A005030 -id:A005030 - cp's OEIS frontend

A060816 a(0) = 1; a(n) = (5*3^(n-1) - 1)/2 for n > 0.

1, 2, 7, 22, 67, 202, 607, 1822, 5467, 16402, 49207, 147622, 442867, 1328602, 3985807, 11957422, 35872267, 107616802, 322850407, 968551222, 2905653667, 8716961002, 26150883007, 78452649022, 235357947067, 706073841202

Offset: 0

Jason Earls, Apr 29 2001

From Erich Friedman's math magic page 2nd paragraph under "Answers" section.
Let A be the Hessenberg matrix of order n, defined by: A[1,j] = 1, A[i,i] = 2,(i>1),  A[i,i-1] = -1, and A[i,j] = 0 otherwise. Then, for n >= 1, a(n) = (-1)^n*charpoly(A,-1). - Milan Janjic, Jan 26 2010
If n > 0 and H = hex number (A003215), then 9*H(a(n)) - 2 = H(a(n+1)), for example 9*H(2) - 2 = 9*19 - 2 = 169 = H(7). For n > 2, this is a subsequence of A017209, see formula. - Klaus Purath, Mar 31 2021
Consider the Tower of Hanoi puzzle of order n (i.e., with n rings to be moved). Label each ring with a distinct symbol from an alphabet of size n. Construct words by performing moves according to the standard rules of the puzzle, recording the corresponding symbol each time a ring is moved. To ensure finiteness, we forbid returning to any previously encountered state. Additionally, we impose the constraint that the same ring cannot be moved twice in succession. Then, a(n) is the number of distinct words that can be formed under these rules. - Thomas Baruchel, Jul 22 2025

Harry J. Smith, Table of n, a(n) for n = 0..200
Erich Friedman, Math. Magic
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A005030 (first differences), A244762 (partial sums).

Cf. A003215, A003462, A007051, A008343, A017209, A057198, A134931.

LinearRecurrence[{4,-3},{1,2,7},30] (* Harvey P. Dale, Nov 15 2022 *)

A060816(n)=if(n, 3^n*5\6, 1) \\ M. F. Hasler, Apr 06 2019

The following is a summary of formulas added over the past 18 years.
a(n) = A057198(n) - 1.
a(n) = 3*a(n-1) + 1; with a(0)=1, a(1)=2. - Jason Earls, Apr 29 2001
From Henry Bottomley, May 01 2001: (Start)
For n>0, a(n) = a(n-1)+5*3^(n-2) = a(n-1)+A005030(n-2).
For n>0, a(n) = (5*A003462(n)+1)/3. (End)
From Colin Barker, Apr 24 2012: (Start)
a(n) = 4*a(n-1) - 3*a(n-2) for n > 2.
G.f.: (1-2*x+2*x^2)/((1-x)*(1-3*x)). (End)
a(n+1) = A134931(n) + 1. - Philippe Deléham, Apr 14 2013
For n > 0, A008343(a(n)) = 0. - Dmitry Kamenetsky, Feb 14 2017
For n > 0, a(n) = floor(3^n*5/6). - M. F. Hasler, Apr 06 2019
From Klaus Purath, Mar 31 2021: (Start)
a(n) = A017209(a(n-2)), n > 2.
a(n) = 2 + Sum_{i = 0..n-2} A005030(i).
a(n+1)*a(n+2) = a(n)*a(n+3) + 20*3^n, n > 1.
a(n) = 3^n - A007051(n-1). (End)
E.g.f.: (5*exp(3*x) - 3*exp(x) + 4)/6. - Stefano Spezia, Aug 28 2023

Edited by M. F. Hasler, Apr 06 2019 and by N. J. A. Sloane, Apr 09 2019

A134931 a(n) = (5*3^n-3)/2.

