A001018 -id:A001018 - cp's OEIS frontend

A016177 a(n) = 8^n - 7^n.

0, 1, 15, 169, 1695, 15961, 144495, 1273609, 11012415, 93864121, 791266575, 6612607849, 54878189535, 452866803481, 3719823438255, 30436810578889, 248242046141055, 2019169299698041, 16385984911571535, 132716292890482729, 1073129238309234975, 8664826172771491801

Offset: 0

N. J. A. Sloane

Number of ways to assign truth values to n ternary conjunctions connected by disjunctions such that the proposition is true. For example, a(2) = 15, since for the proposition '(a & b & c) v (d & e & f)' there are 15 assignments that make the proposition true. - Ori Milstein, Dec 22 2022
Equivalently, the number of length-n words over the alphabet {0,1,...,7} with at least one letter = 7. - Joerg Arndt, Jan 01 2023
a(n) is also the number of n-digit numbers whose smallest decimal digit is 2. - Stefano Spezia, Nov 15 2023

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (15,-56).

Cf. A000420, A001018, A016140.

[8^n -7^n: n in [0..40]]; // G. C. Greubel, Nov 29 2024

Table[8^n - 7^n, {n, 0, 20}] (* Harvey P. Dale, Jan 31 2011 *)

a(n)=8^n-7^n \\ Charles R Greathouse IV, Sep 28 2015

def A016177(n): return pow(8,n) - pow(7,n)
print([A016177(n) for n in range(41)]) # G. C. Greubel, Nov 29 2024

G.f.: x/((1-7x)*(1-8x)).
a(n) = numerator(f(n-1)) where f(n) = Integral_{x=0..1/4} (1-x/2)^n dx. And denominator(f(n)) = 4*(n+1)*8^n. - Al Hakanson (hawkuu(AT)excite.com), Feb 22 2004 [corrected by Michel Marcus, Dec 23 2022]
a(n) = 15*a(n-1) - 56*a(n-2), n > 1. - Philippe Deléham, Jan 01 2009
E.g.f.: e^(8*x) - e^(7*x). - Mohammad K. Azarian, Jan 14 2009
a(n) = 8*a(n-1) + 7^(n-1), a(0)=0. - Vincenzo Librandi, Feb 09 2011

A088138 Generalized Gaussian Fibonacci integers.

Original entry on oeis.org

0, 1, 2, 0, -8, -16, 0, 64, 128, 0, -512, -1024, 0, 4096, 8192, 0, -32768, -65536, 0, 262144, 524288, 0, -2097152, -4194304, 0, 16777216, 33554432, 0, -134217728, -268435456, 0, 1073741824, 2147483648, 0, -8589934592, -17179869184, 0, 68719476736, 137438953472

Offset: 0

Paul Barry, Sep 20 2003

The sequence 0,1,-2,0,8,-16,... has g.f. x/(1+2*x-4*x^2), a(n) = 2^n*sin(2n*Pi/3)/sqrt(3) and is the inverse binomial transform of sin(sqrt(3)*x)/sqrt(3): 0,1,-3,0,9,...
a(n+1) is the Hankel transform of A100192. - Paul Barry, Jan 11 2007
a(n+1) is the trinomial transform of A010892: a(n+1) = Sum_{k=0..2n} trinomial(n,k)*A010892(k+1) where trinomial(n, k) = trinomial coefficients (A027907). - Paul Barry, Sep 10 2007
a(n+1) is the Hankel transform of A100067. - Paul Barry, Jun 16 2009
From Paul Curtz, Oct 04 2009: (Start)
1) a(n) = A131577(n)*A128834(n).
2) Binomial transform of 0,1,0,-3,0,9,0,-27, see A000244.
3) Sequence is identical to every 2n-th difference divided by (-3)^n.
4) a(3n) + a(3n+1) + a(3n+2) = (-1)^n*3*A001018(n) for n >= 1.
5) For missing terms in a(n) see A013731 = 4*A001018. (End)
The coefficient of i of Q^n, where Q is the quaternion 1+i+j+k. Due to symmetry, also the coefficients of j and of k. - Stanislav Sykora, Jun 11 2012 [The coefficients of 1 are in A138230. - Wolfdieter Lang, Jan 28 2016]
With different signs, 0, 1, -2, 0, 8, -16, 0, 64, -128, 0, 512, -1024, ... is the Lucas U(-2,4) sequence. - R. J. Mathar, Jan 08 2013

