A005409 Number of polynomials of height n: a(1)=1, a(2)=1, a(3)=4, a(n) = 2*a(n-1) + a(n-2) + 2 for n >= 4.

1, 1, 4, 11, 28, 69, 168, 407, 984, 2377, 5740, 13859, 33460, 80781, 195024, 470831, 1136688, 2744209, 6625108, 15994427, 38613964, 93222357, 225058680, 543339719, 1311738120, 3166815961, 7645370044, 18457556051, 44560482148, 107578520349, 259717522848

Offset: 1

Author

N. J. A. Sloane, S. M. Diano

Keywords

• nonn
• easy
• nice

Comments

Starting with n=1, the sum of the antidiagonals of the array in a comment from Cloitre regarding A002002. - Gerald McGarvey, Aug 12 2004

Cumulative sum of A001333. - Sture Sjöstedt, Nov 15 2011

a(n) is the number of self-avoiding walks on a 3 rows X n columns grid of squares, starting top-left, ending bottom-left, using moves of R(ight), L(eft), U(p), D(own). E.g., for 3 X 1 there is just the path (D,D), and a(1) = 1. For 3 X 2, there are 4 paths (D,D) (D,R,D,L) (R,D,D,L) and (R,D,L,D) and a(2) = 4. - Toby Gottfried, Mar 04 2013

Define a triangle to have T(n,1) = n*(n-1)+1 and T(n,n) = n; the other terms T(r,c) = T(r-1,c) + T(r-1,c-1) + T(r-2,c-1). The sum of the terms in row(n+1) minus those in row(n) = a(n+2). - J. M. Bergot, Apr 30 2013

Since the terms of the sequence are all finite, it can be used in enumerating all polynomials with integer coefficients. Since each polynomial has only a finite number of roots, this enumeration can be used in turn to enumerate the algebraic numbers. Cantor uses this to derive the existence of transcendental numbers as a corollary of his stronger result that no enumerable sequence of real numbers can include all of them. - Morgan L. Owens, May 15 2022

For n > 1, also the rank of the (n-1)-Pell graph. - Eric W. Weisstein, Aug 01 2023

References

• R. Courant and H. Robbins, What is Mathematics?, Oxford Univ. Press, 1941, p. 103.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..300
• Bill Allombert, Nicolas Brisebarre, and Alain Lasjaunias. On a two-valued sequence and related continued fractions in power series fields, The Ramanujan Journal 45.3 (2018): 859-871. See Theorem 3.
• M. Bicknell, A Primer on the Pell Sequence and related sequences, Fibonacci Quarterly, Vol. 13, No. 4, 1975, pp. 345-349.
• Georg Cantor, Ueber eine Eigenschaft des Inbegriffs aller reellen algebraischen Zahlen. Journal für die reine und angewandte Mathematik 77 (1874), 258-262. English translation by Christopher P. Grant: On a Property of the Class of All Real Algebraic Numbers.
• S. M. Diano, Letter to N. J. A. Sloane
• A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434.
• El-Mehdi Mehiri, Saad Mneimneh, and Hacène Belbachir, The Towers of Fibonacci, Lucas, Pell, and Jacobsthal, arXiv:2502.11045 [math.CO], 2025. See p. 12.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Gy. Tasi and F. Mizukami, Quantum algebraic-combinatoric study of the conformational properties of n-alkanes, J. Math. Chemistry, 25, 1999, 55-64 (see p. 63).
• Eric Weisstein's World of Mathematics, Pell Graph
• Eric Weisstein's World of Mathematics, Graph Rank
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).

Crossrefs

Cf. A000129, A001333, A048654, A048655, A048745.

Cf. A214931 (walks on grids with 4 rows), A006189 (grids with 3 columns).

Cf. A216211 (grids with 4 columns).

Programs

Haskell

a005409 n = a005409_list !! (n-1)
a005409_list = 1 : scanl1 (+) (tail a001333_list)
-- Reinhard Zumkeller, Jul 08 2012

Magma

[1] cat [n le 2 select n^2 else 2*Self(n-1) +Self(n-2) +2: n in [1..30]]; // G. C. Greubel, Apr 22 2021

Mathematica

Join[{1}, RecurrenceTable[{a[1] == 1, a[2] == 4, a[n] == 2 a[n - 1] + a[n - 2] + 2}, a[n], {n, 30}]] (* Harvey P. Dale, Jul 27 2011 *)
Join[{1}, CoefficientList[Series[(x + 1)/((x - 1) (x^2 + 2 x - 1)), {x, 0, 40}], x]] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2012 *)
Join[{1}, Fibonacci[Range[2, 35], 2] -1] (* G. C. Greubel, Apr 22 2021 *)
Join[{1}, LinearRecurrence[{3, -1, -1}, {1, 4, 11}, 20]] (* Eric W. Weisstein, Aug 01 2023 *)

PARI

a(n)=polcoeff(1+x*(1+x)/(1-3*x+x^2+x^3)+x*O(x^n),n) \\ Paul D. Hanna

Sage

[1]+[lucas_number1(n,2,-1) -1 for n in (2..35)] # G. C. Greubel, Apr 22 2021

Formula

a(n) = A000129(n) - 1, n > 1.

a(n) = ((1+sqrt(2))^n - (1-sqrt(2))^n)/(2*sqrt(2))-1 for n > 1, a(1)=1.

G.f.: 1 + x*(1+x)/( (1-x)*(1-2*x-x^2) ). - Simon Plouffe in his 1992 dissertation.

a(n) = 3*a(n-1) - a(n-2) - a(n-3). - Toby Gottfried, Mar 08 2013

(1, 4, 11, 28, ...) = (1, 2, 2, 2, ...) * the Pell sequence starting (1, 2, 5, 12, 29, ...); such that, for example: a(5) = (2, 2, 2, 1) dot (1, 2, 5, 12) = (2 + 4 + 10 + 12) = 48. - Gary W. Adamson May 21 2013

E.g.f.: 1 + exp(x)*(2*(cosh(sqrt(2)*x) - 1) + sqrt(2)*sinh(sqrt(2)*x))/2. - Stefano Spezia, Jun 26 2022

Extensions

Additional comments from Barry E. Williams