Original entry on oeis.org

1, 6, 21, 66, 201, 606, 1821, 5466, 16401, 49206, 147621, 442866, 1328601, 3985806, 11957421, 35872266, 107616801, 322850406, 968551221, 2905653666, 8716961001, 26150883006, 78452649021, 235357947066, 706073841201, 2118221523606

Offset: 0

Rolf Pleisch, Jan 29 2008

Numbers n where the recurrence s(0)=1, if s(n-1) >= n then s(n) = s(n-1) - n else s(n) = s(n-1) + n produces s(n)=0. - Hugo Pfoertner, Jan 05 2012
A046901(a(n)) = 1. - Reinhard Zumkeller, Jan 31 2013
Binomial transform of A146523: (1, 5, 10, 20, 40, ...) and double binomial transform of A010685: (1, 4, 1, 4, 1, 4, ...). - Gary W. Adamson, Aug 25 2016
Also the number of maximal cliques in the (n+1)-Hanoi graph. - Eric W. Weisstein, Dec 01 2017
a(n) is the least k such that f(a(n-1)+1) + ... + f(k) > f(a(n-2)+1) + ... + f(a(n-1)) for n > 1, where f(n) = 1/(n+1). Because Sum_{k=1..5*3^(n-1)} 1/(a(n)+3*k-1) + 1/(a(n)+3*k) + 1/(a(n)+3*k+1) - 1/((a(n)+1+5*3^n)*5*3^(n-1)) < Sum_{k=1..5*3^(n-1)} 1/(a(n-1)+k+1) < Sum_{k=1..5*3^(n-1)} 1/(a(n)+3*k-1) + 1/(a(n)+3*k) + 1/(a(n)+3*k+1), we have 1 < 1/3 + 1/4 + ... + 1/7 < 1/8 + 1/9 + ... + 1/22 < ... . - Jinyuan Wang, Jun 15 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Hanoi Graph
Eric Weisstein's World of Mathematics, Maximal Clique
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A003462, A007051, A034472, A024023, A067771, A029858.

Cf. A005030, A010685, A046901, A060816, A116952, A146523, A225918.

[(5*3^n-3)/2: n in [0..30]]; // Vincenzo Librandi, Jun 05 2011

seq((5*3^n-3)/2, n= 0..25); # Gary Detlefs, Jun 22 2010

a=1; lst={a}; Do[a=a*3+3; AppendTo[lst,a], {n,0,100}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 25 2008 *)
Table[(5 3^n - 9)/6, {n, 20}] (* Eric W. Weisstein, Dec 01 2017 *)
(5 3^Range[20] - 9)/6 (* Eric W. Weisstein, Dec 01 2017 *)
LinearRecurrence[{4, -3}, {1, 6}, 20] (* Eric W. Weisstein, Dec 01 2017 *)
CoefficientList[Series[(1 + 2 x)/(1 - 4 x + 3 x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017 *)

a(n) = (5*3^n-3)/2; /* Joerg Arndt, Apr 14 2013 */

a(n) = 3*(a(n-1) + 1), with a(0)=1.
From R. J. Mathar, Jan 31 2008: (Start)
O.g.f.: (5/2)/(1-3*x) - (3/2)/(1-x).
a(n) = (A005030(n) - 3)/2. (End)
a(n) = A060816(n+1) - 1. - Philippe Deléham, Apr 14 2013
E.g.f.: exp(x)*(5*exp(2*x) - 3)/2. - Stefano Spezia, Aug 28 2023

More terms from Vladimir Joseph Stephan Orlovsky, Dec 25 2008

A200251 T(n,k)=Number of 0..k arrays x(0..n-1) of n elements with each no smaller than the sum of its previous elements modulo (k+1).