Robert Israel, Table of n, a(n) for n = 0..3300
Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
Wikipedia, Lucas sequence
Index entries for linear recurrences with constant coefficients, signature (2,-4).
Index entries for Lucas sequences

Cf. A084102, A088137, A045873, A088139, A138230.

a:=[0,1];; for n in [3..40] do a[n]:=2*a[n-1]-4*a[n-2]; od; a; # Muniru A Asiru, Oct 23 2018

I:=[0,1]; [n le 2 select I[n] else 2*Self(n-1) - 4*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 15 2018

M:= <<1+I,1+I>|>:
T:= <<-I/2,0>|<0,I/2>>:
seq(LinearAlgebra:-Trace(T.M^n),n=0..100); # Robert Israel, Jan 28 2016

Join[{a=0,b=1},Table[c=2*b-4*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011 *)
LinearRecurrence[{2, -4}, {0, 1}, 40] (* Vincenzo Librandi, Jan 29 2016 *)
Table[2^(n-2)*((-1)^Quotient[n-1,3]+(-1)^Quotient[n,3]), {n,0,40}] (*Federico Provvedi,Apr 24 2022*)

/* lists powers of any quaternion */
QuaternionToN(a,b,c,d,nmax) = {local (C);C = matrix(nmax+1,4);C[1,1]=1;for(n=2,nmax+1,C[n,1]=a*C[n-1,1]-b*C[n-1,2]-c*C[n-1,3]-d*C[n-1,4];C[n,2]=b*C[n-1,1]+a*C[n-1,2]+d*C[n-1,3]-c*C[n-1,4];C[n,3]=c*C[n-1,1]-d*C[n-1,2]+a*C[n-1,3]+b*C[n-1,4];C[n,4]=d*C[n-1,1]+c*C[n-1,2]-b*C[n-1,3]+a*C[n-1,4];);return (C);} /* Stanislav Sykora, Jun 11 2012 */

my(x='x+O('x^30)); concat([0], Vec(x/(1-2*x+4*x^2))) \\ G. C. Greubel, Oct 22 2018

a(n) = 2^(n-1)*polchebyshev(n-1, 2, 1/2); \\ Michel Marcus, May 02 2022

[lucas_number1(n,2,4) for n in range(0, 39)] # Zerinvary Lajos, Apr 23 2009

G.f.: x/(1-2*x+4*x^2).
E.g.f.: exp(x)*sin(sqrt(3)*x)/sqrt(3).
a(n) = 2*a(n-1) - 4*a(n-2), a(0)=0, a(1)=1.
a(n) = ((1+i*sqrt(3))^n - (1-i*sqrt(3))^n)/(2*i*sqrt(3)).
a(n) = Im( (1+i*sqrt(3))^n/sqrt(3) ).
a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k+1)*(-3)^k.
From Paul Curtz, Oct 04 2009: (Start)
a(n) = a(n-1) + a(n-2) + 2*a(n-3).
a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3).
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4). (End)
E.g.f.: exp(x)*sin(sqrt(3)*x)/sqrt(3) = G(0)*x^2 where G(k)= 1 + (3*k+2)/(2*x - 32*x^5/(16*x^4 - 3*(k+1)*(3*k+2)*(3*k+4)*(3*k+5)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jul 26 2012
G.f.: x/(1-2*x+4*x^2) = 2*x^2*G(0) where G(k)= 1 + 1/(2*x - 32*x^5/(16*x^4 - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jul 27 2012
a(n) = -2^(n-1)*Product_{k=1..n}(1 + 2*cos(k*Pi/n)) for n >= 1. - Peter Luschny, Nov 28 2019
a(n) = 2^(n-1) * U(n-1, 1/2), where U(n, x) is the Chebyshev polynomial of the second kind. - Federico Provvedi, Apr 24 2022

A016140 Expansion of 1/((1-3x)(1-8*x)).

