A218752 -id:A218752 - cp's OEIS frontend

A064108 a(n) = (20^n - 1)/19.

0, 1, 21, 421, 8421, 168421, 3368421, 67368421, 1347368421, 26947368421, 538947368421, 10778947368421, 215578947368421, 4311578947368421, 86231578947368421, 1724631578947368421, 34492631578947368421, 689852631578947368421, 13797052631578947368421, 275941052631578947368421

Offset: 0

Jason Earls, Sep 17 2001

Partial sums of powers of 20 (A009964), q-integers for q=20: diagonal k=1 in triangle A022184.
Partial sums are in A014904. Also, the sequence is related to A014937 by A014937(n) = n*a(n)-Sum_{i=0..n-1} a(i), for n>0. - Bruno Berselli, Nov 06 2012
For n >= 1, a(n) is the total number of holes in a certain box fractal (start with 20 boxes, 1 hole) after n iterations. See illustration in links. - Kival Ngaokrajang, Jan 28 2015

			From _N. J. A. Sloane_, Nov 04 2014: Can also be obtained by writing powers of 2 in a staggered array and adding them (cf. A249604). For example, a(9) is:
..........1
.........2
........4
.......8
.....16
....32
...64
.128
256
-----------
26947368421

M. F. Hasler, Table of n, a(n) for n = 0..100
Kival Ngaokrajang, Illustration of initial terms
Index entries related to partial sums
Index entries related to q-numbers
Index entries for linear recurrences with constant coefficients, signature (21,-20).

Cf. A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218722, A064108, A218724, ..., A218733, ..., A218743, ..., A218752, A094028.
Cf. also A249604.
Cf. A009964, A014904, A014937, A022184.

a:=n->sum(20^(n-j), j=0..n): seq(a(n), n=0..15); # Zerinvary Lajos, Feb 11 2007

(20^Range[20]-1)/19 (* or *) NestList[20#+1&,1,20] (* Harvey P. Dale, Oct 04 2012 *)

A064108(n):=(20^n-1)/19$ makelist(A064108(n),n,1,30); /* Martin Ettl, Nov 05 2012 */

for (n=0, 100, write("b064108.txt", n, " ", (20^n - 1)/19))  \\ Harry J. Smith, Sep 07 2009

A064108(n)=20^n\19  \\ M. F. Hasler, Nov 04 2012

[gaussian_binomial(n,1,20) for n in range(1,17)] # Zerinvary Lajos, May 29 2009

a(n) = 20*a(n-1) + 1, with a(0)=0. - Vincenzo Librandi, Aug 07 2010
a(0)=0, a(1)=1, a(n) = 21*a(n-1) - 20*a(n-2). - Harvey P. Dale, Oct 04 2012
a(n) = floor(20^n/19). - M. F. Hasler, Nov 04 2012
G.f.: x/((1 - x)*(1 - 20*x)). - Bruno Berselli, Nov 06 2012
E.g.f.: exp(x)*(exp(19*x) - 1)/19. - Stefano Spezia, Mar 23 2023

Edited and extended to offset 0 by M. F. Hasler, Nov 04 2012

A218721 a(n) = (18^n-1)/17.

Original entry on oeis.org

0, 1, 19, 343, 6175, 111151, 2000719, 36012943, 648232975, 11668193551, 210027483919, 3780494710543, 68048904789775, 1224880286215951, 22047845151887119, 396861212733968143, 7143501829211426575, 128583032925805678351

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 18 (A001027), q-integers for q=18: diagonal k=1 in triangle A022182.
Partial sums are in A014901. Also, the sequence is related to A014935 by A014935(n) = n*a(n) - Sum_{i=0..n-1} a(i), for n>0. - Bruno Berselli, Nov 06 2012
From Bernard Schott, May 06 2017: (Start)
Except for 0, 1 and 19, all terms are Brazilian repunits numbers in base 18, and so belong to A125134. From n = 3 to n = 8286, all terms are composite. See link "Generalized repunit primes".
As explained in the extensions of A128164, a(25667) = (18^25667 - 1)/17 would be (is) the smallest prime in base 18. (End)

			a(3) = (18^3 - 1)/17 = 343 = 7 * 49; a(6) = (18^6 - 1)/17 = 2000719 = 931 * 2149. - _Bernard Schott_, May 01 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..800
Harvey Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
Index entries related to partial sums
Index entries for linear recurrences with constant coefficients, signature (19,-18).

