A013730 -id:A013730 - cp's OEIS frontend

A022027 Define the sequence T(a(0),a(1)) by a(n+2) is the greatest integer such that a(n+2)/a(n+1) < a(n+1)/a(n) for n >= 0. This is T(2,16).

2, 16, 127, 1008, 8000, 63492, 503904, 3999232, 31739888, 251903488, 1999230976, 15866888256, 125927492096, 999423012864, 7931916549888, 62951622430720, 499615287394304, 3965194632954880, 31469750573916160, 249759543441752064, 1982215569002196992, 15731845549721911296

Offset: 0

R. K. Guy

nonn

Not to be confused with the Pisot T(2,16) sequence, which is A013730. - R. J. Mathar, Feb 13 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for Pisot sequences

Cf. A022018 - A022025, A022026 - A022032.

I:=[2,16]; [n le 2 select I[n] else Ceiling(Self(n-1)^2/Self(n-2))-1: n in [1..30]]; // Vincenzo Librandi, Feb 12 2016

RecurrenceTable[{a[1] == 2, a[2] == 16, a[n] == Ceiling[a[n-1]^2 / a[n-2] - 1]}, a, {n, 30}] (* Vincenzo Librandi, Feb 12 2016 *)

a=[2, 16]; for(n=2, 1000, a=concat(a, ceil(a[n]^2/a[n-1])-1)); A022027(n)=a[n+1] \\ M. F. Hasler, Feb 11 2016

Conjectures: a(n) = 8*a(n-1)-4*a(n-3). G.f.: -(x^2-2) / (4*x^3-8*x+1). - Colin Barker, Sep 18 2015

a(n+1) = ceiling(a(n)^2/a(n-1))-1 for n>0. - M. F. Hasler, Feb 11 2016

Double-checked (original definition agrees with g.f. / recurrence for n=0..1000), extended and edited by M. F. Hasler, Feb 11 2016

A100628 a(n) = 2^(3*prime(n) + 1).

Original entry on oeis.org

128, 1024, 65536, 4194304, 17179869184, 1099511627776, 4503599627370496, 288230376151711744, 1180591620717411303424, 309485009821345068724781056, 19807040628566084398385987584, 5192296858534827628530496329220096

Offset: 1

Parthasarathy Nambi, Dec 02 2004

nonn

			a(1) = 2^(3*2 + 1) = 128.

Cf. A004171, A013730.

[2^(3*NthPrime(n) + 1): n in [1..20]]; // Vincenzo Librandi, Aug 27 2015

Table[2^(3Prime[n] + 1), {n, 1, 15}] (* Stefan Steinerberger, Feb 28 2006 *)

More terms from Stefan Steinerberger, Feb 28 2006

A267088 Perfect powers of the form x^3 + y^3 where x and y are positive integers.

Original entry on oeis.org

9, 16, 128, 243, 576, 1024, 6561, 8192, 9604, 11664, 28224, 36864, 51984, 65536, 97344, 140625, 177147, 250000, 275625, 345744, 419904, 450241, 524288, 614656, 717409, 746496, 1028196, 1058841, 1399489, 1500625, 1590121, 1750329, 1806336, 1882384, 2359296, 3326976, 4194304

Offset: 1

Altug Alkan, Jan 10 2016

nonn

Intersection of A001597 and A003325.
Motivation for this sequence is the equation m^k = x^3 + y^3 where x,y,m > 0 and k >= 2.
Obviously, because of Fermat's Last Theorem, a(n) cannot be a cube.
A050802 is a subsequence.
Obviously, this sequence contains all numbers of the form 2^(3*n+1) and 3^(3*n-1), for n > 0.

			9 is a term because 9 = 3^2 = 1^3 + 2^3.
16 is a term because 16 = 2^4 = 2^3 + 2^3.
243 is a term because 243 = 3^5 = 3^3 + 6^3.

Chai Wah Wu, Table of n, a(n) for n = 1..5012

Cf. A001597, A003325, A050802, A085332, A013730, A013733, A266927.

T = thueinit('z^3+1);
is(n) = #select(v->min(v[1], v[2])>0, thue(T, n))>0;
for(n=2, 1e7, if(ispower(n) && is(n), print1(n, ", ")))

A003222 a(n) = 2^(3n+1) - 2n(2n+1).

Original entry on oeis.org

2, 10, 108, 982, 8120, 65426, 524132, 4194094, 33554160, 268435114, 2147483228, 17179868678, 137438952872, 1099511627074, 8796093021396, 70368744176734, 562949953420256, 4503599627369306, 36028797018962636, 288230376151710262, 2305843009213692312, 18446744073709549810

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (11,-27,25,-8).

