A067417 -id:A067417 - cp's OEIS frontend

A067411 Third column of triangle A067410 and second column of A067417.

1, 4, 24, 144, 864, 5184, 31104, 186624, 1119744, 6718464, 40310784, 241864704, 1451188224, 8707129344, 52242776064, 313456656384, 1880739938304, 11284439629824, 67706637778944, 406239826673664

Offset: 0

Wolfdieter Lang, Jan 25 2002

Let f(k) be the sum of the smallest three positive divisors of k, g(k) be the sum of the largest two positive divisors of k, this sequence from a(2) onwards contains the numbers k for which g(k) is a positive integer power of f(k). - Yifan Xie, Jan 27 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry and A. Hennessy, The Euler-Seidel Matrix, Hankel Matrices and Moment Sequences, J. Int. Seq. 13 (2010) # 10.8.2, Example 14.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Craig Knecht, Number of tilings for a stackable six sphinx tile shape.
Index entries for linear recurrences with constant coefficients, signature (6).

A002001, A067412 (second and fourth column of A067410), A000244, A067403 (first and third column of A067417), A000400 (powers of 6).

Row sums of A038195.

CoefficientList[Series[(1-2x)/(1-6x),{x,0,30}],x] (* Harvey P. Dale, Feb 26 2015 *)

a(n) = if(n<=0, 0, 4*6^(n-1) ); \\ Joerg Arndt, Feb 23 2014

a(n) = A067410(n+2, 2) = A067417(n+1, 1).
a(n) = 4 * 6^(n-1), for n >= 1, a(0)=1.
G.f.: (1-2*x)/(1-6*x).
E.g.f.: (2*exp(6*x)+1) / 3 = exp(3*x)*(cosh(3*x) + sinh(3*x)/3). - Paul Barry, Nov 20 2003
a(n) = Sum_{k=0..n} C(n,k) * A001045(n+k+1). - Paul Barry, Apr 19 2010

Incorrect formula deleted by Harvey P. Dale, Feb 26 2015

Formula restored by Sean A. Irvine, Jan 10 2021

A067419 Fourth column of triangle A067417.

Original entry on oeis.org

1, 6, 72, 864, 10368, 124416, 1492992, 17915904, 214990848, 2579890176, 30958682112, 371504185344, 4458050224128, 53496602689536, 641959232274432, 7703510787293184, 92442129447518208, 1109305553370218496, 13311666640442621952, 159739999685311463424

Offset: 0

Wolfdieter Lang, Jan 25 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (12).

Cf. A067403 (third column), A067420 (fifth column), A001021 (powers of 12).

[Ceiling(6*(3*4)^(n-1)): n in [0..20]]; // Vincenzo Librandi, Oct 02 2011

Join[{1}, NestList[12*# &, 6, 20]] (* Paolo Xausa, Sep 03 2024 *)

a(n) = A067417(n+3, 3).
a(n) = 6*(3*4)^(n-1), n >= 1, a(0)=1.
G.f.: (1-6*x)/(1-12*x).
a(n) = Sum_{k=0..n} A134309(n,k)*6^k = Sum_{k=0..n} A055372(n,k)*5^k. - Philippe Deléham, Feb 04 2012

A067420 Fifth column of triangle A067417.

Original entry on oeis.org

1, 7, 105, 1575, 23625, 354375, 5315625, 79734375, 1196015625, 17940234375, 269103515625, 4036552734375, 60548291015625, 908224365234375, 13623365478515625, 204350482177734375, 3065257232666015625

Offset: 0

Wolfdieter Lang, Jan 25 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..800
Index entries for linear recurrences with constant coefficients, signature (15).

Cf. A067419 (fourth column), A067421 (sixth column), A001024 (powers of 15).

[Ceiling(7*(3*5)^(n-1)): n in [0..20]]; // Vincenzo Librandi, Oct 02 2011

a(n) = A067417(n+4, 4).
a(n) = 7*(3*5)^(n-1), n >= 1, a(0)=1.
G.f.: (1-8*x)/(1-15*x).

A067421 Sixth column of triangle A067417.

Original entry on oeis.org

1, 8, 144, 2592, 46656, 839808, 15116544, 272097792, 4897760256, 88159684608, 1586874322944, 28563737812992, 514147280633856, 9254651051409408, 166583718925369344, 2998506940656648192

Offset: 0

Wolfdieter Lang, Jan 25 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..800
Index entries for linear recurrences with constant coefficients, signature (18).

Cf. A067420 (fifth column), A067422 (seventh column), A001027 (powers of 18).

