A001018 -id:A001018 - cp's OEIS frontend

A055274 First differences of 8^n (A001018).

1, 7, 56, 448, 3584, 28672, 229376, 1835008, 14680064, 117440512, 939524096, 7516192768, 60129542144, 481036337152, 3848290697216, 30786325577728, 246290604621824, 1970324836974592, 15762598695796736, 126100789566373888, 1008806316530991104

Offset: 0

Barry E. Williams, May 28 2000

For n>=1, a(n) is equal to the number of functions f:{1,2...,n}->{1,2,3,4,5,6,7,8} such that for a fixed x in {1,2,...,n} and a fixed y in {1,2,3,4,5,6,7,8} we have f(x)<>y. - Aleksandar M. Janjic and Milan Janjic, Mar 27 2007
a(n) is the number of compositions of n when there are 7 types of each natural number. - Milan Janjic, Aug 13 2010
For n>0, a(n) is not the sum of two nonnegative cubes. - Bruno Berselli, Mar 20 2012

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
F. Conti, R. Dvornicich, T. Franzoni and S. Mortola, Il Fibonacci N. 0 (included in Il Fibonacci, Unione Matematica Italiana, 2011), 1990, Problem 0.12.4 (see Berselli's comment).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (8).

Cf. A001018.

[1] cat [7*8^(n-1): n in [1..20]]; // G. C. Greubel, Mar 16 2020

1, seq(7*8^(n-1), n=1..20); # G. C. Greubel, Mar 16 2020

q = 8; Join[{a = 1}, Table[If[n == 0, a = q*a - 1, a = q*a], {n, 0, 25}]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)

my(x='x+O('x^66)); Vec((1-x)/(1-8*x)) /* Joerg Arndt, Jun 25 2011 */

[1]+[7*8^(n-1) for n in (1..20)] # G. C. Greubel, Mar 16 2020

G.f.: (1-x)/(1-8*x).
G.f.: 1/( 1 - 7*Sum_{k>=1} x^k ).
a(n) = 7*8^(n-1); a(0)=1.
a(n) = 8*a(n-1) + (-1)^n * C(1, 1-n).
a(n) = 7*Sum_{k=0..n-1} a(k), for n>=1. - Adi Dani, Jun 24 2011
E.g.f.: (1 + 7*exp(8*x))/8. - G. C. Greubel, Mar 16 2020

More terms from James Sellers, May 29 2000

A000007 The characteristic function of {0}: a(n) = 0^n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

Changing the offset to 1 gives the arithmetical function a(1) = 1, a(n) = 0 for n > 1, the identity function for Dirichlet multiplication (see Apostol). - N. J. A. Sloane
Changing the offset to 1 makes this the decimal expansion of 1. - N. J. A. Sloane, Nov 13 2014
Hankel transform (see A001906 for definition) of A000007 (powers of 0), A000012 (powers of 1), A000079 (powers of 2), A000244 (powers of 3), A000302 (powers of 4), A000351 (powers of 5), A000400 (powers of 6), A000420 (powers of 7), A001018 (powers of 8), A001019 (powers of 9), A011557 (powers of 10), A001020 (powers of 11), etc. - Philippe Deléham, Jul 07 2005
This is the identity sequence with respect to convolution. - David W. Wilson, Oct 30 2006
a(A000004(n)) = 1; a(A000027(n)) = 0. - Reinhard Zumkeller, Oct 12 2008
The alternating sum of the n-th row of Pascal's triangle gives the characteristic function of 0, a(n) = 0^n. - Daniel Forgues, May 25 2010
The number of maximal self-avoiding walks from the NW to SW corners of a 1 X n grid. - Sean A. Irvine, Nov 19 2010
Historically there has been some disagreement as to whether 0^0 = 1. Graphing x^0 seems to support that conclusion, but graphing 0^x instead suggests that 0^0 = 0. Euler and Knuth have argued in favor of 0^0 = 1. For some calculators, 0^0 triggers an error, while in Mathematica, 0^0 is Indeterminate. - Alonso del Arte, Nov 15 2011
Another consequence of changing the offset to 1 is that then this sequence can be described as the sum of Moebius mu(d) for the divisors d of n. - Alonso del Arte, Nov 28 2011
With the convention 0^0 = 1, 0^n = 0 for n > 0, the sequence a(n) = 0^|n-k|, which equals 1 when n = k and is 0 for n >= 0, has g.f. x^k. A000007 is the case k = 0. - George F. Johnson, Mar 08 2013
A fixed point of the run length transform. - Chai Wah Wu, Oct 21 2016

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 30.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

David Wasserman, Table of n, a(n) for n = 0..1000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.4.
Dr. Math, 0^0 (zero to the zero power)
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.
Donald E. Knuth, Two notes on notation, arXiv:math/9205211 [math.HO], 1992. See page 6 on 0^0.
Robert Price, Comments on A000007, Jan 27 2016
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for "core" sequences
Index entries for characteristic functions
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (1).