Original entry on oeis.org

2, 3, 3, 4, 6, 5, 5, 10, 12, 8, 6, 15, 26, 24, 13, 7, 21, 45, 69, 48, 21, 8, 28, 75, 135, 181, 96, 34, 9, 36, 112, 267, 405, 476, 192, 55, 10, 45, 164, 448, 951, 1215, 1252, 384, 89, 11, 55, 225, 750, 1792, 3387, 3645, 3292, 768, 144, 12, 66, 305, 1125, 3434, 7168, 12063

Offset: 1

R. H. Hardin Nov 15 2011

Table starts
...2....3.....4.....5......6.......7.......8........9.......10........11
...3....6....10....15.....21......28......36.......45.......55........66
...5...12....26....45.....75.....112.....164......225......305.......396
...8...24....69...135....267.....448.....750.....1125.....1690......2376
..13...48...181...405....951....1792....3434.....5625.....9365.....14256
..21...96...476..1215...3387....7168...15724....28125....51895.....85536
..34..192..1252..3645..12063...28672...71970...140625...287570....513216
..55..384..3292.10935..42963..114688..329455...703125..1593535...3079296
..89..768..8657.32805.153015..458752.1508139..3515625..8830385..18475776
.144.1536.22765.98415.544971.1835008.6903702.17578125.48932530.110854656

			Some solutions for n=7 k=6
..1....2....4....0....1....0....4....0....1....4....2....3....1....3....3....3
..3....5....6....3....2....6....5....4....1....5....5....4....2....5....3....4
..4....5....5....6....5....6....5....6....6....2....1....0....5....6....6....1
..6....5....3....2....6....5....3....6....2....5....3....0....3....0....6....1
..2....5....6....6....0....6....4....3....5....2....5....5....4....2....5....5
..6....2....5....6....2....2....2....5....2....4....5....5....4....5....6....2
..5....4....1....3....4....4....2....5....3....3....1....3....6....1....6....5

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A000045(n+2)
Column 2 is A003945
Column 3 is A099234(n+1)
Column 4 is A005030(n-1)
Column 6 is A002042(n-1)
Row 2 is A000217(n+1)

Empirical: T(n,2k) = (2*k+1)*(k+1)^(n-1)

A338486 Numbers n whose symmetric representation of sigma(n) consists of 3 regions with maximum width 2.

Original entry on oeis.org

15, 35, 45, 70, 77, 91, 110, 130, 135, 143, 154, 170, 182, 187, 190, 209, 221, 225, 238, 247, 266, 286, 299, 322, 323, 350, 374, 391, 405, 418, 437, 442, 493, 494, 506, 527, 550, 551, 572, 589, 598, 638, 646, 650, 667, 682, 703, 713, 748, 754, 782, 806, 814, 836, 850