[Ceiling(8*(3*6)^(n-1)): n in [0..20]]; // Vincenzo Librandi, Oct 02 2011

a(n) = A067417(n+5, 5).
a(n) = 8*(3*6)^(n-1), n >= 1, a(0)=1.
G.f.: (1-10*x)/(1-18*x).

A067422 Seventh column of triangle A067417.

Original entry on oeis.org

1, 9, 189, 3969, 83349, 1750329, 36756909, 771895089, 16209796869, 340405734249, 7148520419229, 150118928803809, 3152497504879989, 66202447602479769, 1390251399652075149, 29195279392693578129

Offset: 0

Wolfdieter Lang, Jan 25 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..700
Index entries for linear recurrences with constant coefficients, signature (21).

Cf. A067421 (sixth column), A067423 (eighth column), A009965 (powers of 21).

[Ceiling(9*(3*7)^(n-1)): n in [0..20]]; // Vincenzo Librandi, Oct 02 2011

a(n) = A067417(n+6, 6).
a(n) = 9*(3*7)^(n-1), n >= 1, a(0)=1.
G.f.: (1-12*x)/(1-21*x).

A067423 Eighth column of triangle A067417.

Original entry on oeis.org

1, 10, 240, 5760, 138240, 3317760, 79626240, 1911029760, 45864714240, 1100753141760, 26418075402240, 634033809653760, 15216811431690240, 365203474360565760, 8764883384653578240, 210357201231685877760

Offset: 0

Wolfdieter Lang, Jan 25 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..700
Index entries for linear recurrences with constant coefficients, signature (24).

Cf. A067422 (seventh column), A067424 (ninth column), A009968 (powers of 24).

[Ceiling(10*(3*8)^(n-1)): n in [0..20]]; // Vincenzo Librandi, Oct 02 2011

a(n) = A067417(n+7, 7).
a(n) = 10*(3*8)^(n-1), n >= 1, a(0)=1.
G.f.: (1-14*x)/(1-24*x).

A067424 Ninth column of triangle A067417.

Original entry on oeis.org

1, 11, 297, 8019, 216513, 5845851, 157837977, 4261625379, 115063885233, 3106724901291, 83881572334857, 2264802453041139, 61149666232110753, 1651040988266990331, 44578106683208738937, 1203608880446635951299

Offset: 0

Wolfdieter Lang, Jan 25 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..700
Index entries for linear recurrences with constant coefficients, signature (27).

Cf. A067417, A067423 (eighth column), A009971 (powers of 27).

[Ceiling(11*(3*9)^(n-1)): n in [0..20]]; // Vincenzo Librandi, Oct 02 2011

CoefficientList[Series[(1-16x)/(1-27x),{x,0,30}],x] (* or *) LinearRecurrence[{27},{1,11},20] (* Harvey P. Dale, Apr 20 2022 *)

a(n) = A067417(n+8, 8).
a(n) = 11*(3*9)^(n-1), n >= 1, a(0)=1.
G.f.: (1-16*x)/(1-27*x).
E.g.f.: (16 + 11*exp(27*x))/27. - Stefano Spezia, Sep 30 2022

A001018 Powers of 8: a(n) = 8^n.

Original entry on oeis.org

1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, 134217728, 1073741824, 8589934592, 68719476736, 549755813888, 4398046511104, 35184372088832, 281474976710656, 2251799813685248, 18014398509481984, 144115188075855872, 1152921504606846976, 9223372036854775808, 73786976294838206464, 590295810358705651712, 4722366482869645213696

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 8), L(1, 8), P(1, 8), T(1, 8). Essentially same as Pisot sequences E(8, 64), L(8, 64), P(8, 64), T(8, 64). See A008776 for definitions of Pisot sequences.
If X_1, X_2, ..., X_n is a partition of the set {1..2n} into blocks of size 2 then, for n>=1, a(n) is equal to the number of functions f : {1..2n} -> {1,2,3} such that for fixed y_1,y_2,...,y_n in {1,2,3} we have f(X_i)<>{y_i}, (i=1..n). - Milan Janjic, May 24 2007
This is the auto-convolution (convolution square) of A059304. - R. J. Mathar, May 25 2009
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=1, a(n) equals the number of 8-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
a(n) is equal to the determinant of a 3 X 3 matrix with rows 2^(n+2), 2^(n+1), 2^n; 2^(n+3), 2^(n+4), 2(n+3); 2^n, 2^(n+1), 2^(n+2) when it is divided by 144. - J. M. Bergot, May 07 2014
a(n) gives the number of small squares in the n-th iteration of the Sierpinski carpet fractal. Equivalently, the number of vertices in the n-Sierpinski carpet graph. - Allan Bickle, Nov 27 2022

			For n=1, the 1st order Sierpinski carpet graph is an 8-cycle.

