A023001 -id:A023001 - cp's OEIS frontend

A218737 a(n) = (34^n - 1)/33.

0, 1, 35, 1191, 40495, 1376831, 46812255, 1591616671, 54114966815, 1839908871711, 62556901638175, 2126934655697951, 72315778293730335, 2458736461986831391, 83597039707552267295, 2842299350056777088031, 96638177901930420993055, 3285698048665634313763871

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 34 (A009978).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries related to partial sums.
Index entries related to q-numbers.
Index entries for linear recurrences with constant coefficients, signature (35,-34).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

[n le 2 select n-1 else 35*Self(n-1)-34*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{35, -34}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218737(n):=(34^n-1)/33$
makelist(A218737(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218737(n)=34^n\33
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1 - x)*(1 - 34*x)).
a(n) = 35*a(n-1) - 34*a(n-2).
a(n) = floor(34^n/33). (End)
E.g.f.: exp(x)*(exp(33*x) - 1)/33. - Stefano Spezia, Mar 26 2023

A218738 a(n) = (35^n - 1)/34.

Original entry on oeis.org

0, 1, 36, 1261, 44136, 1544761, 54066636, 1892332261, 66231629136, 2318107019761, 81133745691636, 2839681099207261, 99388838472254136, 3478609346528894761, 121751327128511316636, 4261296449497896082261, 149145375732426362879136, 5220088150634922700769761

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 35 (A009979).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries related to partial sums.
Index entries related to q-numbers.
Index entries for linear recurrences with constant coefficients, signature (36,-35).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009979.

[n le 2 select n-1 else 36*Self(n-1)-35*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{36, -35}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218738(n):=(35^n-1)/34$
makelist(A218738(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218738(n)=35^n\34
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1 - x)*(1 - 35*x)).
a(n) = 36*a(n-1) - 35*a(n-2).
a(n) = floor(35^n/34). (End)
E.g.f.: exp(x)*(exp(34*x) - 1)/34. - Stefano Spezia, Mar 28 2023

A218745 a(n) = (42^n - 1)/41.

Original entry on oeis.org

0, 1, 43, 1807, 75895, 3187591, 133878823, 5622910567, 236162243815, 9918814240231, 416590198089703, 17496788319767527, 734865109430236135, 30864334596069917671, 1296302053034936542183, 54444686227467334771687, 2286676821553628060410855, 96040426505252378537255911

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 42 (A009986).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries related to partial sums.
Index entries related to q-numbers.
Index entries for linear recurrences with constant coefficients, signature (43,-42).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009986.

[n le 2 select n-1 else 43*Self(n-1) - 42*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{43, -42}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)
(42^Range[0,20]-1)/41 (* Harvey P. Dale, May 08 2024 *)

A218745(n):=(42^n-1)/41$
makelist(A218745(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218745(n)=42^n\41
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-42*x)).
a(n) = 43*a(n-1) - 42*a(n-2).
a(n) = floor(42^n/41). (End)
E.g.f.: exp(x)*(exp(41*x) - 1)/41. - Elmo R. Oliveira, Aug 29 2024

A258643 Irregular triangle read by rows, n >= 1, k >= 0: T(n,k) is the number of distinct patterns of n X n squares with k holes that are squares (see the construction rule in comments).

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 9, 7, 4, 4, 5, 2, 25, 11, 40, 8, 33, 3, 16, 0, 4