Cf. A002943, A013730.

[2^(3*n+1)-2*n*(2*n+1): n in [0..20]]; // Vincenzo Librandi, Jun 26 2011

a(n) = 2^(3*n+1)-2*n*(2*n+1); \\ Altug Alkan, Sep 21 2018

a(n) = A013730(n) - A002943(n).
From R. J. Mathar, May 05 2010: (Start)
a(n) = 11*a(n-1) - 27*a(n-2) + 25*a(n-3) - 8*a(n-4).
G.f.: 2*(1 - 6*x + 26*x^2 + 7*x^3)/((8*x-1)*(x-1)^3). (End)
E.g.f.: 2*exp(x)*(exp(7*x) - x*(3 + 2*x)). - Elmo R. Oliveira, Mar 06 2025

A013766 a(n) = 20^(3*n + 1).

Original entry on oeis.org

20, 160000, 1280000000, 10240000000000, 81920000000000000, 655360000000000000000, 5242880000000000000000000, 41943040000000000000000000000, 335544320000000000000000000000000, 2684354560000000000000000000000000000, 21474836480000000000000000000000000000000

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (8000).

Subsequence of A009964.

Cf. A013730, A013746, A016777.

[20^(3*n+1): n in [0..10]]; // Vincenzo Librandi, Jun 27 2011

20^(3Range[0,10]+1) (* Harvey P. Dale, Jun 23 2011 *)

makelist(20^(3*n+1),n,0,20); /* Martin Ettl, Oct 21 2012 */

a(n)=20^(3*n+1) \\ Charles R Greathouse IV, Jul 10 2016

From Elmo R. Oliveira, Feb 27 2025: (Start)
G.f.: 20/(1 - 8000*x).
E.g.f.: 20*exp(8000*x).
a(n) = A013730(n)*A013746(n) = A009964(A016777(n)). (End)

A198850 a(n) = 2*8^n - 1.

Original entry on oeis.org

1, 15, 127, 1023, 8191, 65535, 524287, 4194303, 33554431, 268435455, 2147483647, 17179869183, 137438953471, 1099511627775, 8796093022207, 70368744177663, 562949953421311, 4503599627370495, 36028797018963967, 288230376151711743, 2305843009213693951, 18446744073709551615

Offset: 0

Vincenzo Librandi, Oct 31 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-8).

Cf. A013730.

Magma
```
[2*8^n-1: n in [0..30]];
```

a(n) = 8*a(n-1) + 7.
a(n) = 9*a(n-1) - 8*a(n-2), n > 1.
G.f.: (1+6*x)/((8*x-1)*(x-1)). - R. J. Mathar, Oct 31 2011
From Elmo R. Oliveira, Sep 13 2024: (Start)
E.g.f.: exp(x)*(2*exp(7*x) - 1).
a(n) = A013730(n) - 1. (End)

A379530 a(n) = (A135318(3n) + A135318(3n+1) + A135318(3*n+2))/3.

Original entry on oeis.org

1, 3, 8, 23, 64, 185, 512, 1479, 4096, 11833, 32768, 94663, 262144, 757305, 2097152, 6058439, 16777216, 48467513, 134217728, 387740103, 1073741824, 3101920825, 8589934592, 24815366599, 68719476736, 198522932793, 549755813888, 1588183462343, 4398046511104, 12705467698745

Offset: 0

Paul Curtz, Dec 24 2024

Index entries for linear recurrences with constant coefficients, signature (0,7,0,8).

Cf. A001018, A013730, A015565, A135318.

LinearRecurrence[{0, 7, 0, 8}, {1, 3, 8, 23}, 30] (* Amiram Eldar, Dec 31 2024 *)

a(n) = 7*a(n-2) + 8*a(n-4) with a(0)=1, a(1)=3, a(2)=8, a(3)=23 for n >= 4.
a(2*n) = A001018(n).
a(2*n+1) = A015565(n+1) + A013730(n).

A386874 Irregular triangle read by rows: row n consists of the coefficients in the expansion of the polynomial (1/(2w)) (x^2 + x) * ((((v + w)/2)^(n - 1)) * (x^2 + 2x + 4 + w) - (((v - w)/2)^(n - 1)) (x^2 + 2x + 4 - w)), where v = x^2 + 4x + 4 and w = sqrt(x^4 + 4x^3 + 12x^2 + 20*x + 12).