Characteristic function of {g}: this sequence (g = 0), A063524 (g = 1), A185012 (g = 2), A185013 (g = 3), A185014 (g = 4), A185015 (g = 5), A185016 (g = 6), A185017 (g = 7). - Jason Kimberley, Oct 14 2011
Characteristic function of multiples of g: this sequence (g = 0), A000012 (g = 1), A059841 (g = 2), A079978 (g = 3), A121262 (g = 4), A079998 (g = 5), A079979 (g = 6), A082784 (g = 7). - Jason Kimberley, Oct 14 2011
Cf. A074909, A027641, A057427.

a000007 = (0 ^)
a000007_list = 1 : repeat 0
-- Reinhard Zumkeller, May 07 2012, Mar 27 2012

[1] cat [0:n in [1..100]]; // Sergei Haller, Dec 21 2006

A000007 := proc(n) if n = 0 then 1 else 0 fi end: seq(A000007(n), n=0..20);
spec := [A, {A=Z} ]: seq(combstruct[count](spec, size=n+1), n=0..20);

Table[If[n == 0, 1, 0], {n, 0, 99}]
Table[Boole[n == 0], {n, 0, 99}] (* Michael Somos, Aug 25 2012 *)
Join[{1},LinearRecurrence[{1},{0},102]] (* Ray Chandler, Jul 30 2015 *)
PadRight[{1},120,0] (* Harvey P. Dale, Jul 18 2024 *)

PARI
```
{a(n) = !n};
```

def A000007(n): return int(n==0) # Chai Wah Wu, Feb 04 2022

Multiplicative with a(p^e) = 0. - David W. Wilson, Sep 01 2001
a(n) = floor(1/(n + 1)). - Franz Vrabec, Aug 24 2005
As a function of Bernoulli numbers (cf. A027641: (1, -1/2, 1/6, 0, -1/30, ...)), triangle A074909 (the beheaded Pascal's triangle) * B_n as a vector = [1, 0, 0, 0, 0, ...]. - Gary W. Adamson, Mar 05 2012
a(n) = Sum_{k = 0..n} exp(2*Pi*i*k/(n+1)) is the sum of the (n+1)th roots of unity. - Franz Vrabec, Nov 09 2012
a(n) = (1-(-1)^(2^n))/2. - Luce ETIENNE, May 05 2015
a(n) = 1 - A057427(n). - Alois P. Heinz, Jan 20 2016
From Ilya Gutkovskiy, Sep 02 2016: (Start)
Binomial transform of A033999.
Inverse binomial transform of A000012. (End)

A000420 Powers of 7: a(n) = 7^n.

Original entry on oeis.org

1, 7, 49, 343, 2401, 16807, 117649, 823543, 5764801, 40353607, 282475249, 1977326743, 13841287201, 96889010407, 678223072849, 4747561509943, 33232930569601, 232630513987207, 1628413597910449, 11398895185373143, 79792266297612001, 558545864083284007

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 7), L(1, 7), P(1, 7), T(1, 7). Essentially same as Pisot sequences E(7, 49), L(7, 49), P(7, 49), T(7, 49). See A008776 for definitions of Pisot sequences.
Sum of coefficients of expansion of (1+x+x^2+x^3+x^4+x^5+x^6)^n.
a(n) is number of compositions of natural numbers into n parts < 7.
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 7-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
Numbers n such that sigma(7n) = 7n + sigma(n). - Jahangeer Kholdi, Nov 23 2013
Number of ways to assign truth values to n ternary disjunctions connected by conjunctions such that the proposition is true. For example, a(2) = 49, since for the proposition '(a v b v c) & (d v e v f)' there are 49 assignments that make the proposition true. - Ori Milstein, Dec 31 2022
Equivalently, the number of length-n words over an alphabet with seven letters. - Joerg Arndt, Jan 01 2023

			a(2)=49 there are 49 compositions of natural numbers into 2 parts < 7.

