OFFSET
0,2
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..2385
Karl Dilcher and Larry Ericksen, Polynomials and algebraic curves related to certain binary and b-ary overpartitions, Ramanujan J. (2025) Vol. 68, No. 44. See also arXiv:2405.12024 [math.CO], 2024. See p. 10.
Sergi Elizalde, Rigoberto Flórez, and José Luis Ramírez, Enumerating symmetric peaks in non-decreasing Dyck paths, Ars Mathematica Contemporanea (2021).
David Eppstein, Non-crossing Hamiltonian Paths and Cycles in Output-Polynomial Time, arXiv:2303.00147 [cs.CG], 2023, p. 20.
Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, Counting Subwords in Non-Decreasing Dyck Paths, J. Int. Seq. (2025) Vol. 28, Art. No. 25.1.6. See pp. 6, 15, 17, 19.
Valentin Ovsienko, Shadow sequences of integers, from Fibonacci to Markov and back, arXiv:2111.02553 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).
FORMULA
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4) for n>3, a(0)=1, a(1)=4, a(2)=13, a(3)=40.
a(n) = 3*a(n-1) - a(n-2) + A001519(n) for n>1, a(0)=1, a(1)=4.
a(n) = A238731(n+1, 1).
a(n) = -A124037(n+1, 1).
a(n) = (-1)^n*A126126(n+1, 1).
a(n) = ( (3 + sqrt(5))^(1+n)*(8 - (1 - sqrt(5))*(13 + 5*n)) + (3 - sqrt(5))^(1+n)*(8 - (1 + sqrt(5))*(13 + 5*n)) ) / (25*2^(2+n)). - Bruno Berselli, Mar 06 2014
From Philippe Deléham, Mar 06 2014: (Start)
0 = 2 + a(n)*(a(n+1) - a(n+3)) + a(n+1)*(-6*a(n+1) + 12*a(n+2)) + a(n+2)*(-6*a(n+2) + a(n+3)) for all n in Z. - Michael Somos, Nov 23 2021
E.g.f.: exp(3*x/2)*(5*(5 + 4*x)*cosh(sqrt(5)*x/2) + sqrt(5)*(17 + 10*x)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Mar 04 2025
EXAMPLE
a(0) = 1*1 = 1;
a(1) = 1*3 + 1*1 = 4;
a(2) = 1*8 + 1*3 + 2*1 = 13;
a(3) = 1*21 + 1*8 + 2*3 + 5*1 = 40;
a(4) = 1*55 + 1*21 + 2*8 + 5*3 + 13*1 = 120; etc. (from first recurrence formula).
a(0) = 3*0 - 0 + 1 = 1;
a(1) = 3*1 - 0 + 1 = 4;
a(2) = 3*4 - 1 + 2 = 13;
a(3) = 3*13 - 4 + 5 = 40;
a(4) = 3*40 - 13 + 13 = 120; etc (from second recurrence formula).
G.f. = 1 + 4*x + 13*x^2 + 40*x^3 + 120*x^4 + 354*x^5 + 1031*x^6 + ... - Michael Somos, Nov 23 2021
MATHEMATICA
LinearRecurrence[{6, -11, 6, -1}, {1, 4, 13, 40}, 30] (* Bruno Berselli, Mar 06 2014 *)
a[ n_] := If[n < 0, SeriesCoefficient[ x^3*(2 - x)/(1 - 3*x + x^2)^2, {x, 0, -n}], SeriesCoefficient[ (1 - 2*x)/(1 - 3*x + x^2)^2, {x, 0, n}]]; (* Michael Somos, Nov 23 2021 *)
PROG
(PARI) {a(n) = if(n<0, polcoeff( x^3*(2-x)/(1-3*x+x^2)^2 + x*O(x^-n), -n), polcoeff( (1-2*x)/(1-3*x+x^2)^2 + x*O(x^n), n))}; /* Michael Somos, Nov 23 2021 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Philippe Deléham, Mar 05 2014
STATUS
approved