A219233 Alternating row sums of Riordan triangle A110162.

1, -3, 7, -18, 47, -123, 322, -843, 2207, -5778, 15127, -39603, 103682, -271443, 710647, -1860498, 4870847, -12752043, 33385282, -87403803, 228826127, -599074578, 1568397607, -4106118243, 10749957122, -28143753123, 73681302247, -192900153618, 505019158607

Offset: 0

Author

Wolfdieter Lang, Nov 16 2012

Keywords

• sign
• easy

Comments

If a(0) is put to 2 instead of 1 this becomes a(n) = (-1)^n*A005248(n), n >= 0. These are then the alternating row sums of triangle A127677.

Also abs(a(n)) is the number of rounded area of pentagon or pentagram in series arrangement. - Kival Ngaokrajang, Mar 27 2013

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Richard M. Low and Ardak Kapbasov, Non-Attacking Bishop and King Positions on Regular and Cylindrical Chessboards, Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.1, Table 8.
• Kival Ngaokrajang, Pentagram for n = 1..6
• Eric Weisstein's World of Mathematics, Pentagram
• Index entries for linear recurrences with constant coefficients, signature (-3,-1).

Crossrefs

Cf. A099837 (row sums of A110162).

Cf. A000032, A000045, A001254, A005248, A075150, A127677.

Programs

Magma

A219233:= func< n | n eq 0 select 1 else (-1)^n*Lucas(2*n) >; // G. C. Greubel, Jun 13 2025

Mathematica

A219233[n_]:= (-1)^n*LucasL[2*n] - Boole[n==0]; (* G. C. Greubel, Jun 13 2025 *)

PARI

Vec((1-x^2)/(1+3*x+x^2) + O(x^40)) \\ Colin Barker, Oct 14 2015

SageMath

def A219233(n): return (-1)**n*lucas_number2(2*n,1,-1) - int(n==0) # G. C. Greubel, Jun 13 2025

Formula

a(0) = 1 and a(n) = (-1)^n*(F(2*(n+1)) - F(2*(n-1))) = (-1)^n*L(2*n), n>=1, with F=A000045 (Fibonacci) and L=A000032 (Lucas).

O.g.f.: (1-x^2)/(1+3*x+x^2).

G.f.: (W(0) -6)/(5*x) -1 , where W(k) = 5*x*k + x + 6 - 6*x*(5*k-9)/W(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Aug 19 2013

From Colin Barker, Oct 14 2015: (Start)

a(n) = -3*a(n-1) - a(n-2) for n>2.

a(n) = (1/2*(-3-sqrt(5)))^n + (1/2*(-3+sqrt(5)))^n for n>0. (End)

E.g.f.: 2*exp(-3*x/2)*cosh(sqrt(5)*x/2) - 1. - Stefano Spezia, Dec 26 2021

From G. C. Greubel, Jun 13 2025: (Start)

a(-n) = a(n).

a(n) = (-1)^n*A001254(n) - 2 - [n=0] = A075150(n) - 2 - [n=0]. (End)