A001018 -id:A001018 - cp's OEIS frontend

A308584 Number of ways to write n as a(a+1)/2 + b(b+1)/2 + 5^c*8^d, where a,b,c,d are nonnegative integers with a <= b.

1, 1, 1, 1, 2, 1, 3, 3, 2, 2, 4, 3, 1, 4, 2, 2, 4, 2, 2, 2, 4, 2, 3, 2, 3, 5, 2, 3, 5, 3, 3, 5, 2, 2, 4, 4, 4, 3, 4, 3, 5, 3, 5, 5, 2, 6, 7, 1, 3, 6, 4, 4, 4, 4, 2, 9, 3, 2, 4, 3, 7, 4, 4, 5, 5, 4, 6, 5, 3, 6, 8, 2, 5, 7, 3, 5, 7, 3, 3, 7, 5, 7, 3, 5, 5, 8, 1, 4, 8, 1, 7, 6, 3, 3, 9, 5, 4, 6, 4, 5

Offset: 1

Zhi-Wei Sun, Jun 08 2019

nonn

Conjecture: a(n) > 0 for all n > 0. Equivalently, each n = 1,2,3,... can be written as w^2 + x*(x+1) + 5^y*8^z with w,x,y,z nonnegative integers.
We have verified a(n) > 0 for all n = 1..4*10^8.
See also A308566 for a similar conjecture.
a(n) > 0 for all 0 < n < 10^10. - Giovanni Resta, Jun 10 2019

			a(13) = 1 with 13 = 3*4/2 + 3*4/2 + 5^0*8^0.
a(48) = 1 with 48 = 5*6/2 + 7*8/2 + 5^1*8^0.
a(87) = 1 with 87 = 1*2/2 + 12*13/2 + 5^0*8^1.
a(90) = 1 with 90 = 4*5/2 + 10*11/2 + 5^2*8^0.
a(423) = 1 with 423 = 9*10/2 + 22*23/2 + 5^3*8^0.
a(517) = 1 with 517 = 17*18/2 + 24*25/2 + 5^0*8^2.
a(985) = 1 with 985 = 19*20/2 + 34*35/2 + 5^2*8^1.
a(2694) = 1 with 2694 = 7*8/2 + 68*69/2 + 5^1*8^2.
a(42507) = 1 with 42507 = 178*179/2 + 223*224/2 + 5^2*8^2.
a(544729) = 1 with 544729 = 551*552/2 + 857*858/2 + 5^5*8^1.
a(913870) = 1 with 913870 = 559*560/2 + 700*701/2 + 5^3*8^4.
a(1843782) = 1 with 1843782 = 808*809/2 + 1668*1669/2 + 5^6*8^1.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.

Cf. A000217, A000351, A001018, A303656, A303637, A308411, A308547, A308566.

TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
tab={};Do[r=0;Do[If[TQ[n-5^k*8^m-x(x+1)/2],r=r+1],{k,0,Log[5,n]},{m,0,Log[8,n/5^k]},{x,0,(Sqrt[4(n-5^k*8^m)+1]-1)/2}];tab=Append[tab,r],{n,1,100}];Print[tab]

A008569 Digits of powers of 8.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

The constant whose decimal expansion is this sequence is irrational (Mahler, 1981). - Amiram Eldar, Mar 23 2025

			Triangle begins:
  1;
  8;
  6, 4;
  5, 1, 2;
  4, 0, 9, 6;
  3, 2, 7, 6, 8;
  2, 6, 2, 1, 4, 4;
  2, 0, 9, 7, 1, 5, 2;
  ...

Seiichi Manyama, Rows n = 0..100, flattened
Kurt Mahler, On some irrational decimal fractions, Journal of Number Theory, Vol. 13, No. 2 (1981), pp. 268-269.

Cf. A001018.

Cf. A000455, A008564, A008565, A008566, A008567, A008568, A008570, A008571, A008572, A008573, A010054.