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
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 272
Tanya Khovanova, Recursive Sequences
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Index entries for linear recurrences with constant coefficients, signature (7).

Cf. A000079 (powers of 2), A000244 (powers of 3), A000302 (powers of 4), A000351 (powers of 5), A000400 (powers of 6), A001018 (powers of 8), ..., A001029 (powers of 19), A009964 (powers of 20), ..., A009992 (powers of 48), A087752 (powers of 49).

a000420 = (7 ^)
a000420_list = iterate (* 7) 1  -- Reinhard Zumkeller, Apr 29 2015

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

A000420:=-1/(-1+7*z); # Simon Plouffe in his 1992 dissertation. [This is actually the generating function, so convert(series(...),list) would yield the actual sequence. - M. F. Hasler, Apr 19 2015]
A000420 := n -> 7^n; # M. F. Hasler, Apr 19 2015

Table[7^n, {n,0,50}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)

makelist(7^n,n,0,20); /* Martin Ettl, Dec 27 2012 */

a(n)=7^n \\ Charles R Greathouse IV, Jul 28 2015

a(n) = 7^n.
a(0) = 1; a(n) = 7*a(n-1).
G.f.: 1/(1-7*x).
E.g.f.: exp(7*x).
4/7 - 5/7^2 + 4/7^3 - 5/7^4 + ... = 23/48. [Jolley, Summation of Series, Dover, 1961]

A008588 Nonnegative multiples of 6.

Original entry on oeis.org

0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, 210, 216, 222, 228, 234, 240, 246, 252, 258, 264, 270, 276, 282, 288, 294, 300, 306, 312, 318, 324, 330, 336, 342, 348

Offset: 0

N. J. A. Sloane

For n > 3, the number of squares on the infinite 3-column half-strip chessboard at <= n knight moves from any fixed point on the short edge.
Second differences of A000578. - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004
A008615(a(n)) = n. - Reinhard Zumkeller, Feb 27 2008
A157176(a(n)) = A001018(n). - Reinhard Zumkeller, Feb 24 2009
These numbers can be written as the sum of four cubes (i.e., 6*n = (n+1)^3 + (n-1)^3 + (-n)^3 + (-n)^3). - Arkadiusz Wesolowski, Aug 09 2013
A122841(a(n)) > 0 for n > 0. - Reinhard Zumkeller, Nov 10 2013
Surface area of a cube with side sqrt(n). - Wesley Ivan Hurt, Aug 24 2014
a(n) is representable as a sum of three but not two consecutive nonnegative integers, e.g., 6 = 1 + 2 + 3, 12 = 3 + 4 + 5, 18 = 5 + 6 + 7, etc. (see A138591). - Martin Renner, Mar 14 2016 (Corrected by David A. Corneth, Aug 12 2016)
Numbers with three consecutive divisors: for some k, each of k, k+1, and k+2 divide n. - Charles R Greathouse IV, May 16 2016
Numbers k for which {phi(k),phi(2k),phi(3k)} is an arithmetic progression. - Ivan Neretin, Aug 12 2016

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 81.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 318
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Essentially the same as A008458.
Cf. A016921, A016933, A016945, A016957, A016969, A138591.
Cf. A044102 (subsequence).

a008588 = (* 6)
a008588_list = [0, 6 ..]  -- Reinhard Zumkeller, Nov 10 2013

[6*n: n in [0..60] ]; // Vincenzo Librandi, Jul 16 2011

Maple
```
[ seq(6*n,n=0..45) ];
```

Range[0, 500, 6] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
NestList[6+#&,0,60] (* Harvey P. Dale, Nov 17 2024 *)

makelist(6*n,n,0,30); /* Martin Ettl, Nov 12 2012 */

a(n)=6*n \\ Charles R Greathouse IV, Feb 08 2012

[6*n for n in range(59)] # Stefano Spezia, Mar 09 2025

From Vincenzo Librandi, Dec 24 2010: (Start)
a(n) = 6*n = 2*a(n-1) - a(n-2).
G.f.: 6*x/(1-x)^2. (End)
a(n) = Sum_{k>=0} A030308(n,k)*6*2^k. - Philippe Deléham, Oct 24 2011
a(n) = Sum_{k=2n-1..2n+1} k. - Wesley Ivan Hurt, Nov 22 2015
From Ilya Gutkovskiy, Aug 12 2016: (Start)
E.g.f.: 6*x*exp(x).
Convolution of A010722 and A057427.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/6 = A002162*A020793. (End)
a(n) = 6 * A001477(n). - David A. Corneth, Aug 12 2016

A001025 Powers of 16: a(n) = 16^n.