Offset: 1

Hartmut F. W. Hoft, Oct 30 2020

nonn

This sequence is a subsequence of A279102. The definition of the sequence excludes squares of primes, A001248, since the 3 regions of their symmetric representation of sigma have width 1 (first column in the irregular triangle of A247687).
Table of numbers in this sequence arranged by the number of prime factors, counting multiplicities:
     2     3     4      5      6      7  ...
  ------------------------------------------
    15    45   135    405   1215   3645
    35    70   225   1125   5625    ...
    77   110   350   1750   8750 744795
    91   130   550   2584    ...    ...
   143   154   572   2750  85455
   187   170   650   3128    ...
   209   182   748   3250
   221   190   836   3496
   247   238   850   3944
   299   266   884   4216
   ...   ...   ...    ...
              1035   9585
               ...    ...
The numbers in the first row of the table above are b(k) = 5*3^k, k>=1, (see A005030) so that infinitely many odd numbers occur outside of the first column. The central region of the symmetric representation of sigma(b(k)) contains 2*k-1 separate contiguous sections consisting of sequences of entire legs of width 2, k>=1 (see Lemma 2 in the link).
Conjecture: The combined extent of these sections in sigma(b(k)) is 2*3^(k-1) - 1 = A048473(k-1), k>=1.
Since each number n in the first column and first row has a prime factor of odd exponent a contiguous section of the symmetric representation of sigma(n) centered on the diagonal has width 2. For odd numbers n not in the first row or column in which all prime factors have even powers, such as 225 and 5625 in the second row, a contiguous section of the symmetric representation of sigma(n) centered on the diagonal has width 1 (see Lemma 1 in the link).
For each k>=3 and every prime p such that b(k-1) < 2*p < 4*b(k-2), the odd number p*b(k-1) is in the column of b(k). The two inequalities are equivalent to b(k-1) <= row(p*b(k-1)) < 2*b(k-1) ensuring that the symmetric representation of sigma(p*b(k-1)) consists of 3 regions.
45 is the only odd number in its column (see Lemma 3 in the link).
Since the factors of n = p*q satisfy 2 < p < q < 2*p the first column in the table above is a subsequence of A082663 and of A087718 (see Lemma 4 in the link). Each of the two outer regions consists of a single leg of width 1 and length (1 + p*q)/2. The center region of size p+q consists of two subparts (see A196020 & A280851) of width 1 of sizes 2*p-q and 2*q-p, respectively (see Lemma 5 in the link). The table below arranges the first column in the table above according to the length 2*p-q of their single contiguous extent of width 2 in the center region:
     1     3      5      7      9     11     13     15  ...
------------------------------------------------------
    15    35    187    247    143    391   2257    323
    91    77    493    589    221   1363   3139    437
   703   209    943   2479    551   2911   6649    713
  1891   299   1537   3397    851   3901    ...   1247
  2701   527   4183   8509   1643   6313          1457
  ...    ...    ...    ...    ...    ...          ....
A129521: p*q satisfies 2*p - q = 1 (excluding A129521(1)=6)
A226755: p*q satisfies 2*p - q = 3 (excluding A226755(1)=9)
Sequences with larger differences 2*p - q are not in OEIS.

			a(6) = 91 = 7*13 is in the sequence and in the 2-column of the first table since 1 < 2 < 7 < 13 = row(91) representing the 4 odd divisors 1 - 91 - 7 - 13 (see A237048) results in the following pattern for the widths of the legs (see A249223): 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2 for 3 regions with width not exceeding 2. It also is in the 1-column of the second table since it has a single area of width 2 which is 1 unit long.
a(29) = 405 = 5*3^4 is in the sequence and in the 5-column of the first table since 1 < 2 < 3 < 5 < 6 < 9 < 10 < 15 < 18 < 27 = row(405) representing the 10 odd divisors 1 - 405 - 3 - 5 - 135 - 9 - 81 - 15 - 45 - 27 results in the following pattern for the widths of the legs: 1, 0, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2 for 3 regions with width not exceeding 2, and 7 = 2*4 - 1 sections of width 2 in the central region.
a(35) = 506 = 2*11*23 is in the sequence since positions 1 < 4 < 11 < 23 < row(506) = 31 representing the 4 odd divisors 1 - 253 - 11 - 23 results in the following pattern for the widths of the legs: 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2 for 3 regions with width not exceeding 2, with the two outer regions consisting of 3 legs of width 1, and a single area of width 2 in the central region.

Hartmut F. W. Hoft, proofs for quoted lemmas

Cf. A001248, A005030, A048473, A082663, A087718, A129521, A196020, A226755, A235791, A237048, A237270, A237271, A237591, A237593, A247687, A249223, A279102, A280107, A280851.

(* Functions path and a237270 are defined in A237270 *)
maxDiagonalLength[n_] := Max[Map[#[[1]]-#[[2]]&, Transpose[{Drop[Drop[path[n], 1], -1], path[n-1]}]]]
a338486[m_, n_] := Module[{r, list={}, k}, For[k=m, k<=n, k++, r=a237270[k]; If[Length[r]== 3 && maxDiagonalLength[k]==2,AppendTo[list, k]]]; list]
a338486[1, 850]

A166465 a(n) = 3*a(n-2) for n > 2; a(1) = 1, a(2) = 5.

