a(n+1)*(-2*a(n+1) +22*a(n+2) +20*a(n+3) -7*a(n+4)) +a(n+2)*(+30*a(n+2) +22*a(n+3) -8*a(n+4)) +a(n+3)*(-2*a(n+3) +a(n+4)) for all n in Z. - Michael Somos, Jun 27 2017

Extensions

More terms from Vladeta Jovovic, May 13 2003

A208034 G.f.: exp( Sum_{n>=1} 2*Pell(n)^2 * x^n/n ), where Pell(n) = A000129(n).

Original entry on oeis.org

1, 2, 6, 26, 122, 602, 3062, 15906, 83906, 447842, 2412566, 13094490, 71513210, 392592410, 2164815590, 11982792386, 66548673282, 370672213826, 2069974290726, 11586244722202, 64986102400122, 365183031749722, 2055594717395926, 11588727763937506, 65425688924696002
Offset: 0

Views

Author

Paul D. Hanna, Feb 22 2012

Keywords

Comments

Conjectures: For all positive integers k,
(1) exp( Sum_{n>=1} 2*Pell(n)^(2*k) * x^n/n ) is an integer series;
(2) exp( Sum_{n>=1} 2*Pell(n)^(2*k-1) * x^n/n ) is NOT an integer series;
(3) exp( Sum_{n>=1} Pell(n)^(2*k) * x^n/n ) is NOT an integer series.

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 26*x^3 + 122*x^4 + 602*x^5 + 3062*x^6 + ...
such that, by definition,
log(A(x))/2 = x + 2^2*x^2/2 + 5^2*x^3/3 + 12^2*x^4/4 + 29^2*x^5/5 + 70^2*x^6/6 + 169^2*x^7/7 + 408^2*x^8/8 + ... + Pell(n)^2*x^n/n + ...

Links

Crossrefs

Cf. A000129, A208055, A208056, A204061, A204062.

Programs

  • Mathematica
    CoefficientList[Series[Sqrt[1 + x] / (1 - 6 x + x^2)^(1/4), {x, 0, 33}], x] (* Vincenzo Librandi, Feb 26 2018 *)
  • PARI
    {Pell(n)=polcoeff(x/(1-2*x-x^2 +x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(m=1,n,2*Pell(m)^2*x^m/m) +x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    seq(n)={Vec(sqrt(1 + x + O(x^n)) / sqrt(sqrt(1 - 6*x + x^2 + O(x^n))))} \\ Andrew Howroyd, Feb 25 2018

Formula

G.f.: sqrt(1 + x) / (1 - 6*x + x^2)^(1/4).
a(n) ~ 2^(1/8) * (1 + sqrt(2))^(2*n) / (Gamma(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 31 2024

A208056 G.f.: exp( Sum_{n>=1} 2*Pell(n)^(2*n) * x^n/n ), where Pell(n) = A000129(n).

Original entry on oeis.org

1, 2, 18, 10450, 215011842, 168283323489554, 4613762736903044410402, 4429409381416783893511092430530, 147401742703370819998531165821635082467298, 169293247178836261713452084817353169649400098579929282
Offset: 0

Views

Author

Paul D. Hanna, Feb 22 2012

Keywords

Comments

Given g.f. A(x), note that A(x)^(1/2) does not yield an integer series.

Examples

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 10450*x^3 + 215011842*x^4 +...
such that, by definition,
log(A(x))/2 = x + 2^4*x^2/2 + 5^6*x^3/3 + 12^8*x^4/4 + 29^10*x^5/5 + 70^12*x^6/6 + 169^14*x^7/7 +...+ Pell(n)^(2*n)*x^n/n +...

Crossrefs

Cf. A000129, A211891, A208034, A208055, A204061, A204062.

Programs

  • PARI
    {Pell(n)=polcoeff(x/(1-2*x-x^2 +x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(m=1,n,2*Pell(m)^(2*m)*x^m/m) +x*O(x^n)),n)}
for(n=0,15,print1(a(n),", "))

A208055 G.f.: exp( Sum_{n>=1} 2*Pell(n)^4 * x^n/n ), where Pell(n) = A000129(n).

Original entry on oeis.org

1, 2, 18, 450, 11362, 311426, 8857426, 259072706, 7730804098, 234255654466, 7184570715602, 222512186923010, 6947171244623714, 218374183252085826, 6903938704875627410, 219355658720815861378, 6999679608428089841154, 224210965624588803552642
Offset: 0

Views

Author

Paul D. Hanna, Feb 22 2012

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 450*x^3 + 11362*x^4 + 311426*x^5 +...
such that, by definition,
log(A(x))/2 = x + 2^4*x^2/2 + 5^4*x^3/3 + 12^4*x^4/4 + 29^4*x^5/5 + 70^4*x^6/6 + 169^4*x^7/7 + 408^4*x^8/8 +...+ Pell(n)^4*x^n/n +...

Crossrefs

Cf. A000129, A208034, A208056, A204061, A204062, A207969.

