

A061777


Start with a single triangle; at nth generation add a triangle at each vertex, allowing triangles to overlap; sequence gives total population of triangles at nth generation.


16



1, 4, 10, 22, 40, 70, 112, 178, 268, 406, 592, 874, 1252, 1822, 2584, 3730, 5260, 7558, 10624, 15226, 21364, 30574, 42856, 61282, 85852, 122710, 171856, 245578, 343876, 491326, 687928, 982834, 1376044, 1965862, 2752288, 3931930, 5504788
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OFFSET

0,2


COMMENTS

From the definition, assign label value "1" to an origin triangle; at nth generation add a triangle at each vertex. Each nonoverlapping triangle will have the same label value as that of the predecessor triangle to which it is connected; for the overlapping ones, the label value will be the sum of the label values of predecessors. a(n) is the sum of all label values at the nth generation. The triangle count is A005448. See illustration. For n >= 1, (a(n)  a(n1))/3 is A027383.  Kival Ngaokrajang, Sep 05 2014


REFERENCES

R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 56.


LINKS

Table of n, a(n) for n=0..36.
Kival Ngaokrajang, Illustration of initial terms
R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 56. [Scanned photocopy of pages 5, 6 only, with annotations by R. K. Guy and N. J. A. Sloane]
Index entries for linear recurrences with constant coefficients, signature (2,1,4,2).


FORMULA

From Colin Barker, May 08 2012: (Start)
a(n) = 21*2^(n/2)  6*n  20 if n is even.
a(n) = 30*2^((n1)/2)  6*(n  1)  26 if n is odd.
a(n) = 2*a(n1) + a(n2)  4*a(n3) + 2*a(n4).
G.f.: (1 + 2*x)*(1 + x^2)/((1  x)^2*(1  2*x^2)). (End)
From Robert Israel, Sep 14 2014: (Start)
a(n) = 20  6*n + (21 + 15*sqrt(2))*sqrt(2)^(n2) + (21  15*sqrt(2))*(sqrt(2))^(n2).
a(n) = 2*a(n2) + ((3*n2)/(3*n5))*(a(n1)2*a(n3)). (End)
E.g.f.: 21*cosh(sqrt(2)*x) + 15*sqrt(2)*sinh(sqrt(2)*x)  2*exp(x)*(10 + 3*x).  Stefano Spezia, Aug 13 2022


MAPLE

seq(`if`(n::even, 21*2^(n/2)  6*n20, 30*2^((n1)/2)6*n20), n=0..100); # Robert Israel, Sep 14 2014


MATHEMATICA

Table[If[EvenQ[n], 21 2^(n/2)6n20, 30 2^((n1)/2)6(n1)26], {n, 0, 40}] (* Harvey P. Dale, Nov 06 2011 *)


PROG

(PARI) a(n)=if(n%2, 30, 21)<<(n\2)  6*n  20 \\ Charles R Greathouse IV, Sep 19 2014


CROSSREFS

Partial sums of A061776.
Cf. A005448, A027383.
Sequence in context: A008248 A301243 A177736 * A298030 A155369 A155404
Adjacent sequences: A061774 A061775 A061776 * A061778 A061779 A061780


KEYWORD

nonn,nice,easy


AUTHOR

N. J. A. Sloane, R. K. Guy, Jun 23 2001


EXTENSIONS

Corrected by T. D. Noe, Nov 08 2006


STATUS

approved



