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 A064412 At stage 1, start with a unit equilateral equiangular triangle. At each successive stage add 3*(n-1) new triangles around outside with edge-to-edge contacts. Sequence gives number of triangles (regardless of size) at n-th stage. 7
 1, 5, 14, 32, 60, 103, 160, 238, 335, 459, 606, 786, 994, 1241, 1520, 1844, 2205, 2617, 3070, 3580, 4136, 4755, 5424, 6162, 6955, 7823, 8750, 9758, 10830, 11989, 13216, 14536, 15929, 17421, 18990, 20664, 22420, 24287, 26240, 28310, 30471, 32755, 35134, 37642 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Number of unit triangles at n-th stage = 3n(n-1)/2 + 1, A005448. REFERENCES Anthony Gardiner, "Mathematical Puzzling," Dover Publications, Inc., Mineola, NY., 1987, page 88. LINKS N. J. A. Sloane, Illustration of initial terms Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,-2,0,2,-1). FORMULA G.f.: (1+x+x^2)(1+2x+x^2+3x^3)/((1-x)^2(1-x^2)(1-x^4)). a(2n+1) = (7n^3+12n^2+7n+2)/2; a(2n) = (28n^3+6n^2+4n+1+(-1)^(n+1))/8. - Len Smiley, Oct 07 2001 a(n) = (14*n^3+6*n^2+5*n+7+3*(n-1)*(-1)^n-2*((-1)^((2*n-1+(-1)^n)/4)+(-1)^((6*n-1+(-1)^n)/4)))/32. - Luce ETIENNE, Jun 27 2014 EXAMPLE a(4) = 32: 19 triangles of side 1, 10 of side 2 and 3 of side 3. MAPLE A064412:=n->(14*n^3+6*n^2+5*n+7+3*(n-1)*(-1)^n-2*((-1)^((2*n-1+(-1)^n)/4)+(-1)^((6*n-1+(-1)^n)/4)))/32; seq(A064412(n), n=1..30); # Wesley Ivan Hurt, Jun 27 2014 MATHEMATICA CoefficientList[Series[(1 + x + x^2) (1 + 2 x + x^2 + 3 x^3)/((1 - x)^2 (1 - x^2) (1 - x^4)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jun 27 2014 *) LinearRecurrence[{2, 0, -2, 2, -2, 0, 2, -1}, {1, 5, 14, 32, 60, 103, 160, 238}, 50] (* Harvey P. Dale, Apr 12 2016 *) PROG (PARI) a(n)=polcoeff(x*(1+x+x^2)*(1+2*x+x^2+3*x^3)/((1-x)^2*(1-x^2)*(1-x^4))+x*O(x^n), n) CROSSREFS Cf. A056640. Sequence in context: A070134 A295344 A219902 * A211803 A299275 A266759 Adjacent sequences:  A064409 A064410 A064411 * A064413 A064414 A064415 KEYWORD nonn,easy AUTHOR Robert G. Wilson v, Sep 29 2001 STATUS approved

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Last modified February 19 10:03 EST 2020. Contains 332041 sequences. (Running on oeis4.)