Programs

  • PARI
    {Pell(n)=polcoeff(x/(1-2*x-x^2 +x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(m=1,n,2*Pell(m)^4*x^m/m) +x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))

Formula

The o.g.f. A(x) = 1 + 2*x + 18*x^2 + 450*x^3 + ... is an algebraic function: A(x)^32 = (1 + 6*x + x^2)^4/( (1 - 34*x + x^2)*(1 - 2*x + x^2)^3 ). Cf. A207969. - Peter Bala, Apr 03 2014
From Vaclav Kotesovec, Oct 31 2024: (Start)
G.f.: (1 + x*(6 + x))^(1/8) / ((1 - x)^(3/16)*(1 + (-17 + 12*sqrt(2))*x)^(1/32) * (1 - (17 + 12*sqrt(2))*x)^(1/32)).
a(n) ~ 5^(1/8) * (1 + sqrt(2))^(4*n) / (2^(13/64) * 3^(1/32) * Gamma(1/32) * n^(31/32)). (End)

A384853 Squared length of interior diagonal of n-th (U, V)-crossbox, where U = (1, 0, 1) and V = (0, 1, 0), as in Comments.

Original entry on oeis.org

1, 5, 9, 21, 57, 165, 489, 1461, 4377, 13125, 39369, 118101, 354297, 1062885, 3188649, 9565941, 28697817, 86093445, 258280329, 774840981, 2324522937, 6973568805, 20920706409, 62762119221, 188286357657, 564859072965, 1694577218889, 5083731656661
Offset: 1

Views

Author

Clark Kimberling, Jul 02 2025

Keywords

Comments

Suppose that U and V are 3-dimensional vectors over the field of real numbers. Define f(1) = U, f(2) = V, f(3) = UxV, where x = cross product, and for n>=2, define f(n) = h(n - 1), g(n) = f(n - 1) + g(n - 1) - h(n - 1), h(n) = f(n) x g(n).
The parallelopiped having edge vectors f(n), g(n), h(n) is the n-th (U,V)-crossbox, with volume |f(n).(g(n) x h(n))|, where . = dot product, and interior diagonal length ||g(n)||. These two sequences, after removal of their first 2 terms, are given for selected U and V by the following table, except for the 3 initial terms:
U V volume squared diagonal length, ||g(n)||^2
(1, 0, 0) (0, 1, 0) A000079 A052548
(1, 0, 0) (0, 1, 1) A008776 3*A052919
(1, 0, 0) (1, 0, 1) A000351 A178676
(1, 0, 0) (1, 1, 1) A167747 5*A204061
(1, 0, 0) (0, 2, 0) A005054 4*A199215
(1, 0, 0) (1, 2, 0) A013731 8*A199552
(1, 0, 0) (2, 1, 0) A011557 10*A000533
(1, 0, 0) (1, 1, 2) A067403 18*A135423
(1, 0, 0) (2, 1, 1) A334603 11*A199750
(1, 0, 1) (0, 1, 0) A008776 this sequence
(1, 1, 0) (0, 1, 1) A081341 6*A199318
(1, 1, 0) (1, 1, 1) A270369 9*A199559
(1, 2, 3) (3, 2, 1) 2*A009992 48 + 96*A009992

Examples

			Taking U = (1, 0, 1) and V = (0, 1, 0), successive edge vectors are given by
(f(n)) = ( (1, 0, 1), (-1,0,1), (-1,2,-1), (3,0,-3), (3,-6,3), ...)
(g(n)) = ( (0,1,0), (2,1,0), (2,-1,2), (-2,1,4), (-2,7,-2), (10,1,-8), ...)
(h(n)) = ( (-1.0,1), (-1,2,-1), (3,0,-3), (3,-6,3), (-9,0,9),...)
The successive volumes are (2, 6, 18, 54, 162, 486, 1458, 4374, 13122,...).
The lengths of diagonals of the first five crossboxes are 1, sqrt(5), 3, sqrt(21), sqrt(57), so the first five squared lengths are 1, 5, 9, 21, 57.

Links

Crossrefs

Cf. A000079, A052548, A008776.

Programs

  • Mathematica
    f[1] = {1, 0, 1}; g[1] = {0, 1, 0}; h[1] = Cross[f[1], g[1]];
f[n_] := f[n] = h[n - 1];
g[n_] := g[n] = f[n - 1] + g[n - 1] - h[n - 1];
h[n_] := h[n] = Cross[f[n], g[n]];
v[n_] := f[n] . Cross[g[n], h[n]] (* signed volume of nth parallelopiped P(n) *)
d[n_] := Norm[g[n]] (* length of interior diagonal of P(n) *)
Column[Table[{f[n], g[n], h[n]}, {n, 1, 16}]]  (* edge vectors of P(n) *)
Table[v[n], {n, 1, 16}]  (* A008776 *)
u = Table[d[n]^2, {n, 1, 30}] (* A384853 *)
Join[{1},Table[1+2*(3^(n-1)+1),{n,40}]] (* or *) LinearRecurrence[{4,-3},{1,5,9},50] (* Harvey P. Dale, Jul 20 2025 *)

Formula

a(0) = 1, a(n) = 1 + 2 * (3^(n-1)+1) for n>=1.
a(n) = 4*a(n-1) - 3*a(n-2) for n>=4.
In general, suppose that U = (a,b,c) and V = (s,t,u), and let D = -(a^2 + b^2 + c^2 + s^2 + t^2 + u^2 + 2 (a s + b t + c u)). Then, linear recurrences are given for n>=3 by f(n) = D*f (n - 2), g(n) = g(n - 1) + D*g(n - 2) - D*g(n - 3), h(n) = D*h(n - 2). If w(n) denotes the volume of the n-th (U,V)-crossbox, then w(n) = D*w(n-1) for n>=2.
Showing 1-6 of 6 results.

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