K. H. Rosen et al., eds., Handbook of Discrete and Combinatorial Mathematics, CRC Press, 2017; p. 15.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 273
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Tanya Khovanova, Recursive Sequences
Caroline Nunn, A Proof of a Generalization of Niven's Theorem Using Algebraic Number Theory, Rose-Hulman Undergraduate Mathematics Journal: Vol. 22, Iss. 2, Article 3 (2021). See table at p. 9.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Sierpiński Carpet
Index entries for linear recurrences with constant coefficients, signature (8).

Cf. A013730, A103333, A013731, A067417, A083233, A055274.
Cf. A008588, A016969, A157176.
Cf. A000079 (powers of 2), A000244 (powers of 3), A000302 (powers of 4), A000351 (powers of 5), A000400 (powers of 6), A000420 (powers of 7), A001019 (powers of 9), ..., A001029 (powers of 19), A009964 (powers of 20), ..., A009992 (powers of 48), A087752 (powers of 49), A165800 (powers of 50), A159991 (powers of 60).
Cf. A032766 (floor(3*n/2)).
Cf. A271939 (number of edges in the n-Sierpinski carpet graph).

a001018 = (8 ^)
a001018_list = iterate (* 8) 1  -- Reinhard Zumkeller, Apr 29 2015

[8^n : n in [0..30]]; // Wesley Ivan Hurt, Sep 27 2016

seq(8^n, n=0..23); # Nathaniel Johnston, Jun 26 2011
A001018 := n -> 8^n; # M. F. Hasler, Apr 19 2015

Table[8^n, {n,0,50}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)

makelist(8^n,n,0,20); /* Martin Ettl, Nov 12 2012 */

a(n)=8^n \\ Charles R Greathouse IV, May 10 2014

print([8**n for n in range(25)]) # Michael S. Branicky, Dec 29 2021

a(n) = 8^n.
a(0) = 1; a(n) = 8*a(n-1) for n > 0.
G.f.: 1/(1-8*x).
E.g.f.: exp(8*x).
Sum_{n>=0} 1/a(n) = 8/7. - Gary W. Adamson, Aug 29 2008
a(n) = A157176(A008588(n)); a(n+1) = A157176(A016969(n)). - Reinhard Zumkeller, Feb 24 2009
From Stefano Spezia, Dec 28 2021: (Start)
a(n) = (-1)^n*(1 + sqrt(-3))^(3*n) (see Nunn, p. 9).
a(n) = (-1)^n*Sum_{k=0..floor(3*n/2)} (-3)^k*binomial(3*n, 2*k) (see Nunn, p. 9). (End)

A067425 Triangle with columns built from certain power sequences.

Original entry on oeis.org

1, 4, 1, 16, 5, 1, 64, 40, 6, 1, 256, 320, 72, 7, 1, 1024, 2560, 864, 112, 8, 1, 4096, 20480, 10368, 1792, 160, 9, 1, 16384, 163840, 124416, 28672, 3200, 216, 10, 1, 65536, 1310720, 1492992, 458752, 64000

Offset: 0

Wolfdieter Lang, Jan 25 2002

The fifth column (m=4) gives [1, 8, 160, 3200, 64000, 1280000, 25600000, ...].

			Triangle starts:
   1;
   4,  1;
  16,  5,  1;
  64, 40,  6,  1;
  ...

Indranil Ghosh, Rows 0..125, flattened

Cf. A009998, A067410, A067417.
Columns 0..3 are A000302 (powers of 4), A067412, A067419, A067404.
Columns 5..8 are A067426, A067427, A067428, A067429.

A067425[n_, m_] := If[n == m, 1, (m + 4)*(4*(m + 1))^(n - m - 1)];
Table[A067425[n, m], {n, 0, 10}, {m, 0, n}] (* Paolo Xausa, Oct 16 2024 *)

T(n,m) = 1 if n = m; T(n,m) = (m+4)*(4*(m+1))^(n-m-1) if n > m >= 0, else 0.

G.f. for column m: (x^m)*(1-3*m*x)/(1-4*(m+1)*x).

cp's OEIS Frontend

A067411 Third column of triangle A067410 and second column of A067417.

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A067419 Fourth column of triangle A067417.

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A067420 Fifth column of triangle A067417.

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A067421 Sixth column of triangle A067417.

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A067422 Seventh column of triangle A067417.

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A067423 Eighth column of triangle A067417.

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A067424 Ninth column of triangle A067417.

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A001018 Powers of 8: a(n) = 8^n.

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