Cf. A000225, A001027, A002275, A002450, A002452, A003462, A003463, A003464, A014901, A014935, A016123, A016125, A022182, A023000, A023001, A064108, A091030, A091045, A094028, A125134, A128164, A131865, A135518, A135519, A218722, A218724, A218733, A218743, A218752.

[n le 2 select n-1 else 19*Self(n-1)-18*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{19, -18}, {0, 1}, 40] (* Vincenzo Librandi, Nov 07 2012 *)
Join[{0},Accumulate[18^Range[0,20]]] (* Harvey P. Dale, Nov 08 2012 *)

A218721(n):=(18^n-1)/17$ makelist(A218721(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
A218721(n)=18^n\17
```

a(n) = floor(18^n/17).
G.f.: x/((1-x)*(1-18*x)). - Bruno Berselli, Nov 06 2012
a(n) = 19*a(n-1) - 18*a(n-2). - Vincenzo Librandi, Nov 07 2012
E.g.f.: exp(x)*(exp(17*x) - 1)/17. - Stefano Spezia, Mar 11 2023

A218753 a(n) = (49^n - 1)/48.

Original entry on oeis.org

0, 1, 50, 2451, 120100, 5884901, 288360150, 14129647351, 692352720200, 33925283289801, 1662338881200250, 81454605178812251, 3991275653761800300, 195572507034328214701, 9583052844682082520350, 469569589389422043497151, 23008909880081680131360400

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 49 (A087752).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries related to partial sums
Index entries for linear recurrences with constant coefficients, signature (50,-49)

Cf. A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218752, A133853, A094028, A218723.

Cf. A087752.

[n le 2 select n-1 else 50*Self(n-1) - 49*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 08 2012

LinearRecurrence[{50, -49}, {0, 1}, 30] (* Vincenzo Librandi, Nov 08 2012 *)
Join[{0},Accumulate[49^Range[0,20]]] (* Harvey P. Dale, Apr 14 2023 *)

A218753(n):=floor(49^n/48)$ makelist(A218753(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
A218753(n)=49^n\48
```

G.f.: x/((1-x)*(1-49*x)). - Vincenzo Librandi, Nov 08 2012
a(n) = 50*a(n-1) - 49*a(n-2) with a(0)=0, a(1)=1. - Vincenzo Librandi, Nov 08 2012
a(n) = 49*a(n-1) + 1 with a(0)=0. - Vincenzo Librandi, Nov 08 2012
a(n) = floor(49^n/48). - Vincenzo Librandi, Nov 08 2012
E.g.f.: exp(25*x)*sinh(24*x)/24. - Elmo R. Oliveira, Aug 27 2024

A125118 Triangle read by rows: T(n,k) = value of the n-th repunit in base (k+1) representation, 1<=k<=n.

Original entry on oeis.org

1, 3, 4, 7, 13, 21, 15, 40, 85, 156, 31, 121, 341, 781, 1555, 63, 364, 1365, 3906, 9331, 19608, 127, 1093, 5461, 19531, 55987, 137257, 299593, 255, 3280, 21845, 97656, 335923, 960800, 2396745, 5380840, 511, 9841, 87381, 488281, 2015539, 6725601, 19173961, 48427561, 111111111

Offset: 1

Reinhard Zumkeller, Nov 21 2006

			First 4 rows:
1: [1]_2
2: [11]_2 ........ [11]_3
3: [111]_2 ....... [111]_3 ....... [111]_4
4: [1111]_2 ...... [1111]_3 ...... [1111]_4 ...... [1111]_5
_
1: 1
2: 2+1 ........... 3+1
3: (2+1)*2+1 ..... (3+1)*3+1 ..... (4+1)*4+1
4: ((2+1)*2+1)*2+1 ((3+1)*3+1)*3+1 ((4+1)*4+1)*4+1 ((5+1)*5+1)*5+1.

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Eric Weisstein's World of Mathematics, Repunit

This triangle shares some features with triangle A104878.
This triangle is a portion of rectangle A055129.
Each term of A110737 comes from the corresponding row of this triangle.
Diagonals (adjusting offset as necessary): A060072, A023037, A031973, A173468.
Columns (adjusting offset as necessary): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724, A218725, A218726, A218727, A218728, A218729, A218730, A218731, A218732, A218733, A218734, A132469, A218736, A218737, A218738, A218739, A218740, A218741, A218742, A218743, A218744, A218745, A218746, A218747, A218748, A218749, A218750, A218751, A218753, A218752.
Cf. A023037, A031973, A125119, A125120 (row sums).