Offset: 1

Kival Ngaokrajang, Jun 06 2015

The sequence of row lengths is A261243. - Wolfdieter Lang, Aug 18 2015
The construction rules are: (o) The n X n square has horizontal and vertical diagonals. (i) A pattern must be symmetric with respect to both vertical and horizontal axes. (ii) For n >= 2, each pattern must have four squares at the corners. (iii) The squares must have continuity contact to each other either by sides or corners. (iv) The hole(s) must be square(s). Mirror patterns with respect to the main diagonal are not considered as different. See illustration in the links.
Each pattern can be a seed of a box fractal; e.g., the second pattern of T(3,0), consisting of 5 squares and 0 holes, is a seed of the Vicsek fractal (see a link below); the second pattern of T(4,2), consisting of 10 squares and 2 holes, is a seed of the fractal in a link of A002276.
If the figures are rotated by 45 degrees in the clockwise direction they can be considered as binary bisymmetric n X n matrices B_n if a red square stand for 1 and an empty square for 0. The four corners have entries 1, that is B_n[1, 1] = 1 = B_n[1, n]. The continuity of the red squares, mentioned above in point (iii), means that there is no rectangular path of 0's (no diagonal steps) in the matrix B_n that dissects it into two parts. See A261242 for more details, where also the figures with nonsquare holes and the mirrors (row reversion in the B_n matrix) are considered. - Wolfdieter Lang, Aug 18 2015

			Irregular triangle begins:
n\k  0   1   2  3   4  5   6  7  8 ...
1    1
2    1
3    2   1
4    3   1   2
5    9   7   4  4   5  2
6   25  11  40  8  33  3  16  0  4
...

Kival Ngaokrajang, Illustration of pattern T(n,k) for n = 1..5, k >= 0, T(6,k) patterns
Wikipedia, Vicsek fractal

Cf. A002276 (10 squares, 2 holes), A016203 (8 squares, 0 holes), A023001 (8 squares, 1 hole), A218724 (21 squares, 4 holes).

Cf. A261242, A261243.

A261544 a(n) = Sum_{k=0..n} 1000^k.

Original entry on oeis.org

1, 1001, 1001001, 1001001001, 1001001001001, 1001001001001001, 1001001001001001001, 1001001001001001001001, 1001001001001001001001001, 1001001001001001001001001001, 1001001001001001001001001001001, 1001001001001001001001001001001001

Offset: 0

Ilya Gutkovskiy, Aug 24 2015

A sequence of palindromic numbers.

			From _Bruno Berselli_, Aug 25 2015: (Start)
a(n)   is the binary representation of    A023001
-------------------------------------------------
1  ...........................................  1
1001  ........................................  9
1001001 .....................................  73
1001001001  ................................  585
1001001001001  ............................  4681
1001001001001001  ........................  37449
1001001001001001001  ....................  299593
1001001001001001001001  ................  2396745
1001001001001001001001001  ............  19173961, etc.
(End)

Colin Barker, Table of n, a(n) for n = 0..333 (corrected by Michel Marcus, Jan 19 2019)
John Rafael M. Antalan, A Recreational Application of Two Integer Sequences and the Generalized Repetitious Number Puzzle, arXiv:1908.06014 [math.HO], 2019.
Eric Weisstein's World of Mathematics, Palindromic Number.
Index entries for linear recurrences with constant coefficients, signature (1001,-1000).

Cf. A000533, A002113, A023001.
Subsequence of A033146.
Sums of 100^k: A094028; sums of 10^k: A000042.
Cf. similar sequences of the form (k^n-1)/(k-1) listed in A269025.

[(1000^(n+1)-1)/999: n in [0..30]]; // Vincenzo Librandi, Aug 24 2015

Table[(1000^(n + 1) - 1)/999, {n, 0, 15}]
LinearRecurrence[{1001, -1000}, {1, 1001}, 20] (* Vincenzo Librandi, Aug 24 2015 *)

Vec(1 / ((x-1)*(1000*x-1)) + O(x^20)) \\ Colin Barker, Aug 24 2015

a(n) = (1000^(n + 1) - 1)/999.
a(n) = 1001*a(n-1) - 1000*a(n-2). - Colin Barker, Aug 24 2015
G.f.: 1 / ((x-1)*(1000*x-1)). - Colin Barker, Aug 24 2015
E.g.f.: (1/999)*(1000000*exp(1000*x) - exp(x)). - G. C. Greubel, Aug 29 2015

A318935 a(n) = Sum_{2^m divides n} 2^(3*m).