Original entry on oeis.org

1, 11, 97, 803, 6505, 52283, 418993, 3354131, 26839609, 214736555, 1717951489, 13743789059, 109950843913, 879608345627, 7036871547985, 56294986732787, 450359936909017, 3602879624412299, 28823037382718881, 230584300224012515, 1844674405278884521, 14757395252691429371, 118059162052912494577, 944473296517443135443

Offset: 0

N. J. A. Sloane

In general, for expansion of 1/((1-b*x)*(1-c*x)): a(n) = (c^(n+1) - b^(n+1))/(c-b) = (b+c)*a(n-1) - b*c*a(n-2) = b*a(n-1) + c^n = c*a(n-1) + b^n = Sum_{i=0..n} b^i*c^(n-i). - Henry Bottomley, Jul 20 2000

8*a(n) gives the number of edges in the n-th-order Sierpiński carpet graph. - Eric W. Weisstein, Aug 19 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (11,-24).

Sequences with g.f. 1/((1-n*x)*(1-8*x)): A001018 (n=0), A023001 (n=1), A016131 (n=2), this sequence (n=3), A016152 (n=4), A016162 (n=5), A016170 (n=6), A016177 (n=7), A053539 (n=8), A016185 (n=9), A016186 (n=10), A016187 (n=11), A016188 (n=12), A060195 (n=16).

Cf. A190543.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1-8*x)))); // Vincenzo Librandi, Jun 24 2013

Table[(8^(n+1)-3^(n+1))/5, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2011 *)
CoefficientList[Series[1/((1-3 x)(1-8 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 24 2013 *)
LinearRecurrence[{11,-24},{1,11},30] (* Harvey P. Dale, Feb 03 2022 *)

Vec(1/((1-3*x)*(1-8*x))+O(x^30)) \\ Charles R Greathouse IV, Sep 23 2012

[lucas_number1(n,11,24) for n in range(1, 30)] # Zerinvary Lajos, Apr 27 2009

a(n) = (8^(n+1) - 3^(n+1))/5.
a(n) = 11*a(n-1) - 24*a(n-2).
a(n) = 3*a(n-1) + 8^n.
a(n) = 8*a(n-1) + 3^n.
a(n) = Sum_{i=0..n} 3^i*8^(n-i).
E.g.f.: (1/5)*(8*exp(8*x) - 3*exp(3*x)). - G. C. Greubel, Nov 14 2024

A066004 Sum of digits of 8^n.

Original entry on oeis.org

1, 8, 10, 8, 19, 26, 19, 26, 37, 35, 37, 62, 64, 71, 46, 62, 73, 80, 82, 80, 82, 89, 109, 89, 109, 125, 100, 107, 118, 107, 118, 125, 127, 107, 118, 125, 145, 143, 145, 152, 172, 170, 172, 188, 181, 170, 190, 215, 172, 215, 235, 233, 217, 215

Offset: 0

N. J. A. Sloane, Dec 11 2001

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. sum of digits of k^n: A001370 (k=2), A004166 (k=3), A065713 (k=4), A066001 (k=5), A066002 (k=6), A066003(k=7), this sequence (k=8), A065999 (k=9), A066005 (k=11), A066006 (k=12), A175527 (k=13).

Table[Total[IntegerDigits[8^n]], {n, 0, 60}] (* Vincenzo Librandi, Oct 08 2013 *)

a(n) = sumdigits(8^n); \\ Michel Marcus, Nov 01 2013

a(n) = A007953(A001018(n)). - Michel Marcus, Nov 01 2013

A009992 Powers of 48: a(n) = 48^n.

Original entry on oeis.org

1, 48, 2304, 110592, 5308416, 254803968, 12230590464, 587068342272, 28179280429056, 1352605460594688, 64925062108545024, 3116402981210161152, 149587343098087735296, 7180192468708211294208, 344649238497994142121984, 16543163447903718821855232

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 48), L(1, 48), P(1, 48), T(1, 48). Essentially same as Pisot sequences E(48, 2304), L(48, 2304), P(48, 2304), T(48, 2304). See A008776 for definitions of Pisot sequences.
If X_1, X_2, ..., X_n is a partition of the set {1,2,...,2*n} into blocks of size 2 then, for n>=1, a(n) is equal to the number of functions f : {1,2,..., 2*n}->{1,2,3,4,5,6,7} such that for fixed y_1,y_2,...,y_n in {1,2,3,4,5,6,7} we have f(X_i)<>{y_i}, (i=1,2,...,n). - Milan Janjic, May 24 2007
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 48-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

T. D. Noe, Table of n, a(n) for n = 0..100
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (48).

Cf. A001018 (powers of 8), ..., A001029 (powers of 19), A009964 (powers of 20), ..., A009991 (powers of 47), A087752 (powers of 49).