Original entry on oeis.org

1, 16, 256, 4096, 65536, 1048576, 16777216, 268435456, 4294967296, 68719476736, 1099511627776, 17592186044416, 281474976710656, 4503599627370496, 72057594037927936, 1152921504606846976, 18446744073709551616, 295147905179352825856, 4722366482869645213696, 75557863725914323419136, 1208925819614629174706176

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 16), L(1, 16), P(1, 16), T(1, 16). Essentially same as Pisot sequences E(16, 256), L(16, 256), P(16, 256), T(16, 256). See A008776 for definitions of Pisot sequences.
Convolution-square (auto-convolution) of A098430. - R. J. Mathar, May 22 2009
Subsequence of A161441: A160700(a(n)) = 1. - Reinhard Zumkeller, Jun 10 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 16-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

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).

Muniru A Asiru, Table of n, a(n) for n = 0..820 (terms n = 0..100 from T. D. Noe)
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 280
Tanya Khovanova, Recursive Sequences
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Index entries for linear recurrences with constant coefficients, signature (16).

Partial sums give A131865.

Cf. A000079, A000302, A001018, A005843, A008586.

List([0..20],n->16^n); # Muniru A Asiru, Nov 07 2018

a001025 = (16 ^)
a001025_list = iterate (* 16) 1  -- Reinhard Zumkeller, Nov 07 2012

A001025:=-1/(-1+16*z); # Simon Plouffe in his 1992 dissertation

Table[4^(2*n), {n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Mar 01 2009 *)

A001025(n):=16^n$
makelist(A001025(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

a(n)=1<<(4*n) \\ Charles R Greathouse IV, Feb 01 2012

print([16**n for n in range(20)]) # Stefano Spezia, Nov 10 2018

[lucas_number1(n,16,0) for n in range(1, 18)] # Zerinvary Lajos, Apr 29 2009

G.f.: 1/(1-16*x).
E.g.f.: exp(16*x).
From Muniru A Asiru, Nov 07 2018: (Start)
a(n) = 16^n.
a(0) = 1, a(n) = 16*a(n-1). (End)
a(n) = 4^A005843(n) = 2^A008586(n) = A000302(n)^2 = A000079(n)*A001018(n). - Muniru A Asiru, Nov 10 2018
a(n) = ( Sum_{k = 0..n} (2*k + 1)*binomial(2*n + 1, n - k) ) * ( Sum_{k = 0..n} (-1)^k/(2*k + 1)*binomial(2*n + 1, n - k) ). - Peter Bala, Feb 12 2019
a(n) = Sum_{k = 0..2*n} A000984(k) * A000984(2*n-k). - Peter Bala, Aug 23 2025

A062395 a(n) = 8^n + 1.

Original entry on oeis.org

2, 9, 65, 513, 4097, 32769, 262145, 2097153, 16777217, 134217729, 1073741825, 8589934593, 68719476737, 549755813889, 4398046511105, 35184372088833, 281474976710657, 2251799813685249, 18014398509481985, 144115188075855873

Offset: 0

Henry Bottomley, Jun 22 2001

Any number of the form b^k+1 is composite for b>2 and k odd since b+1 algebraically divides b^k+1. - Robert G. Wilson v, Aug 25 2002

D. M. Burton, Elementary Number Theory, Allyn and Bacon, Boston, MA, 1976, pp. 51.
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

Vincenzo Librandi, Table of n, a(n) for n = 0..140
Index entries for linear recurrences with constant coefficients, signature (9,-8).

Cf. A054977, A007395, A000051, A034472, A052539, A034474, A062394, A034491, A062396, A062397, A007689, A063376, A063481, A074600 - A074624, A034524, A178248, A228081 for numbers one more than powers.