Original entry on oeis.org

1, 9, 1, 73, 1, 9, 1, 585, 1, 9, 1, 73, 1, 9, 1, 4681, 1, 9, 1, 73, 1, 9, 1, 585, 1, 9, 1, 73, 1, 9, 1, 37449, 1, 9, 1, 73, 1, 9, 1, 585, 1, 9, 1, 73, 1, 9, 1, 4681, 1, 9, 1, 73, 1, 9, 1, 585, 1, 9, 1, 73, 1, 9, 1, 299593, 1, 9, 1, 73, 1, 9, 1, 585, 1, 9, 1, 73, 1, 9, 1, 4681

Offset: 1

N. J. A. Sloane, Sep 14 2018

Sum of cubes of powers of 2 that divide n.

The high-water marks are (8^m - 1)/7, see A023001.

Antti Karttunen, Table of n, a(n) for n = 1..16383
P. J. C. Lamont, The number of Cayley integers of given norm, Proceedings of the Edinburgh Mathematical Society, 25.1 (1982): 101-103. See T, p. 102.

Cf. A007814, A023001.

A007814 := n -> padic[ordp](n, 2):
T:= n -> add(2^(3*m),m=0..A007814(n));
[seq(T(n),n=1..100)];

Array[DivisorSum[#, 2^(3 Log2@ #) &, IntegerQ@ Log2@ # &] &, 80] (* or *)
Array[Total[2^(3 Select[Log2@ Divisors@ #, IntegerQ])] &, 80] (* Michael De Vlieger, Nov 07 2018 *)
a[n_] := (8^(IntegerExponent[n, 2] + 1) - 1) / 7; Array[a, 100] (* Amiram Eldar, Oct 23 2023 *)

A318935(n) = { my(s=1,w=8); while(!(n%2), s += w; n /= 2; w *= 8); (s); }; \\ Antti Karttunen, Nov 07 2018

from _future_ import division
def A318935(n):
    s = bin(n)
    return (8**(len(s)-len(s.rstrip('0'))+1) - 1)//7 # Chai Wah Wu, Sep 14 2018

a(n) = (8^(m+1)-1)/7 where m is the 2-adic valuation of n (A007814). - Chai Wah Wu, Sep 14 2018
Thus multiplicative with a(2^m) = (8^(m+1)-1)/7, and a(p^e) = 1 for odd primes p. - Antti Karttunen, Nov 07 2018
Dirichlet g.f.: zeta(s) / (1 - 1/2^(s-3)). - Amiram Eldar, Oct 23 2023

Keyword:mult added by Antti Karttunen, Nov 07 2018

A016203 Expansion of g.f. 1/((1-x)(1-2x)(1-8x)).

Original entry on oeis.org

1, 11, 95, 775, 6231, 49911, 399415, 3195575, 25565111, 204521911, 1636177335, 13089422775, 104715390391, 837723139511, 6701785148855, 53614281256375, 428914250182071, 3431314001718711, 27450512014273975, 219604096115240375, 1756832768924020151, 14054662151396355511

Offset: 0

N. J. A. Sloane

4*a(n) is the total number of holes in a certain box fractal (start with 8 boxes, 0 hole) after n iterations. See illustration in link. - Kival Ngaokrajang, Jan 27 2015

Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (11,-26,16).

Cf. A016131, A023001.

a:=n->sum((8^(n-j)-2^(n-j))/6,j=0..n): seq(a(n), n=1..19); # Zerinvary Lajos, Jan 15 2007

Vec(1/((1-x)*(1-2*x)*(1-8*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = (4*8^(n+1) - 7*2^(n+1) + 3)/21. - Mitch Harris, Jun 27 2005; corrected by Yahia Kahloune, May 06 2013
a(0) = 1, a(n) = 8*a(n-1) + 2^(n+1) - 1. - Vincenzo Librandi, Feb 09 2011
From Elmo R. Oliveira, Mar 26 2025: (Start)
E.g.f.: exp(x)*(32*exp(7*x) - 14*exp(x) + 3)/21.
a(n) = 11*a(n-1) - 26*a(n-2) + 16*a(n-3).
a(n) = A016131(n+1) - A023001(n+2). (End)

More terms from Elmo R. Oliveira, Mar 26 2025

A218727 a(n) = (24^n - 1)/23.