[((k+1)^n -1)/k : k in [1..n], n in [1..12]]; // G. C. Greubel, Aug 15 2022

Table[((k+1)^n -1)/k, {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Aug 15 2022 *)

def A125118(n,k): return ((k+1)^n -1)/k
flatten([[A125118(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Aug 15 2022

T(n, k) = Sum_{i=0..n-1} (k+1)^i.
T(n+1, k) = (k+1)*T(n, k) + 1.
Sum_{k=1..n} T(n, k) = A125120(n).
T(2*n-1, n) = A125119(n).
T(n, 1) = A000225(n).
T(n, 2) = A003462(n) for n>1.
T(n, 3) = A002450(n) for n>2.
T(n, 4) = A003463(n) for n>3.
T(n, 5) = A003464(n) for n>4.
T(n, 9) = A002275(n) for n>8.
T(n, n) = A060072(n+1).
T(n, n-1) = A023037(n) for n>1.
T(n, n-2) = A031973(n) for n>2.
T(n, k) = A055129(n, k+1) = A104878(n+k, k+1), 1<=k<=n. - Mathew Englander, Dec 19 2020

A104878 A sum-of-powers number triangle.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 15, 13, 5, 1, 1, 6, 31, 40, 21, 6, 1, 1, 7, 63, 121, 85, 31, 7, 1, 1, 8, 127, 364, 341, 156, 43, 8, 1, 1, 9, 255, 1093, 1365, 781, 259, 57, 9, 1, 1, 10, 511, 3280, 5461, 3906, 1555, 400, 73, 10, 1, 1, 11, 1023, 9841, 21845

Offset: 0

Paul Barry, Mar 28 2005

Columns are partial sums of the columns of A004248. Row sums are A104879. Diagonal sums are A104880.

The rows of this triangle (apart from the initial "1" in each row) are the antidiagonals of rectangle A055129. The diagonals of this triangle (apart from the initial "1") are the rows of rectangle A055129. The columns of this triangle (apart from the leftmost column) are the same as the columns of rectangle A055129 but shifted downward. - Mathew Englander, Dec 21 2020

			Triangle starts:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3,  1;
  1,  4,  7,  4,  1;
  1,  5, 15, 13,  5,  1;
  1,  6, 31, 40, 21,  6,  1;
  ...

Cf. A004248 (first differences by column), A104879 (row sums), A104880 (antidiagonal sums), A125118 (version of this triangle with fewer terms).
This triangle (ignoring the leftmost column) is a rotation of rectangle A055129.
Columns (adjusting offset as necessary): A000012, A000027, A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724, A218725, A218726, A218727, A218728, A218729, A218730, A218731, A218732, A218733, A218734, A132469, A218736, A218737, A218738, A218739, A218740, A218741, A218742, A218743, A218744, A218745, A218746, A218747, A218748, A218749, A218750, A218751, A218753, A218752.
T(2n,n) gives A031973.

A104878 :=proc(n,k): if k = 0 then 1 elif k=1 then n elif k>=2 then (k^(n-k+1)-1)/(k-1) fi: end: for n from 0 to 7 do seq(A104878(n,k), k=0..n) od; seq(seq(A104878(n,k), k=0..n), n=0..10); # Johannes W. Meijer, Aug 21 2011

T(n, k) = if(k=1, n, if(k<=n, (k^(n-k+1)-1)/(k-1), 0));
G.f. of column k: x^k/((1-x)(1-k*x)). [corrected by Werner Schulte, Jun 05 2019]
T(n, k) = A069777(n+1,k)/A069777(n,k). [Johannes W. Meijer, Aug 21 2011]
T(n, k) = A055129(n+1-k, k) for n >= k > 0. - Mathew Englander, Dec 19 2020

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A064108 a(n) = (20^n - 1)/19.

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A218721 a(n) = (18^n-1)/17.

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A218753 a(n) = (49^n - 1)/48.

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A125118 Triangle read by rows: T(n,k) = value of the n-th repunit in base (k+1) representation, 1<=k<=n.

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A104878 A sum-of-powers number triangle.

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