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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A081016 a(n) = (Lucas(4*n+3) + 1)/5, or Fibonacci(2*n+1)*Fibonacci(2*n+2), or A081015(n)/5.

Original entry on oeis.org

1, 6, 40, 273, 1870, 12816, 87841, 602070, 4126648, 28284465, 193864606, 1328767776, 9107509825, 62423800998, 427859097160, 2932589879121, 20100270056686, 137769300517680, 944284833567073, 6472224534451830
Offset: 0

Views

Author

R. K. Guy, Mar 01 2003

Keywords

Comments

a(n-1) is, together with b(n) := A089508(n), n >= 1, the solution to a binomial problem; see A089508.
Numbers k such that 1 - 2*k + 5*k^2 is a square. - Artur Jasinski, Oct 26 2008
Also solution y of Diophantine equation x^2 + 4*y^2 = h^2 for which x = y-1. - Carmine Suriano, Jun 23 2010

References

  • Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.
  • A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 26.

Links

Crossrefs

Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers), A081015.
Partial sums of A033889. Bisection of A001654. Equals A003482 + 1.
Cf. A145995, A178898.

Programs

  • GAP
    List([0..30], n-> (Lucas(1,-1,4*n+3)[2] +1)/5 ); # G. C. Greubel, Jul 13 2019
  • Magma
    [(Lucas(4*n+3) +1)/5: n in [0..30]]; // G. C. Greubel, Dec 18 2017
  • Maple
    luc := proc(n) option remember: if n=0 then RETURN(2) fi: if n=1 then RETURN(1) fi: luc(n-1)+luc(n-2): end: for n from 0 to 25 do printf(`%d,`,(luc(4*n+3)+1)/5) od: # James Sellers, Mar 03 2003
  • Mathematica
    LinearRecurrence[{8,-8,1}, {1,6,40}, 30] (* Bruno Berselli, Aug 31 2017 *)
  • PARI
    a(n)=([0,1,0; 0,0,1; 1,-8,8]^n*[1;6;40])[1,1] \\ Charles R Greathouse IV, Sep 28 2015
  • PARI
    first(n) = Vec((1-2*x)/((1-x)*(1-7*x+x^2)) + O(x^n)) \\ Iain Fox, Dec 19 2017
  • Sage
    [(lucas_number2(4*n+3,1,-1) +1)/5 for n in (0..30)] # G. C. Greubel, Jul 13 2019

Formula

a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).
G.f.: (1 - 2*x)/((1 - x)*(1 - 7*x + x^2)).
a(n) = F(1) + F(5) + F(9) +...+ F(4*n+1) = F(2*n)*F(2*n+3) + 1, where F(j) = Fibonacci(j).
a(n) = 7*a(n-1) - a(n-2) - 1, n >= 2. - R. J. Mathar, Nov 07 2015

A202670 Symmetric matrix based on A000290 (the squares), by antidiagonals.

Original entry on oeis.org

1, 4, 4, 9, 17, 9, 16, 40, 40, 16, 25, 73, 98, 73, 25, 36, 116, 184, 184, 116, 36, 49, 169, 298, 354, 298, 169, 49, 64, 232, 440, 584, 584, 440, 232, 64, 81, 305, 610, 874, 979, 874, 610, 305, 81, 100, 388, 808, 1224, 1484, 1484, 1224, 808, 388, 100, 121
Offset: 1

Views

Author

Clark Kimberling, Dec 22 2011

Keywords

Comments

Let s=(1,4,9,16,...) and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A202670 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A202671 for characteristic polynomials of principal submatrices of M.
...
row 1 (1,4,9,16,...) A000290
row 2 (4,17,40,73,...) A145995
diagonal (1,17,98,354,...) A000538
antidiagonal sums (1,8,35,112,...) A040977
...
The n-th "square border sum" m(n,1)+m(n,2)+...+m(n,n)+m(n-1,n)+m(n-2,n)+...+m(1,n) is a squared square pyramidal number: [n*(n+1)*(2*n+1)/6]^2; see A000330.

Examples

			Northwest corner:
1.....4......9....16....25
4....17.....40....73...116
9....40.....98...184...298
16...73....184...354...584
25...116...298...584...979

Crossrefs

Cf. A000290, A202671, A193722.

Programs

  • Mathematica
    U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[ Table[k^2, {k, 1, 12}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
Showing 1-2 of 2 results.

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