Original entry on oeis.org

0, 1, 25, 601, 14425, 346201, 8308825, 199411801, 4785883225, 114861197401, 2756668737625, 66160049703001, 1587841192872025, 38108188628928601, 914596527094286425, 21950316650262874201, 526807599606308980825, 12643382390551415539801, 303441177373233972955225

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 24 (A009968); q-integers for q=24: diagonal k=1 in triangle A022188.

Partial sums are in A014913. Also, the sequence is related to A014942 by A014942(n) = n*a(n) - Sum_{i=0..n-1} a(i) for n > 0. [Bruno Berselli, Nov 07 2012]

Vincenzo Librandi, Table of n, a(n) for n = 0..700
Index entries related to partial sums.
Index entries for linear recurrences with constant coefficients, signature (25,-24).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009968, A014942, A022188.

[n le 2 select n-1 else 25*Self(n-1)-24*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{25, -24}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218727(n):=(24^n-1)/23$
makelist(A218727(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218727(n)=24^n\23
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-24*x)).
a(n) = floor(24^n/23).
a(n) = 25*a(n-1) - 24*a(n-2). (End)
E.g.f.: exp(x)*(exp(23*x) - 1)/23. - Elmo R. Oliveira, Aug 29 2024

A218729 a(n) = (26^n - 1)/25.

Original entry on oeis.org

0, 1, 27, 703, 18279, 475255, 12356631, 321272407, 8353082583, 217180147159, 5646683826135, 146813779479511, 3817158266467287, 99246114928149463, 2580398988131886039, 67090373691429037015, 1744349715977154962391, 45353092615406029022167, 1179180408000556754576343

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 26 (A009970); q-integers for q=26.

Vincenzo Librandi, Table of n, a(n) for n = 0..700
Index entries related to partial sums.
Index entries for linear recurrences with constant coefficients, signature (27,-26).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009970.

[n le 2 select n-1 else 27*Self(n-1)-26*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{27, -26}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218729(n):=(26^n-1)/25$
makelist(A218729(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218729(n)=26^n\25
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-26*x)).
a(n) = floor(26^n/25).
a(n) = 27*a(n-1) - 26*a(n-2). (End)
E.g.f.: exp(x)*(exp(25*x) - 1)/25. - Elmo R. Oliveira, Aug 29 2024

A218730 a(n) = (27^n - 1)/26.

Original entry on oeis.org

0, 1, 28, 757, 20440, 551881, 14900788, 402321277, 10862674480, 293292210961, 7918889695948, 213810021790597, 5772870588346120, 155867505885345241, 4208422658904321508, 113627411790416680717, 3067940118341250379360, 82834383195213760242721, 2236528346270771526553468

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 27 (A009971); q-integers for q=27.

Vincenzo Librandi, Table of n, a(n) for n = 0..700
Index entries related to partial sums
Index entries for linear recurrences with constant coefficients, signature (28,-27).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009971.

[n le 2 select n-1 else 28*Self(n-1)-27*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{28, -27}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218730(n):=(27^n-1)/26$
makelist(A218730(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
a(n)=27^n\26
```

G.f.: x/((1-x)*(1-27*x)). - Vincenzo Librandi, Nov 07 2012
a(n) = floor(27^n/26). - Vincenzo Librandi, Nov 07 2012
a(n) = 28*a(n-1) - 27*a(n-2). - Vincenzo Librandi, Nov 07 2012
E.g.f.: exp(14*x)*sinh(13*x)/13. - Elmo R. Oliveira, Aug 27 2024

cp's OEIS Frontend

A218737 a(n) = (34^n - 1)/33.

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A218738 a(n) = (35^n - 1)/34.

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A218745 a(n) = (42^n - 1)/41.

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A258643 Irregular triangle read by rows, n >= 1, k >= 0: T(n,k) is the number of distinct patterns of n X n squares with k holes that are squares (see the construction rule in comments).

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A261544 a(n) = Sum_{k=0..n} 1000^k.

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A318935 a(n) = Sum_{2^m divides n} 2^(3*m).

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A016203 Expansion of g.f. 1/((1-x)*(1-2*x)*(1-8*x)).

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A016203 Expansion of g.f. 1/((1-x)(1-2x)(1-8x)).