A086462 -id:A086462 - cp's OEIS frontend

A211019 Triangle read by rows: T(n,k) = number of squares and rectangles of area 2^(k-1) after 2^n stages in the toothpick structure of A139250, divided by 4, n>=1, k>=1, assuming the toothpicks have length 2.

0, 0, 1, 2, 3, 1, 10, 13, 3, 1, 42, 53, 13, 3, 1, 170, 213, 53, 13, 3, 1, 682, 853, 213, 53, 13, 3, 1, 2730, 3413, 853, 213, 53, 13, 3, 1, 10922, 13653, 3413, 853, 213, 53, 13, 3, 1, 43690, 54613, 13653, 3413, 853, 213, 53, 13, 3, 1, 174762, 218453

Offset: 1

Omar E. Pol, Sep 24 2012

All internal regions in the toothpick structure are squares and rectangles.

			Triangle begins:
0;
0,         1;
2,         3,     1;
10,       13,     3,    1;
42,       53,    13,    3,   1;
170,     213,    53,   13,   3,   1;
682,     853,   213,   53,  13,   3,  1;
2730,   3413,   853,  213,  53,  13,  3,  1;
10922, 13653,  3413,  853, 213,  53, 13,  3, 1;
43690, 54613, 13653, 3413, 853, 213, 53, 13, 3, 1;

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Row sums give 0 together with A014825.

Cf. A002450, A020988, A072197, A086462, A139250, A160124, A211016, A211017, A211018.

T(n,k) = A211016(n,k)/4.

T(n,1) = A020988(n-2), n>=2.

A321421 a(n) = 10*(4^n - 1)/3 + 1.

Original entry on oeis.org

1, 11, 51, 211, 851, 3411, 13651, 54611, 218451, 873811, 3495251, 13981011, 55924051, 223696211, 894784851, 3579139411, 14316557651, 57266230611, 229064922451, 916259689811, 3665038759251, 14660155037011, 58640620148051, 234562480592211, 938249922368851

Offset: 0

Paul Curtz, Nov 09 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A000302, A002450, A010727.

List([0..25],n->10*(4^n-1)/3+1); # Muniru A Asiru, Nov 10 2018

seq(coeff(series((1+6*x)/((1-x)*(1-4*x)),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Nov 10 2018

a[n_]:=10*(4^n - 1)/3 + 1 ; Array[a, 20, 0] (* or *)
CoefficientList[Series[-((7 E^x)/3) + (10 E^(4 x))/3 , {x, 0, 20}], x]*Table[n!, {n, 0, 20}] (* Stefano Spezia, Nov 10 2018 *)
LinearRecurrence[{5,-4},{1,11},30] (* Harvey P. Dale, Aug 22 2020 *)

Vec((1 + 6*x) / ((1 - x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Nov 10 2018

a(n) = 4*a(n-1) + 7, a(0) = 1 for n > 0.
a(n) = 5*a(n-1) - 4*a(n-2), a(0) = 1, a(1) = 11, n > 1.
a(n) = a(n-1) + 10*4^(n-1), a(0) = 1, n > 0.
a(n) = A086462(n) + 1 for n > 0. - Michel Marcus, Nov 09 2018
G.f.: (1 + 6*x) / ((1 - x)*(1 - 4*x)). - Colin Barker, Nov 10 2018
E.g.f.: (-7*exp(x) + 10*exp(4*x))/3. - Stefano Spezia, Nov 10 2018
a(n) = 10*A002450(n) + 1. - Omar E. Pol, Nov 10 2018

More terms from Colin Barker, Nov 10 2018

cp's OEIS Frontend

A211019 Triangle read by rows: T(n,k) = number of squares and rectangles of area 2^(k-1) after 2^n stages in the toothpick structure of A139250, divided by 4, n>=1, k>=1, assuming the toothpicks have length 2.

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A321421 a(n) = 10*(4^n - 1)/3 + 1.

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