Cf. A000079 (2^n), A000244 (3^n), A000302 (4^n), A000400 (6^n), A001018 (8^n), A001021 (12^n), A001025 (16^n), A009968 (24^n).

List([0..20],n->48^n); # Muniru A Asiru, Nov 21 2018

[48^n: n in [0..20]] // Vincenzo Librandi, Nov 21 2010

A009992 := n -> 48^n: seq(A009992(n), n=0..20); # M. F. Hasler, Apr 19 2015

48^Range[0, 15] (* Michael De Vlieger, Jan 13 2018 *)

A009992(n)=48^n \\ M. F. Hasler, Apr 19 2015

for n in range(0,20): print(48**n, end=', ') # Stefano Spezia, Nov 21 2018

[(48)^n for n in range(20)] # G. C. Greubel, Nov 21 2018

G.f.: 1/(1-48*x). - Philippe Deléham, Nov 24 2008
a(n) = 48^n; a(n) = 48*a(n-1), a(0)=1. - Vincenzo Librandi, Nov 21 2010
E.g.f.: exp(48*x). - Muniru A Asiru, Nov 21 2018

Edited by M. F. Hasler, Apr 19 2015

A013731 a(n) = 2^(3*n+2).

Original entry on oeis.org

4, 32, 256, 2048, 16384, 131072, 1048576, 8388608, 67108864, 536870912, 4294967296, 34359738368, 274877906944, 2199023255552, 17592186044416, 140737488355328, 1125899906842624, 9007199254740992, 72057594037927936, 576460752303423488, 4611686018427387904

Offset: 0

N. J. A. Sloane

Starting rank of the (j-1)-Washtenaw series for the fixed ratio 2^(-j-1) (see Griess). - J. Taylor (jt_cpp(AT)yahoo.com), Apr 03 2004
1/4 + 1/32 + 1/256 + 1/2048 + ... = 2/7. - Gary W. Adamson, Aug 29 2008
Independence number of the (n+1)-Sierpinski carpet graph. - Eric W. Weisstein, Sep 06 2017
Clique covering number of the (n+1)-Sierpinski carpet graph. - Eric W. Weisstein, Apr 22 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Robert L. Griess Jr. Pieces of 2^d: Existence and uniqueness for Barnes-Wall and Ypsilanti lattices, arXiv:math/0403480 [math.GR], Mar 28 2004. See Definition 14.21.
Tanya Khovanova, Recursive Sequences
Eric Weisstein's World of Mathematics, Clique Covering Number
Eric Weisstein's World of Mathematics, Independence Number
Eric Weisstein's World of Mathematics, Sierpinski Carpet Graph
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (8).

Cf. A001018, A013730, A198852.

Cf. A092811 (same sequence with 1 prepended).

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

seq(2^(3*n+2),n=0..19); # Nathaniel Johnston, Jun 26 2011

(* Start from Eric W. Weisstein, Sep 06 2017 *)
Table[2^(3 n + 2), {n, 0, 20}]
2^(3 Range[0, 20] + 2)
LinearRecurrence[{8}, {4}, 20]
CoefficientList[Series[-(4/(-1 + 8 x)), {x, 0, 20}], x]
(* End *)

a(n)=4<<(3*n) \\ Charles R Greathouse IV, Apr 07 2012

[lucas_number1(3*n, 2, 0) for n in range(1, 20)] # Zerinvary Lajos, Oct 27 2009

From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 8*a(n-1), n > 0; a(0)=4.
G.f.: 4/(1-8x). (End)
a(n) = A198852(n) + 1. - Michel Marcus, Aug 23 2013
a(n) = A092811(n+1). - Eric W. Weisstein, Sep 06 2017
a(n) = 4*A001018(n). - R. J. Mathar, May 21 2024
E.g.f.: 4*exp(8*x). - Stefano Spezia, May 29 2024

A084058 a(n) = 2a(n-1) + 7a(n-2) for n>1, a(0)=1, a(1)=1.

