A023001 -id:A023001 - cp's OEIS frontend

A135519 Generalized repunits in base 14.

1, 15, 211, 2955, 41371, 579195, 8108731, 113522235, 1589311291, 22250358075, 311505013051, 4361070182715, 61054982558011, 854769755812155, 11966776581370171, 167534872139182395, 2345488209948553531

Offset: 1

Julien Peter Benney (jpbenney(AT)gmail.com), Feb 19 2008

Primes are given in A006032.

Let A be the Hessenberg matrix of the order n, defined by: A[1,j]=1, A[i,i]:=14, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - Milan Janjic, Feb 21 2010

			a(4) = 2955 because (14^4-1)/13 = 38416/13 = 2955.
For n=6, a(6) = 1*6 + 13*15 + 169*20 + 2197*15 + 28561*6 + 371293*1 = 579195. - _Bruno Berselli_, Nov 12 2015

Harvey P. Dale, Table of n, a(n) for n = 1..873
Index entries for linear recurrences with constant coefficients, signature (15,-14).

Cf. A000225, A001022, A002450, A002452, A003462, A003463, A003464, A016123, A016125, A023000, A023001, A135278.

Table[FromDigits[PadRight[{}, n, 1], 14], {n, 20}] (* or *) LinearRecurrence[{15, -14}, {1, 15}, 20] (* Harvey P. Dale, Aug 29 2016 *)

A135519(n):=(14^n-1)/13$ makelist(A135519(n),n,1,30); /* Martin Ettl, Nov 05 2012 */

[gaussian_binomial(n,1,14) for n in range(1,15)] # Zerinvary Lajos, May 28 2009

[(14^n-1)/13 for n in (1..30)] # Bruno Berselli, Nov 12 2015

a(n) = (14^n - 1)/13.
a(n) = 14*a(n-1) + 1 for n>1, a(1)=1. - Vincenzo Librandi, Aug 03 2010
a(n) = Sum_{i=0..n-1} 13^i*binomial(n,n-1-i). - Bruno Berselli, Nov 12 2015
From G. C. Greubel, Oct 17 2016: (Start)
G.f.: x/((1-x)*(1-14*x)).
E.g.f.: (1/13)*(exp(14*x) - exp(x)). (End)

A190958 a(n) = 2a(n-1) - 10a(n-2), with a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 2, -6, -32, -4, 312, 664, -1792, -10224, -2528, 97184, 219648, -532544, -3261568, -1197696, 30220288, 72417536, -157367808, -1038910976, -504143872, 9380822016, 23803082752, -46202054656, -330434936832, -198849327104, 2906650714112, 7801794699264

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

For the difference equation a(n) = c*a(n-1) - d*a(n-2), with a(0) = 0, a(1) = 1, the solution is a(n) = d^((n-1)/2) * ChebyshevU(n-1, c/(2*sqrt(d))) and has the alternate form a(n) = ( ((c + sqrt(c^2 - 4*d))/2)^n - ((c - sqrt(c^2 - 4*d))/2)^n )/sqrt(c^2 - 4*d). In the case c^2 = 4*d then the solution is a(n) = n*d^((n-1)/2). The generating function is x/(1 - c*x + d^2) and the exponential generating function takes the form (2/sqrt(c^2 - 4*d))*exp(c*x/2)*sinh(sqrt(c^2 - 4*d)*x/2) for c^2 > 4*d, (2/sqrt(4*d - c^2))*exp(c*x/2)*sin(sqrt(4*d - c^2)*x/2) for 4*d > c^2, and x*exp(sqrt(d)*x) if c^2 = 4*d. - G. C. Greubel, Jun 10 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..190
Index entries for linear recurrences with constant coefficients, signature (2,-10).

