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

Author

R. K. Guy

Keywords

• nonn

Links

• 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]

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

Programs

GAP

List([0..40], 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

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 *)

Maxima

A045949(n):=floor(n*(14*n^2+9*n+2)/4)$
makelist(A045949(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

PARI

{a(n) = (28*n^3 +18*n^2 +4*n -1 +(-1)^n)/8}; \\ G. C. Greubel, Apr 05 2019

R

floor(1:25*(14*(1:25)^2+9*(1:25)+2)/4) # Christian N. K. Anderson, Apr 27 2013

Sage

[(28*n^3 +18*n^2 +4*n -1 +(-1)^n)/8 for n in (0..40)] # G. C. Greubel, Apr 05 2019

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

Extensions

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