Original entry on oeis.org

1, 1, 9, 25, 113, 401, 1593, 5993, 23137, 88225, 338409, 1294393, 4957649, 18976049, 72655641, 278143625, 1064876737, 4076758849, 15607654857, 59752621657, 228758827313, 875786006225, 3352883803641, 12836269650857, 49142725927201

Offset: 0

Paul Barry, May 10 2003

Binomial transform of expansion of cosh(sqrt(8)x) (A001018 with interpolated zeros : 1, 0, 8, 0, 64, 0, 512, 0, ...); inverse binomial transform of A084128.

The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 8 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(8). - Cino Hilliard, Sep 25 2005

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,7).

Essentially a duplicate of A083100.

a:=[1,1];; for n in [3..30] do a[n]:=2*a[n-1]+7*a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-8); S:=[ ((1+r8)^n+(1-r8)^n)/2: n in [0..30] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 16 2008

I:=[1,1]; [n le 2 select I[n] else 2*Self(n-1) +7*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 01 2019

a[n_]:= Simplify[((1 + Sqrt[8])^n + (1 - Sqrt[8])^n)/2]; Array[a, 30, 0] (* Or *) CoefficientList[Series[(1-x)/(1-2x-7x^2), {x,0,30}], x] (* Or *) LinearRecurrence[{2, 7}, {1, 1}, 30] (* Robert G. Wilson v, Sep 18 2013 *)

my(x='x+O('x^30)); Vec((1-x)/(1-2*x-7*x^2)) \\ G. C. Greubel, Aug 01 2019

((1-x)/(1-2*x-7*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

a(n) = ((1+sqrt(8))^n + (1-sqrt(8))^n)/2.
G.f.: (1-x)/(1-2*x-7*x^2).
E.g.f.: exp(x) * cosh(sqrt(8)*x).
a(n) = Sum_{k=0..n} A098158(n,k)*8^(n-k). - Philippe Deléham, Dec 26 2007
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(8*k-1)/(x*(8*k+7) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
Satisfies recurrence relation system a(n) = 2*b(n-1) - a(n-1), b(n) = 3*b(n-1) + 2*a(n-1), a(0)=1, b(0)=1. - Ilya Gutkovskiy, Apr 11 2017

A087752 Powers of 49.

Original entry on oeis.org

1, 49, 2401, 117649, 5764801, 282475249, 13841287201, 678223072849, 33232930569601, 1628413597910449, 79792266297612001, 3909821048582988049, 191581231380566414401, 9387480337647754305649, 459986536544739960976801, 22539340290692258087863249, 1104427674243920646305299201

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Oct 02 2003

Same as Pisot sequences E(1, 49), L(1, 49), P(1, 49), T(1, 49). Essentially same as Pisot sequences E(49, 2401), L(49, 2401), P(49, 2401), T(49, 2401). See A008776 for definitions of Pisot sequences.

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 49-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

T. D. Noe, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (49).

Bisection of A000420.
Cf. A001018 (powers of 8), ..., A001029 (powers of 19), A009964 (powers of 20), ..., A009992 (powers of 48).
Cf. A005843, A008776.

[49^n: n in [0..20]]; // Vincenzo Librandi, Nov 21 2010

49^Range[0,20] (* or *) Join[{1},NestList[49#&,49,20]] (* Harvey P. Dale, May 10 2019 *)

makelist(49^n,n,0,20); /* Martin Ettl, Nov 12 2012 */

PARI
```
a(n)=49^n \\ M. F. Hasler, Apr 19 2015
```

G.f.: 1/(1-49*x). - Philippe Deléham, Nov 24 2008
From Vincenzo Librandi, Nov 21 2010: (Start)
a(n) = 49^n.
a(n) = 49*a(n-1), a(0)=1. (End)
From Elmo R. Oliveira, Jul 08 2025: (Start)
E.g.f.: exp(49*x).
a(n) = A000420(A005843(n)). (End)

Edited by M. F. Hasler, Apr 19 2015

A089357 a(n) = 2^(6*n).