IntegerDigits/@(8^Range[0,20])//Flatten (* Harvey P. Dale, Nov 14 2020 *)

A271939 Number of edges in the n-th order Sierpinski carpet graph.

Original entry on oeis.org

8, 88, 776, 6424, 52040, 418264, 3351944, 26833048, 214716872, 1717892440, 13743611912, 109950312472, 879606751304, 7036866765016, 56294972383880, 450359893862296, 3602879495272136, 28823036995298392, 230584299061751048, 1844674401792100120

Offset: 1

Emeric Deutsch, Apr 17 2016

Also the number of maximal and maximum cliques in the n-Sierpinski carpet graph. - Eric W. Weisstein, Dec 01 2017

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

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.
Eric Weisstein's World of Mathematics, Edge Count
Eric Weisstein's World of Mathematics, Maximal Clique
Eric Weisstein's World of Mathematics, Maximum Clique
Eric Weisstein's World of Mathematics, Sierpiński Carpet Graph
Index entries for linear recurrences with constant coefficients, signature (11,-24).

Cf. A016140.

Cf. A001018 (number of vertices in the n-Sierpinski carpet graph).

Maple
```
seq((1/5)*(8*(8^n-3^n)), n = 1 .. 20);
```

Table[8 (8^n - 3^n)/5, {n, 20}] (* Eric W. Weisstein, Jun 17 2017 *)
LinearRecurrence[{11, -24}, {8, 88}, 20] (* Eric W. Weisstein, Jun 17 2017 *)
CoefficientList[Series[8/(1 - 11 x + 24 x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Jun 17 2017 *)

x='x+O('x^99); Vec(8/((1-3*x)*(1-8*x))) \\ Altug Alkan, Apr 17 2016

a(n) = 8 * (8^n - 3^n)/5.
a(n) = 8 * A016140(n).
G.f.: 8*x / ( (8*x-1)*(3*x-1) ). - R. J. Mathar, Apr 17 2016
a(n) = 8*a(n-1) + 8*3^(n-1). - Allan Bickle, Nov 27 2022

A223331 T(n,k)=Rolling cube footprints: number of nXk 0..7 arrays starting with 0 where 0..7 label vertices of a cube and every array movement to a horizontal or antidiagonal neighbor moves along a corresponding cube edge.

Original entry on oeis.org

1, 3, 8, 9, 27, 64, 27, 189, 243, 512, 81, 1323, 3969, 2187, 4096, 243, 9261, 64827, 83349, 19683, 32768, 729, 64827, 1059723, 3176523, 1750329, 177147, 262144, 2187, 453789, 17324685, 121264857, 155649627, 36756909, 1594323, 2097152, 6561

Offset: 1

R. H. Hardin Mar 19 2013

Table starts
.........1..........3.............9................27....................81
.........8.........27...........189..............1323..................9261
........64........243..........3969.............64827...............1059723
.......512.......2187.........83349...........3176523.............121264857
......4096......19683.......1750329.........155649627...........13876429707
.....32768.....177147......36756909........7626831723.........1587890407761
....262144....1594323.....771895089......373714754427.......181703507374179
...2097152...14348907...16209796869....18312022966923.....20792470582897209
..16777216..129140163..340405734249...897289125379227...2379298227030964827
.134217728.1162261467.7148520419229.43967167143582123.272264906211251105313
Horizontal or vertical instead of horizontal or antidiagonal gives A222444

			Some solutions for n=3 k=4
..0..4..5..1....0..4..0..1....0..4..6..4....0..2..0..4....0..4..6..4
..5..4..0..1....5..1..5..1....0..2..0..2....6..2..6..4....6..2..6..7
..6..2..3..1....5..7..3..2....3..2..3..1....6..4..0..4....0..2..6..7
Vertex neighbors:
0 -> 1 2 4
1 -> 0 3 5
2 -> 0 3 6
3 -> 1 2 7
4 -> 0 5 6
5 -> 1 4 7
6 -> 2 4 7
7 -> 3 5 6