Original entry on oeis.org

1, 5, 3, 15, 9, 45, 27, 135, 81, 405, 243, 1215, 729, 3645, 2187, 10935, 6561, 32805, 19683, 98415, 59049, 295245, 177147, 885735, 531441, 2657205, 1594323, 7971615, 4782969, 23914845, 14348907, 71744535, 43046721, 215233605, 129140163

Offset: 1

Klaus Brockhaus, Oct 14 2009

Interleaving of A000244 and A005030.
Second binomial transform is A054485.
Fifth binomial transform is A153596.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,3).

Cf. A000244 (powers of 3), A005030 (5*3^n), A054485, A153596, A162813.

[ n le 2 select 4*n-3 else 3*Self(n-2): n in [1..35] ];

LinearRecurrence[{0,3}, {1,5}, 41] (* G. C. Greubel, Jul 27 2024 *)

[3^(n/2)*(5*((n+1)%2) +sqrt(3)*(n%2))/3 for n in range(1,41)] # G. C. Greubel, Jul 27 2024

a(n) = (4 + (-1)^n) * 3^((2*n - 5 + (-1)^n)/4).
G.f.: x*(1+5*x)/(1-3*x^2).
a(n) = A162813(n-1), for n >= 2.
From G. C. Greubel, Jul 27 2024: (Start)
a(n) = (1/6)*3^(n/2)*( 5*(1+(-1)^n) + sqrt(3)*(1-(-1)^n) ).
E.g.f.: (1/3)*(sqrt(3)*sinh(sqrt(3)*x) + 10*(sinh(sqrt(3)*x/2))^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

A382407 a(n) is the number of partitions n = x + y + z of positive integers such that xy + yz + x*z is a perfect square.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 3, 0, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 3, 0, 5, 1, 1, 2, 3, 3, 2, 1, 1, 3, 6, 1, 4, 2, 7, 4, 4, 0, 3, 5, 3, 4, 2, 1, 7, 2, 1, 5, 9, 3, 5, 3, 4, 1, 9, 2, 6, 3, 5, 6, 5, 4, 7, 5, 1, 5, 6, 3, 13, 7, 8, 4, 6, 0, 4, 4, 11, 5, 13, 2

Offset: 1

Felix Huber, Apr 04 2025

nonn

a(n) is the number of distinct cuboids with edge length 4*n whose surface area is half of a square.

Conjecture: a(k) = 0 iff k is an element of {2, 4, 8, 13} union A000244 union A005030.

			The a(14) = 3 partitions [x, y, z] are [1, 1, 12], [1, 4, 9] and [4, 4, 6] because 1*1 + 1*12 + 1*12 = 5^2, 1*4 + 4*9 + 1*9 = 7^2 and 4*4 + 4*6 + 4*6 = 8^2.

Felix Huber, Table of n, a(n) for n = 1..10000
Felix Huber, Maple program to calculate the partitions

Cf. A000244, A005030, A066955, A069905, A338939, A375512, A375576, A375580, A375731.

A382407:=proc(n)
    local a,x,y,z;
    a:=0;
    for x to n/3 do
        for y from x to (n-x)/2 do
            z:=n-x-y;
            if issqr(x*y+x*z+y*z) then
                a:=a+1
            fi
        od
    od;
    return a
end proc;
seq(A382407(n),n=1..87);

cp's OEIS Frontend

A060816 a(0) = 1; a(n) = (5*3^(n-1) - 1)/2 for n > 0.

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A134931 a(n) = (5*3^n-3)/2.

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A200251 T(n,k)=Number of 0..k arrays x(0..n-1) of n elements with each no smaller than the sum of its previous elements modulo (k+1).

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A338486 Numbers n whose symmetric representation of sigma(n) consists of 3 regions with maximum width 2.

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A166465 a(n) = 3*a(n-2) for n > 2; a(1) = 1, a(2) = 5.

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

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A382407 a(n) is the number of partitions n = x + y + z of positive integers such that x*y + y*z + x*z is a perfect square.

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A382407 a(n) is the number of partitions n = x + y + z of positive integers such that xy + yz + x*z is a perfect square.