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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A046729 Expansion of 4*x/((1+x)*(1-6*x+x^2)).

Original entry on oeis.org

0, 4, 20, 120, 696, 4060, 23660, 137904, 803760, 4684660, 27304196, 159140520, 927538920, 5406093004, 31509019100, 183648021600, 1070379110496, 6238626641380, 36361380737780, 211929657785304, 1235216565974040, 7199369738058940, 41961001862379596, 244566641436218640
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Related to Pythagorean triples: alternate terms of A001652 and A046090.
Even-valued legs of nearly isosceles right triangles: legs differ by 1. 0 is smaller leg of degenerate triangle with legs 0 and 1 and hypotenuse 1. - Charlie Marion, Nov 11 2003
The complete (nearly isosceles) primitive Pythagorean triple is given by {a(n), a(n)+(-1)^n, A001653(n)}. - Lekraj Beedassy, Feb 19 2004
Note also that A046092 is the even leg of this other class of nearly isosceles Pythagorean triangles {A005408(n), A046092(n), A001844(n)}, i.e., {2n+1, 2n(n+1), 2n(n+1)+1} where longer sides (viz. even leg and hypotenuse) are consecutive. - Lekraj Beedassy, Apr 22 2004
Union of even terms of A001652 and A046090. Sum of legs of primitive Pythagorean triangles is A002315(n) = 2*a(n) + (-1)^n. - Lekraj Beedassy, Apr 30 2004

Examples

			[1,0,1]*[1,2,2; 2,1,2; 2,2,3]^0 gives (degenerate) primitive Pythagorean triple [1, 0, 1], so a(0) = 0. [1,0,1]*[1,2,2; 2,1,2; 2,2,3]^7 gives primitive Pythagorean triple [137903, 137904, 195025] so a(7) = 137904.
G.f. = 4*x + 20*x^2 + 120*x^3 + 696*x^4 + 4060*x^5 + 23660*x^6 + ...

References

  • A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, pp. 122-125, 1964.
  • W. Sierpiński, Pythagorean triangles, Dover Publications, Inc., Mineola, NY, 2003, p. 17. MR2002669.

Links

Crossrefs

Cf. A000129, A001109, A001333, A001652, A002315, A029549, A046090.
Cf. A046727, A047235, A084158, A084159, A089499, A114336.

Programs

  • Magma
    [4*Floor(((Sqrt(2)+1)^(2*n+1)-(Sqrt(2)-1)^(2*n+1)-2*(-1)^n) / 16): n in [0..30]]; // Vincenzo Librandi, Jul 29 2019
  • Mathematica
    LinearRecurrence[{5,5,-1}, {0,4,20}, 25] (* Vincenzo Librandi, Jul 29 2019 *)
  • PARI
    a(n)=n%2+(real((1+quadgen(8))^(2*n+1))-1)/2
  • PARI
    a(n)=if(n<0,-a(-1-n),polcoeff(4*x/(1+x)/(1-6*x+x^2)+x*O(x^n),n))
  • SageMath
    [(lucas_number2(2*n+1,2,-1) -2*(-1)^n)/4 for n in range(41)] # G. C. Greubel, Feb 11 2023