R. H. Hardin, Table of n, a(n) for n = 1..199

Column 1 is A001018(n-1)
Column 2 is A013708(n-1)
Column 3 is 9*21^(n-1)
Column 4 is 27*49^(n-1)
Row 1 is A000244(n-1)
Row 2 is 27*7^(n-2) for n>1

Empirical for column k:
k=1: a(n) = 8*a(n-1)
k=2: a(n) = 9*a(n-1)
k=3: a(n) = 21*a(n-1)
k=4: a(n) = 49*a(n-1)
k=5: a(n) = 117*a(n-1) -294*a(n-2)
k=6: a(n) = 282*a(n-1) -3969*a(n-2) +9604*a(n-3)
k=7: a(n) = 692*a(n-1) -43569*a(n-2) +847042*a(n-3) -6303164*a(n-4) +15731352*a(n-5)
Empirical for row n:
n=1: a(n) = 3*a(n-1)
n=2: a(n) = 7*a(n-1) for n>2
n=3: a(n) = 18*a(n-1) -27*a(n-2) for n>4
n=4: a(n) = 48*a(n-1) -402*a(n-2) +1064*a(n-3) -789*a(n-4) for n>7
n=5: [order 9] for n>13
n=6: [order 20] for n>25
n=7: [order 51] for n>57

A339688 a(n) = Sum_{d|n} 8^(d-1).

Original entry on oeis.org

1, 9, 65, 521, 4097, 32841, 262145, 2097673, 16777281, 134221833, 1073741825, 8589967945, 68719476737, 549756076041, 4398046515265, 35184374186505, 281474976710657, 2251799830495305, 18014398509481985, 144115188210078217, 1152921504607109185

Offset: 1

Ilya Gutkovskiy, Dec 12 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Column 8 of A308813.
Cf. A001018, A320073.
Sums of the form Sum_{d|n} q^(d-1): A034729 (q=2), A034730 (q=3), A113999 (q=10), A339684 (q=4), A339685 (q=5), A339686 (q=6), A339687 (q=7), this sequence (q=8), A339689 (q=9).

A339688:= func< n | (&+[8^(d-1): d in Divisors(n)]) >;
[A339688(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024

Table[Sum[8^(d - 1), {d, Divisors[n]}], {n, 1, 21}]
nmax = 21; CoefficientList[Series[Sum[x^k/(1 - 8 x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, 8^(d-1)); \\ Michel Marcus, Dec 13 2020

def A339688(n): return sum(8^(k-1) for k in (1..n) if (k).divides(n))
[A339688(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024

G.f.: Sum_{k>=1} x^k / (1 - 8*x^k).
G.f.: Sum_{k>=1} 8^(k-1) * x^k / (1 - x^k).
a(n) ~ 8^(n-1). - Vaclav Kotesovec, Jun 05 2021

A365606 Number of degree 2 vertices in the n-Sierpinski carpet graph.

Original entry on oeis.org

8, 20, 84, 500, 3540, 26996, 212052, 1684724, 13442772, 107437172, 859182420, 6872514548, 54977282004, 439809752948, 3518452514388, 28147543587572, 225180119118036, 1801440264196724, 14411520047331156, 115292154179921396, 922337214843187668, 7378697662956950900, 59029581136289955924

Offset: 1

Allan Bickle, Sep 12 2023

The level 0 Sierpinski carpet graph is a single vertex. The level n Sierpinski carpet graph is formed from 8 copies of level n-1 by joining boundary vertices between adjacent copies.

			The level 1 Sierpinski carpet graph is an 8-cycle, which has 8 degree 2 vertices and 0 degree 3 or 4 vertices.  Thus a(1) = 8.

Paolo Xausa, Table of n, a(n) for n = 1..1000
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.
Eric Weisstein's World of Mathematics, Sierpiński Carpet Graph
Index entries for linear recurrences with constant coefficients, signature (12,-35,24).

Cf. A001018 (order), A271939 (size).
Cf. A365606 (degree 2), A365607 (degree 3), A365608 (degree 4).
Cf. A009964, A291066, A359452, A359453, A291066, A083233, A332705 (Menger sponge graph).

