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A219233 Alternating row sums of Riordan triangle A110162. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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).

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)

PROG

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

CROSSREFS

Cf. A099837 (row sums of A110162).

Sequence in context: A173765 A027969 A027971 * A211276 A018028 A045994

Adjacent sequences:  A219230 A219231 A219232 * A219234 A219235 A219236

KEYWORD

sign,easy

AUTHOR

Wolfdieter Lang, Nov 16 2012

STATUS

approved

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Last modified March 25 01:17 EDT 2019. Contains 321450 sequences. (Running on oeis4.)