Formula

a(n) = ((1+sqrt(2))^(2n+1) + (1-sqrt(2))^(2n+1) + 2*(-1)^(n+1))/4.
a(n) = A089499(n)*A089499(n+1).
a(n) = 4*A084158(n). - Lekraj Beedassy, Jul 16 2004
a(n) = ceiling((sqrt(2)+1)^(2*n+1) - (sqrt(2)-1)^(2*n+1) - 2*(-1)^n)/4. - Lambert Klasen (Lambert.Klasen(AT)gmx.net), Nov 12 2004
a(n) is the k-th entry among the complete near-isosceles primitive Pythagorean triple A114336(n), where k = (3*(2n-1) - (-1)^n)/2, i.e., a(n) = A114336(A047235(n)), for positive n. - Lekraj Beedassy, Jun 04 2006
a(n) = A046727(n) - (-1)^n = 2*A114620(n). - Lekraj Beedassy, Aug 14 2006
From George F. Johnson, Aug 29 2012: (Start)
2*a(n)*(a(n) + (-1)^n) + 1 = (A000129(2*n+1))^2;
n > 0, 2*a(n)*(a(n) + (-1)^n) + 1 = ((a(n+1) - a(n-1))/4)^2, a perfect square.
a(n+1) = (3*a(n) + 2*(-1)^n) + 2*sqrt(2*a(n)*(a(n) + (-1)^n)+ 1).
a(n-1) = (3*a(n) + 2*(-1)^n) - 2*sqrt(2*a(n)*(a(n) + (-1)^n)+ 1).
a(n+1) = 6*a(n) - a(n-1) + 4*(-1)^n.
a(n+1) = 5*a(n) + 5*a(n-1) - a(n-2).
a(n+1) *a(n-1) = a(n)*(a(n) + 4*(-1)^n).
a(n) = (sqrt(1 + 8*A029549(n)) - (-1)^n)/2.
a(n) = A002315(n) - A084159(n) = A084159(n) - (-1)^n.
a(n) = A001652(n) + (1 - (-1)^n)/2 = A046090(n) - (1 + (-1)^n)/2.
Limit_{n->oo} a(n)/a(n-1) = 3 + 2*sqrt(2).
Limit_{n->oo} a(n)/a(n-2) = 17 + 12*sqrt(2).
Limit_{n->oo} a(n)/a(n-r) = (3 + 2*sqrt(2))^r.
Limit_{n->oo} a(n-r)/a(n) = (3 - 2*sqrt(2))^r. (End)
From G. C. Greubel, Feb 11 2023: (Start)
a(n) = (A001333(2*n+1) - 2*(-1)^n)/4.
a(n) = (1/2)*(A001109(n+1) + A001109(n) - (-1)^n). (End)
E.g.f.: exp(-x)*(exp(4*x)*(cosh(2*sqrt(2)*x) + sqrt(2)*sinh(2*sqrt(2)*x)) - 1)/2. - Stefano Spezia, Aug 03 2024

A089928 a(n) = 2*a(n-1) + 2*a(n-3) + a(n-4), with a(0)=1, a(1)=2, a(3)=4, a(4)=10.

Original entry on oeis.org

1, 2, 4, 10, 25, 60, 144, 348, 841, 2030, 4900, 11830, 28561, 68952, 166464, 401880, 970225, 2342330, 5654884, 13652098, 32959081, 79570260, 192099600, 463769460, 1119638521, 2703046502, 6525731524, 15754509550, 38034750625, 91824010800
Offset: 0

Views

Author

Paul Barry, Nov 15 2003

Keywords

Comments

a(n) is the number of tilings of an n-board (a board of size n X 1) using white squares, black squares, and white (1,1)-fences. A (1,1)-fence is a tile composed of two squares separated by a gap of width 1. - Michael A. Allen, Mar 12 2021
a(n) is the number of tilings of an n-board using white squares, black squares, white trominoes, black trominoes, and white tetrominoes. - Michael A. Allen, Mar 12 2021

Links

Crossrefs

Cf. A000129, A001333, A006498, A057077, A079291, A114620.

Programs

  • Magma
    [(Evaluate(DicksonFirst(n+2,-1), 2) + 2*(-1)^Binomial(n,2))/8: n in [0..40]]; // G. C. Greubel, Aug 18 2022
  • Mathematica
    CoefficientList[Series[1/(1-2x-2x^3-x^4),{x,0,30}],x] (* Michael A. Allen, Mar 12 2021 *)
LinearRecurrence[{2,0,2,1}, {1,2,4,10}, 41] (* G. C. Greubel, Aug 18 2022 *)
nxt[{a_,b_,c_,d_}]:={b,c,d,2d+2b+a}; NestList[nxt,{1,2,4,10},30][[;;,1]] (* Harvey P. Dale, Jul 18 2024 *)
  • SageMath
    [(lucas_number2(n+2,2,-1) +2*(-1)^binomial(n,2))/8 for n in (0..40)] # G. C. Greubel, Aug 18 2022

Formula

a(n) = ( (1+sqrt(2))^(n+2) + (1-sqrt(2))^(n+2) + 2*(-1)^floor(n/2) )/8.
a(n) = (-i)^n*Sum_{k=0..floor(n/2)} U(n-2*k, i) with i^2 = -1.
a(n) + a(n+2) = A000129(n+3). - Alex Ratushnyak, Aug 06 2012
G.f.: 1/ ( (1+2*x)*(1-2*x-x^2) ). - R. J. Mathar, Apr 26 2013
4*a(n) = A057077(n) + A001333(n+2). - R. J. Mathar, Apr 26 2013
a(2*n) = (A000129(n+1))^2 = A079291(n+1). - Michael A. Allen, Mar 12 2021
a(2*n+1) = A000129(n+1)*A000129(n+2) = A114620(n+1). - Michael A. Allen, Mar 12 2021

Extensions

Formula corrected by Max Alekseyev, Aug 22 2013
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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