[8^n + 1: n in [0..40] ]; // Vincenzo Librandi, Apr 30 2011

Table[8^n + 1, {n, 0, 20}]
LinearRecurrence[{9,-8},{2,9},20] (* Harvey P. Dale, Jan 24 2019 *)

PARI
```
for(n=0,22,print(8^n+1))
```

a(n) = 8a(n-1)-7 = A001018(n)+1 = 9a(n-1) - 8a(n-2).
G.f.: -(-2+9*x)/(-1+x)/(-1+8*x). - R. J. Mathar, Nov 16 2007
E.g.f.: e^x+e^(8*x). - Mohammad K. Azarian, Jan 02 2009

A003992 Square array read by upwards antidiagonals: T(n,k) = n^k for n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 9, 8, 1, 0, 1, 5, 16, 27, 16, 1, 0, 1, 6, 25, 64, 81, 32, 1, 0, 1, 7, 36, 125, 256, 243, 64, 1, 0, 1, 8, 49, 216, 625, 1024, 729, 128, 1, 0, 1, 9, 64, 343, 1296, 3125, 4096, 2187, 256, 1, 0, 1, 10, 81, 512, 2401, 7776, 15625, 16384, 6561, 512, 1, 0

Offset: 0

Marc LeBrun

If the array is transposed, T(n,k) is the number of oriented rows of n colors using up to k different colors. The formula would be T(n,k) = [n==0] + [n>0]*k^n. The generating function for column k would be 1/(1-k*x). For T(3,2)=8, the rows are AAA, AAB, ABA, ABB, BAA, BAB, BBA, and BBB. - Robert A. Russell, Nov 08 2018

T(n,k) is the number of multichains of length n from {} to [k] in the Boolean lattice B_k. - Geoffrey Critzer, Apr 03 2020

			Rows begin:
[1, 0,  0,   0,    0,     0,      0,      0, ...],
[1, 1,  1,   1,    1,     1,      1,      1, ...],
[1, 2,  4,   8,   16,    32,     64,    128, ...],
[1, 3,  9,  27,   81,   243,    729,   2187, ...],
[1, 4, 16,  64,  256,  1024,   4096,  16384, ...],
[1, 5, 25, 125,  625,  3125,  15625,  78125, ...],
[1, 6, 36, 216, 1296,  7776,  46656, 279936, ...],
[1, 7, 49, 343, 2401, 16807, 117649, 823543, ...], ...

T. D. Noe, Rows n = 0..50 of triangle, flattened

Rows 0-49 are A000007, A000012, A000079, A000244, A000302, A000351, A000400, A000420, A001018, A001019, A011557, A001020, A001021, A001022, A001023, A001024, A001025, A001026, A001027, A001029, A009964-A009992, A087752.
Columns 0-26 are A000012, A001477, A000290, A000578, A000583, A000584, A001014, A001015, A001016, A001017, A008454, A008455, A008456, A010801-A010813, A089081.
Main diagonal is A000312. Other diagonals include A000169, A007778, A000272, A008788. Antidiagonal sums are in A026898.
Cf. A099555.
Transpose is A004248. See A051128, A095884, A009999 for other versions.
Cf. A277504 (unoriented), A293500 (chiral).

[[(n-k)^k: k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 08 2018

Table[If[k == 0, 1, (n - k)^k], {n, 0, 11}, {k, 0, n}]//Flatten

T(n,k) = (n-k)^k \\ Charles R Greathouse IV, Feb 07 2017

E.g.f.: Sum T(n,k)*x^n*y^k/k! = 1/(1-x*exp(y)). - Paul D. Hanna, Oct 22 2004

E.g.f.: Sum T(n,k)*x^n/n!*y^k/k! = e^(x*e^y). - Franklin T. Adams-Watters, Jun 23 2006

More terms from David W. Wilson

Edited by Paul D. Hanna, Oct 22 2004

A016185 a(n) = 9^n - 8^n.

Original entry on oeis.org

0, 1, 17, 217, 2465, 26281, 269297, 2685817, 26269505, 253202761, 2413042577, 22791125017, 213710059745, 1992110014441, 18478745943857, 170706760005817, 1571545212141185, 14425381885981321, 132080236787517137, 1206736529597136217, 11004743954450081825

Offset: 0

N. J. A. Sloane

a(n) is also the number of n-digit numbers whose smallest decimal digit is 1. - Stefano Spezia, Nov 15 2023

G. C. Greubel, Table of n, a(n) for n = 0..1000
John Elias, Illustration of initial terms: Sierpinski Carpet
Wikipedia, Sierpinski carpet
Index entries for linear recurrences with constant coefficients, signature (17,-72).

Cf. A001018, A001019, A016140.