Original entry on oeis.org

0, 1, 1, 0, 4, 7, 4, 1, 0, 15, 40, 42, 23, 7, 1, 0, 56, 201, 306, 262, 140, 48, 10, 1, 0, 209, 943, 1877, 2189, 1672, 881, 325, 82, 13, 1, 0, 780, 4239, 10412, 15368, 15276, 10841, 5660, 2194, 624, 125, 16, 1, 0, 2911, 18506, 54051, 96501, 118175, 105495

Offset: 1

Franck Maminirina Ramaharo, Aug 06 2025

T(n,k) is the number of ways to assign horizontal or vertical barriers at each interior construction dot of the 4 X 2n barrier-free Celtic shadow diagram CK_4^(2n) such that the resulting design consists of exactly k connected components.

The n-th row is the coefficients in the expansion of the Kauffman bracket polynomial for the shadow of the Celtic link CK_4^(2n).

			The triangle T(n,k) begins:
  n\k 0    1     2     3     4     5       6     7     8     9   10   11  12 13 14
  1:  0    1     1
  2:  0    4     7     4     1
  3:  0   15    40    42    23      7      1
  4:  0   56   201   306   262    140     48    10     1
  5:  0  209   943  1877  2189   1672    881   325    82    13    1
  6:  0  780  4239 10412 15368  15276  10841  5660  2194   624  125   16   1
  7:  0 2911 18506 54051 96501 118175 105495 71107 36885 14817 4579 1064 177 19  1
  ...

Roger Antonsen and Laura Taalman, Categorizing Celtic Knot Designs, in Proceedings of Bridges 2021: Mathematics, Art, Music, Architecture, Culture, 2021, pp. 87-94.
Jonathan L. Gross and Thomas W. Tucker, A Celtic Framework for Knots and Links, Discrete & Computational Geometry 46 (2011), 86-99.
Franck Ramaharo, The bracket polynomial of the Celtic link shadow CK_4^(2n), arXiv:2508.10410 [math.GT], 2025. See p. 6.

Row sums: A013730.

Cf. A299989, A300192, A300453, A300454, A316659, A316989.

With[{nmax = 15}, CoefficientList[CoefficientList[Series[x*y*(x + 1)*(1 - x*y)/(1 - ((x + 2)^2)*y + ((x + 1)^3)*y^2), {x, 0, 2*nmax}, {y, 0, nmax}], y], x]] // Flatten

nmax: 15$ v: x^2 + 4*x + 4$ w: sqrt(x^4 + 4*x^3 + 12*x^2 + 20*x + 12)$
p(n, x) := expand((1/(2*w))*(x^2 + x)*((((v + w)/2)^(n - 1))*(x^2 + 2*x + 4 + w) - (((v - w)/2)^(n - 1))*(x^2 + 2*x + 4 - w)))$
create_list(ratcoef(p(n, x), x, k), n, 1, nmax, k, 0, 2*n);

T(n,1) = A001353(n).

G.f.: x*y*(x + 1)*(1 - x*y) / (1 - ((x + 2)^2)*y + ((x + 1)^3)*y^2).

cp's OEIS Frontend

A022027 Define the sequence T(a(0),a(1)) by a(n+2) is the greatest integer such that a(n+2)/a(n+1) < a(n+1)/a(n) for n >= 0. This is T(2,16).

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A100628 a(n) = 2^(3*prime(n) + 1).

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A267088 Perfect powers of the form x^3 + y^3 where x and y are positive integers.

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PARI

A003222 a(n) = 2^(3*n+1) - 2*n*(2*n+1).

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Magma

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A013766 a(n) = 20^(3*n + 1).

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Magma

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A198850 a(n) = 2*8^n - 1.

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A379530 a(n) = (A135318(3*n) + A135318(3*n+1) + A135318(3*n+2))/3.

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A386874 Irregular triangle read by rows: row n consists of the coefficients in the expansion of the polynomial (1/(2*w)) * (x^2 + x) * ((((v + w)/2)^(n - 1)) * (x^2 + 2*x + 4 + w) - (((v - w)/2)^(n - 1)) * (x^2 + 2*x + 4 - w)), where v = x^2 + 4*x + 4 and w = sqrt(x^4 + 4*x^3 + 12*x^2 + 20*x + 12).

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A003222 a(n) = 2^(3n+1) - 2n(2n+1).

A379530 a(n) = (A135318(3n) + A135318(3n+1) + A135318(3*n+2))/3.

A386874 Irregular triangle read by rows: row n consists of the coefficients in the expansion of the polynomial (1/(2w)) (x^2 + x) * ((((v + w)/2)^(n - 1)) * (x^2 + 2x + 4 + w) - (((v - w)/2)^(n - 1)) (x^2 + 2x + 4 - w)), where v = x^2 + 4x + 4 and w = sqrt(x^4 + 4x^3 + 12x^2 + 20*x + 12).