LinearRecurrence[{12,-35,24},{8,20,84},30] (* Paolo Xausa, Oct 16 2023 *)

def A365606(n): return ((1<<3*n-1)+(3**(n-1)<<4))//5+4 # Chai Wah Wu, Nov 27 2023

a(n) = (1/10)*8^n + (16/15)*3^n + 4.
a(n) = 8*a(n-1) - 16*3^(n-2) - 28.
a(n) = 8^n - A365607(n) - A365608(n).
2*a(n) = 2*A271939(n) - 3*A365607(n) - 4*A365608(n).
G.f.: 4*x*(2 - 19*x + 31*x^2)/((1 - x)*(1 - 3*x)*(1 - 8*x)). - Stefano Spezia, Sep 12 2023

A047854 a(n) = A047848(6, n).

Original entry on oeis.org

1, 2, 11, 92, 821, 7382, 66431, 597872, 5380841, 48427562, 435848051, 3922632452, 35303692061, 317733228542, 2859599056871, 25736391511832, 231627523606481, 2084647712458322, 18761829412124891, 168856464709124012, 1519708182382116101, 13677373641439044902, 123096362772951404111

Offset: 0

Clark Kimberling

n-th difference of a(n), a(n-1), ..., a(0) is A001018(n-1) for n >= 1.

Also, the cogrowth sequence of the 16-element group D4 X C2 = . - Sean A. Irvine, Nov 10 2024

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A001018, A047848.

[(9^n +7)/8: n in [0..40]]; // G. C. Greubel, Jan 12 2025

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=9*a[n-1]+1 od: seq(a[n]+1, n=0..17); # Zerinvary Lajos, Mar 20 2008

a = {1}; ZZ = 1; Do[ZZ = ZZ + 3^(2x); AppendTo[a, ZZ], {x,0,40}]; a (* Zerinvary Lajos, Apr 03 2007 *)
(9^Range[0,40] +7)/8 (* G. C. Greubel, Jan 12 2025 *)

def A047854(n): return (pow(9,n) +7)//8
print([A047854(n) for n in range(41)]) # G. C. Greubel, Jan 12 2025

a(n) = (9^n + 7)/8. - Ralf Stephan, Feb 14 2004
From Philippe Deléham, Oct 06 2009: (Start)
a(0) = 1, a(1) = 2, a(n) = 10*a(n-1) - 9*a(n-2) for n > 1.
G.f.: (1 - 8*x)/(1 - 10*x + 9*x^2). (End)
a(n) = 9*a(n-1) - 7 (with a(0)=1). - Vincenzo Librandi, Aug 06 2010
E.g.f.: exp(x)*(exp(8*x) + 7)/8. - Elmo R. Oliveira, Aug 29 2024

a(18)-a(22) from Elmo R. Oliveira, Aug 29 2024

A075503 Stirling2 triangle with scaled diagonals (powers of 8).

Original entry on oeis.org

1, 8, 1, 64, 24, 1, 512, 448, 48, 1, 4096, 7680, 1600, 80, 1, 32768, 126976, 46080, 4160, 120, 1, 262144, 2064384, 1232896, 179200, 8960, 168, 1, 2097152, 33292288, 31653888, 6967296, 537600, 17024, 224, 1

Offset: 1

Wolfdieter Lang, Oct 02 2002

This is a lower triangular infinite matrix of the Jabotinsky type. See the Knuth reference given in A039692 for exponential convolution arrays.

The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(8*z) - 1)*x/8) - 1.

			[1]; [8,1]; [64,24,1]; ...; p(3,x) = x(64 + 24*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*       1
*       8        1
*      64       24        1
*     512      448       48       1
*    4096     7680     1600      80      1
*   32768   126976    46080    4160    120     1
*  262144  2064384  1232896  179200   8960   168   1
* 2097152 33292288 31653888 6967296 537600 17024 224 1
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 1-7 are A001018, A060195, A076003-A076007. Row sums are A075507.

