A005052 -id:A005052 - cp's OEIS frontend

A081250 Numbers k such that A081249(m)/m^2 has a local minimum for m = k.

1, 3, 11, 33, 101, 303, 911, 2733, 8201, 24603, 73811, 221433, 664301, 1992903, 5978711, 17936133, 53808401, 161425203, 484275611, 1452826833, 4358480501, 13075441503, 39226324511, 117678973533, 353036920601, 1059110761803

Offset: 0

Klaus Brockhaus, Mar 17 2003

The limit of the local minima, lim_{n->infinity} A081249(n)/n^2 = 1/10. For local maxima cf. A081251.

			11 is a term since A081249(10)/10^2 = 11/100 = 0.110, A081249(11)/11^2 = 13/121 = 0.107, A081249(12)/12^2 = 16/144 = 0.111.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Klaus Brockhaus, Illustration for A081134, A081249, A081250 and A081251
Index entries for linear recurrences with constant coefficients, signature (3,1,-3).

Cf. A005052, A062107, A081134, A081249, A081251.

List([0..30], n-> (5*3^n +(-1)^n -2)/4); # G. C. Greubel, Jul 14 2019

[Floor(3^n*5/4): n in [0..30]]; // Vincenzo Librandi, Jun 10 2011

a[0]:=1:a[1]:=3:for n from 2 to 50 do a[n]:=2*a[n-1]+3*a[n-2]+2 od: seq(a[n], n=0..30); # Zerinvary Lajos, Apr 28 2008

Floor[5*3^Range[0, 30]/4] (* Wesley Ivan Hurt, Mar 30 2017 *)

vector(30, n, n--; (5*3^n +(-1)^n -2)/4) \\ G. C. Greubel, Jul 14 2019

[(5*3^n +(-1)^n -2)/4 for n in (0..30)] # G. C. Greubel, Jul 14 2019

a(n) = floor(3^n*5/4).
G.f.: x*(1+x^2)/((1-x)*(1+x)*(1-3*x)).
a(n) = 3*a(n-1) + 1*a(n-2) - 3*a(n-3).
a(n) = (5*3^n + (-1)^n - 2)/4. - Paul Barry, May 19 2003
a(n) = a(n-2) + 10*3^(n-2) for n > 1.
a(n+2) - a(n) = A005052(n).
a(2*n) = Sum_{j=1..n+1} A062107(2*j).
a(2*n+1) = Sum_{j=1..n+1} A062107(2*j+1).
With a leading 0, this is a(n) = (5*3^n - 6 + 4*0^n - 3*(-1)^n)/12, the binomial transform of A084183. - Paul Barry, May 19 2003
Convolution of 3^n and {1, 0, 2, 0, 2, 0, ...}. a(n) = Sum_{k=0..n} ((1 + (-1)^k) - 0^k)*3^(n-k) = Sum_{k=0..n} ((1 + (-1)^(n-k)) - 0^(n-k))3^k. - Paul Barry, Jul 19 2004
a(n) = 2*a(n-1) + 3*a(n-2) + 2, a(0)=1, a(1)=3. - Zerinvary Lajos, Apr 28 2008

Offset changed from 1 to 0 by Vincenzo Librandi, Jun 10 2011

A062107 Diagonal of table A062104.

Original entry on oeis.org

0, 1, 3, 10, 30, 90, 270, 810, 2430, 7290, 21870, 65610, 196830, 590490, 1771470, 5314410, 15943230, 47829690, 143489070, 430467210, 1291401630, 3874204890, 11622614670, 34867844010, 104603532030, 313810596090, 941431788270

Offset: 1

Antti Karttunen, May 30 2001

nonn

Index entries for linear recurrences with constant coefficients, signature (3).

Except for initial terms, same as A005052.

Maple
```
[seq(ChessPawnTriangle(j,j),j=1..50)];
```

a(n) = 10*3^(n-4) for n >= 4.
From Paul Barry, Oct 15 2004: (Start)
G.f.: x^2(1+x^2)/(1-3x);
a(n) = Sum_{k=0..n-2} 3^(n-k-2)binomial(1, k/2)(1+(-1)^k)/2. (End)

A258597 a(n) = 13*3^n.

Original entry on oeis.org

13, 39, 117, 351, 1053, 3159, 9477, 28431, 85293, 255879, 767637, 2302911, 6908733, 20726199, 62178597, 186535791, 559607373, 1678822119, 5036466357, 15109399071, 45328197213, 135984591639, 407953774917, 1223861324751, 3671583974253, 11014751922759

Offset: 0

Vincenzo Librandi, Jun 05 2015

Also maximum leaf number of the (n+3)-Apollonian network for n >= 0. - Eric W. Weisstein, Jan 17 2018

Eric Weisstein's World of Mathematics, Apollonian Network
Eric Weisstein's World of Mathematics, Maximum Leaf Number
Index entries for linear recurrences with constant coefficients, signature (3).

Cf. k*3^n: A000244 (k=1,3,9), A008776 (k=2,6), A003946 (k=4), A005030 (k=5), A005032 (k=7), A005051 (k=8), A005052 (k=10), A120354 (k=11), A003946 (k=12), this sequence (k=13), A258598 (k=17), A176413 (k=19).

Magma
```
[13*3^n: n in [0..30]];
```

Table[13 3^n, {n, 0, 30}]
13 3^Range[0, 20] (* Eric W. Weisstein, Jan 17 2018 *)
LinearRecurrence[{3}, {13}, 20] (* Eric W. Weisstein, Jan 17 2018 *)
CoefficientList[Series[13/(1 - 3 x), {x, 0, 20}], x] (* Eric W. Weisstein, Jan 17 2018 *)

G.f.: 13/(1-3*x).
a(n) = 3*a(n-1).
a(n) = 13*A000244(n).
E.g.f.: 13*exp(3*x). - Elmo R. Oliveira, Aug 16 2024

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A081250 Numbers k such that A081249(m)/m^2 has a local minimum for m = k.

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A062107 Diagonal of table A062104.

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A258597 a(n) = 13*3^n.

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