Original entry on oeis.org

1, 64, 4096, 262144, 16777216, 1073741824, 68719476736, 4398046511104, 281474976710656, 18014398509481984, 1152921504606846976, 73786976294838206464, 4722366482869645213696, 302231454903657293676544, 19342813113834066795298816

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Dec 26 2003

For n > 0, numbers M such that a(n) is the highest power of 2 in the Collatz (3x+1) iteration are given by 2^k*(a(n)-1)/3 for any k >= 0. Example: For n = 1, the numbers such that 64 is the highest power of 2 in the Collatz (3x+1) iteration are given by 2^k*(64-1)/3 = 21*2^k for any k >= 0. See A008908 for more information on the Collatz (3x+1) iteration. - Derek Orr, Sep 22 2014

Powers of 64. - Alexander Fraebel, Aug 29 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (64).

Cf. A000079, A000302, A001018, A001025, A008588, A008908, A009976.

[2^(6*n): n in [0..20]]; // Vincenzo Librandi, Jun 07 2011

seq(2^(6*n), n=0..14); # Nathaniel Johnston, Jun 26 2011

2^(6 Range[0,20]) (* or *) NestList[64#&,1,20] (* Harvey P. Dale, Sep 28 2011 *)

makelist(2^(6*n),n,0,20); /* Martin Ettl, Nov 12 2012 */

a(n)=64^n \\ Charles R Greathouse IV, Jul 02 2016

G.f.: 1/(1-64*x). - Philippe Deléham, Nov 24 2008
a(n) = 63*a(n-1) + 64^(n-1), a(0)=1. - Vincenzo Librandi, Jun 07 2011
E.g.f.: exp(64*x). - Ilya Gutkovskiy, Jul 02 2016
a(n) = A000079(A008588(n)). - Wesley Ivan Hurt, Jul 02 2016
a(n) = 64*a(n-1). - Miquel Cerda, Oct 27 2016

A171890 Octonomial coefficient array.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 28, 36, 42, 46, 48, 48, 46, 42, 36, 28, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 84, 120, 161, 204, 246, 284, 315, 336, 344, 336, 315, 284, 246, 204, 161, 120, 84, 56, 35

Offset: 0

N. J. A. Sloane, Oct 19 2010

Row lengths are 1,8,15,22,... = 1+7n = A016993(n). Row sums are 1,8,64,... = 8^n = A001018(n). M. F. Hasler, Jun 17 2012

			Array begins:
[1]
[1, 1, 1, 1, 1, 1, 1, 1]
[1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1]
...

T. D. Noe, Rows n = 0..25, flattened

The q-nomial arrays are for q=2..10: A007318 (Pascal), A027907, A008287,A035343, A063260, A063265, A171890, A213652, A213651.

#Define the r-nomial coefficients for r = 1, 2, 3, ...
rnomial := (r,n,k) -> add((-1)^i*binomial(n,i)*binomial(n+k-1-r*i,n-1), i = 0..floor(k/r)):
#Display the 8-nomials as a table
r := 8:  rows := 10:
for n from 0 to rows do
seq(rnomial(r,n,k), k = 0..(r-1)*n)
end do;
# Peter Bala, Sep 07 2013

Flatten[Table[CoefficientList[(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7)^n, x], {n, 0, 10}]] (* T. D. Noe, Apr 04 2011 *)

concat(vector(5, k, Vec(sum(j=0, 7, x^j)^k)))  \\ M. F. Hasler, Jun 17 2012

Row n has g.f. (1+x+...+x^7)^n.

T(n,k) = sum {i = 0..floor(k/8)} (-1)^i*binomial(n,i)*binomial(n+k-1-8*i,n-1) for n >= 0 and 0 <= k <= 7*n. - Peter Bala, Sep 07 2013

cp's OEIS Frontend

A016177 a(n) = 8^n - 7^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Formula

A088138 Generalized Gaussian Fibonacci integers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

PARI

Sage

Formula

A016140 Expansion of 1/((1-3*x)*(1-8*x)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A066004 Sum of digits of 8^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A009992 Powers of 48: a(n) = 48^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Python

Sage

Formula

Extensions

A013731 a(n) = 2^(3*n+2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

A016140 Expansion of 1/((1-3x)(1-8*x)).

A084058 a(n) = 2a(n-1) + 7a(n-2) for n>1, a(0)=1, a(1)=1.