A045949 Number of equilateral triangles formed out of matches that can be found in a hexagonal chunk of side length n in hexagonal array of matchsticks.

0, 6, 38, 116, 262, 496, 840, 1314, 1940, 2738, 3730, 4936, 6378, 8076, 10052, 12326, 14920, 17854, 21150, 24828, 28910, 33416, 38368, 43786, 49692, 56106, 63050, 70544, 78610, 87268, 96540, 106446, 117008, 128246, 140182, 152836, 166230, 180384, 195320, 211058

Offset

0,2

Links

Table of n, a(n) for n=0..39.

N. J. A. Sloane, Illustration of a(1)=6

Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1). [From R. J. Mathar, Sep 03 2010]

Formula

a(n) = floor(n*(14*n^2 + 9*n + 2)/4).

From R. J. Mathar, Sep 03 2010: (Start)

a(n) = +3*a(n-1) -2*a(n-2) -2*a(n-3) +3*a(n-4) -a(n-5).

G.f.: 2*x*(3+10*x+7*x^2+x^3) / ( (1+x)*(1-x)^4 ).

a(n) = (28*n^3 + 18*n^2 + 4*n - 1 + (-1)^n)/8. (End)

a(n) = A033581(n) + A307253(n). - John King, Apr 04 2019

E.g.f.: (x*(25 + 51*x + 14*x^2)*exp(x) - sinh(x))/4. - G. C. Greubel, Apr 05 2019

Mathematica

LinearRecurrence[{3, -2, -2, 3, -1}, {0, 6, 38, 116, 262}, 40] (* or *) CoefficientList[Series[(2*x*(x*(x+2)*(x+5)+3))/((x-1)^4*(x+1)), {x, 0, 40}], x] (* Harvey P. Dale, Jun 11 2011 *)

Prog

(Maxima) A045949(n):=floor(n*(14*n^2+9*n+2)/4)$
makelist(A045949(n), n, 0, 30); /* Martin Ettl, Nov 03 2012 */
(R) floor(1:25*(14*(1:25)^2+9*(1:25)+2)/4) # Christian N. K. Anderson, Apr 27 2013
(PARI) {a(n) = (28*n^3 +18*n^2 +4*n -1 +(-1)^n)/8}; \\ G. C. Greubel, Apr 05 2019
(Magma) [(28*n^3 +18*n^2 +4*n -1 +(-1)^n)/8: n in [0..40]]; // G. C. Greubel, Apr 05 2019
(Sage) [(28*n^3 +18*n^2 +4*n -1 +(-1)^n)/8 for n in (0..40)] # G. C. Greubel, Apr 05 2019
(GAP) List([0..40], n-> (28*n^3 +18*n^2 +4*n -1 +(-1)^n)/8) # G. C. Greubel, Apr 05 2019

Crossrefs

See A008893 for a related sequence.

For hexagons, the number of matches required is A045945, the number of size=1 triangles is A033581, the larger triangles is A307253 and the total number is A045949. For the analogs for triangles see A045943 and for stars see A045946. - John King, Apr 05 2019

Sequence in context: A060454 A060452 A229620 * A189492 A281803 A288030

Adjacent sequences: A045946 A045947 A045948 * A045950 A045951 A045952

Keyword

nonn

Author

R. K. Guy

Extensions

Edited by N. J. A. Sloane, May 29 2012

Status

approved