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

LinearRecurrence[{17,-72}, {0,1}, 31] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2011 *)

def A016185(n): return pow(9,n) - pow(8,n)
print([A016185(n) for n in range(41)]) # G. C. Greubel, Nov 29 2024

G.f.: x/((1-8*x)*(1-9*x)).
E.g.f.: e^(9*x) - e^(8*x). - Mohammad K. Azarian, Jan 14 2009
a(n) = 9*a(n-1) + 8^(n-1), a(0)=0. - Vincenzo Librandi, Feb 09 2011
a(n) = 17*a(n-1) - 72*a(n-2), a(0)=0, a(1)=1. - Vincenzo Librandi, Feb 09 2011
a(n) = A001019(n) - A001018(n). - Alois P. Heinz, Aug 19 2021

A081138 8th binomial transform of (0,0,1,0,0,0, ...).

Original entry on oeis.org

0, 0, 1, 24, 384, 5120, 61440, 688128, 7340032, 75497472, 754974720, 7381975040, 70866960384, 670014898176, 6253472382976, 57724360458240, 527765581332480, 4785074604081152, 43065671436730368, 385057768140177408

Offset: 0

Paul Barry, Mar 08 2003

Starting at 1, the three-fold convolution of A001018 (powers of 8).

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (24,-192,512).

Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), A027472 (q=3), A038845 (q=4), A081135 (q=5), A081136 (q=6), A027474 (q=7), this sequence (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

[8^n*Binomial(n+2, 2): n in [-2..20]]; // Vincenzo Librandi, Oct 16 2011

LinearRecurrence[{24,-192,512},{0,0,1},30] (* Harvey P. Dale, Jun 08 2014 *)

a(n) = 24*a(n-1) - 192*a(n-2) + 512*a(n-3) for n>2, a(0)=a(1)=0, a(2)=1.
a(n) = 8^(n-2)*binomial(n, 2).
G.f.: x^2/(1 - 8*x)^3.
E.g.f.: (x^2/2)*exp(8*x). - G. C. Greubel, May 13 2021
From Amiram Eldar, Jan 06 2022: (Start)
Sum_{n>=2} 1/a(n) = 16 - 112*log(8/7).
Sum_{n>=2} (-1)^n/a(n) = 144*log(9/8) - 16. (End)

A024088 a(n) = 8^n - 1.

Original entry on oeis.org

0, 7, 63, 511, 4095, 32767, 262143, 2097151, 16777215, 134217727, 1073741823, 8589934591, 68719476735, 549755813887, 4398046511103, 35184372088831, 281474976710655, 2251799813685247, 18014398509481983

Offset: 0

N. J. A. Sloane

Numbers whose base 8 or octal representation is 777777.......7. - Zerinvary Lajos, Feb 03 2007

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

Cf. A000225, A001576, A001018, A248725.

List([0..30], n-> 8^n -1); # G. C. Greubel, Aug 03 2019

[8^n -1: n in [0..20]]; // G. C. Greubel, Aug 03 2019

8^Range[0,20]-1 (* or *) LinearRecurrence[{9,-8},{0,7},20] (* Harvey P. Dale, Jan 04 2017 *)

vector(20, n, n--; 8^n -1) \\ G. C. Greubel, Aug 03 2019

[gaussian_binomial(3*n,1,2) for n in range(0,20)] # Zerinvary Lajos, May 28 2009

[stirling_number2(3*n+1,2) for n in range(0,20)] # Zerinvary Lajos, Nov 26 2009

[8^n-1 for n in (0..20)] # Bruno Berselli, Nov 11 2015

From Mohammad K. Azarian, Jan 14 2009: (Start)
G.f.: 1/(1-8*x) - 1/(1-x).
E.g.f.: exp(8*x) - exp(x). (End)
a(n) = A000225(n)*A001576(n). - Reinhard Zumkeller, Feb 15 2009
a(n) = 8*a(n-1) + 7 for n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
a(n) = Sum_{i=1..n} 7^i*binomial(n,n-i) for n>0, a(0)=0. - Bruno Berselli, Nov 11 2015
a(n) = A001018(n) - 1. - Sean A. Irvine, Jun 19 2019
Sum_{n>=1} 1/a(n) = A248725. - Amiram Eldar, Nov 13 2020

cp's OEIS Frontend

A055274 First differences of 8^n (A001018).

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A000007 The characteristic function of {0}: a(n) = 0^n.

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A000420 Powers of 7: a(n) = 7^n.

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A008588 Nonnegative multiples of 6.

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A001025 Powers of 16: a(n) = 16^n.

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