Cf. A075502, A075504.

Flatten[Table[8^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)

for(n=1, 11, for(m=1, n, print1(8^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

a(n, m) = (8^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*8)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 8m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-8k*x), m >= 1.
E.g.f. for m-th column: (((exp(8x)-1)/8)^m)/m!, m >= 1.

A157176 a(n+1) = a(n - n mod 2) + a(n - n mod 3), a(0) = 1.

Original entry on oeis.org

1, 2, 2, 3, 5, 8, 8, 16, 16, 24, 40, 64, 64, 128, 128, 192, 320, 512, 512, 1024, 1024, 1536, 2560, 4096, 4096, 8192, 8192, 12288, 20480, 32768, 32768, 65536, 65536, 98304, 163840, 262144, 262144, 524288, 524288, 786432, 1310720, 2097152, 2097152, 4194304, 4194304

Offset: 0

Reinhard Zumkeller, Feb 24 2009

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

Cf. A001018, A008588, A013730, A016921, A016933, A016945, A016957, A016969, A067412, A103333.

LinearRecurrence[{0,0,0,0,0,8},{1, 2, 2, 3, 5, 8},45] (* Stefano Spezia, May 29 2024 *)

a(n+6) = 8*a(n).
a(6*k) = 8^k; a(A008588(n))=A001018(n);
a(6*k+1) = a(6*k+2) = 2*8^k; a(A016921(n))=a(A016933(n))=A013730(n);
a(6*k+3) = 3*8^k; a(A016945(n))=A103333(n+1);
a(6*k+4) = 5*8^k; a(A016957(n))=A067412(n+1);
a(6*k+5) = 8^(k+1); a(A016969(n))=A001018(n+1).
G.f.: (1 + 2*x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5)/((1 - 2*x^2)*(1 + 2*x^2 + 4*x^4)). - Stefano Spezia, May 29 2024

a(43)-a(44) from Stefano Spezia, May 29 2024

A165829 Totally multiplicative sequence with a(p) = 8.

Original entry on oeis.org

1, 8, 8, 64, 8, 64, 8, 512, 64, 64, 8, 512, 8, 64, 64, 4096, 8, 512, 8, 512, 64, 64, 8, 4096, 64, 64, 512, 512, 8, 512, 8, 32768, 64, 64, 64, 4096, 8, 64, 64, 4096, 8, 512, 8, 512, 512, 64, 8, 32768, 64, 512

Offset: 1

Jaroslav Krizek, Sep 28 2009

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A001018, A001222.

8^PrimeOmega[Range[100]](* G. C. Greubel, Apr 09 2016 *)

a(n) = 8^bigomega(n); \\ Altug Alkan, Apr 09 2016

a(n) = A001018(A001222(n)) = 8^bigomega(n) = 8^A001222(n).

Dirichlet g.f.: Product_{p prime} 1 / (1 - 8 * p^(-s)). - Ilya Gutkovskiy, Oct 30 2019

cp's OEIS Frontend

A308584 Number of ways to write n as a*(a+1)/2 + b*(b+1)/2 + 5^c*8^d, where a,b,c,d are nonnegative integers with a <= b.

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A008569 Digits of powers of 8.

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Mathematica

A271939 Number of edges in the n-th order Sierpinski carpet graph.

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Maple

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PARI

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A223331 T(n,k)=Rolling cube footprints: number of nXk 0..7 arrays starting with 0 where 0..7 label vertices of a cube and every array movement to a horizontal or antidiagonal neighbor moves along a corresponding cube edge.

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A339688 a(n) = Sum_{d|n} 8^(d-1).

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Magma

Mathematica

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SageMath

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A365606 Number of degree 2 vertices in the n-Sierpinski carpet graph.

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Formula

A047854 a(n) = A047848(6, n).

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A308584 Number of ways to write n as a(a+1)/2 + b(b+1)/2 + 5^c*8^d, where a,b,c,d are nonnegative integers with a <= b.