Sequences of the form a(n) = c*a(n-1) - d*a(n-2), with a(0)=0, a(1)=1:
c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10
.A128834,A107920,A106852,A106853,A106854,A145934,A145976,A145978,A146078,A146080
.A000027,A009545,A088137,A088138,A045873,A088139,A089181,A090591,A127357,A190958
.A001906,A000225,A057083,A049072,A190959,A190960,A190961,A190962,A190963,A190964
.A001353,A007070,A003462,A001787,A099456,A190965,A168175,A190966,A190967,A190968
.A004254,A107839,A116415,A002450,A030191,A001047,A099450,A190969,A190970,A190971
.A001109,A154244,A138395,A084326,A003463,A030192,A081179,A006516,A027471,A106392
.A004187,A186446,A190972,A190973,A190974,A003464,A030240,A152268,A099459,A016127
.A001090,A190975,A190976,A099156,A190977,A190978,A023000,A057084,A154245,A154235
.A018913,A190979,A190980,A190981,A190982,A190983,A190984,A023001,A057085,A178869
A004189,A190869,A190985,A190986,A190987,A190988,A190989,A190990,A002452,A057086

I:=[0,1]; [n le 2 select I[n] else 2*Self(n-1)-10*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 17 2011

Mathematica
```
LinearRecurrence[{2,-10}, {0,1}, 50]
```

a(n)=([0,1; -10,2]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Apr 08 2016

[lucas_number1(n,2,10) for n in (0..50)] # G. C. Greubel, Jun 10 2022

G.f.: x / ( 1 - 2*x + 10*x^2 ). - R. J. Mathar, Jun 01 2011
E.g.f.: (1/3)*exp(x)*sin(3*x). - Franck Maminirina Ramaharo, Nov 13 2018
a(n) = 10^((n-1)/2) * ChebyshevU(n-1, 1/sqrt(10)). - G. C. Greubel, Jun 10 2022
a(n) = (1/3)*10^(n/2)*sin(n*arctan(3)) = Sum_{k=0..floor(n/2)} (-1)^k*3^(2*k)*binomial(n,2*k+1). - Gerry Martens, Oct 15 2022

A218724 a(n) = (21^n - 1)/20.

Original entry on oeis.org

0, 1, 22, 463, 9724, 204205, 4288306, 90054427, 1891142968, 39714002329, 833994048910, 17513875027111, 367791375569332, 7723618886955973, 162195996626075434, 3406115929147584115, 71528434512099266416, 1502097124754084594737, 31544039619835776489478

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 21 (A009965); q-integers for q=21: diagonal k=1 in triangle A022185.
Partial sums are in A014905. Also, the sequence is related to A014938 by A014938(n) = n*a(n) - Sum_{i=0..n-1} a(i) for n > 0. - Bruno Berselli, Nov 06 2012
For n >= 1, 4*a(n) is the total number of holes in a certain box fractal (start with 21 boxes, 4 holes) after n iterations. See illustration in links. - Kival Ngaokrajang, Jan 27 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..700
Kival Ngaokrajang, Illustration of initial terms
Index entries related to partial sums
Index entries for linear recurrences with constant coefficients, signature (22,-21).

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, A218725-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009965, A014905, A014938, A022185.

[n le 2 select n-1 else 22*Self(n-1) - 21*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{22, -21}, {0, 1}, 40] (* Vincenzo Librandi, Nov 07 2012 *)

A218724(n):=(21^n-1)/20$ makelist(A218724(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
A218724(n)=21^n\20
```

a(n) = floor(21^n/20).
G.f.: x/((1-x)*(1-21*x)). - Bruno Berselli, Nov 06 2012
a(n) = 22*a(n-1) - 21*a(n-2). - Vincenzo Librandi, Nov 07 2012
a(n) = 21*a(n-1) + 1. - Kival Ngaokrajang, Jan 27 2015
a(n) = a(n-1) + 21^(n-1), n >= 1, a(0) = 0.  - Wolfdieter Lang, Feb 02 2015
E.g.f.: exp(11*x)*sinh(10*x)/10. - Elmo R. Oliveira, Aug 29 2024

A218734 a(n) = (31^n - 1)/30.

Original entry on oeis.org

0, 1, 32, 993, 30784, 954305, 29583456, 917087137, 28429701248, 881320738689, 27320942899360, 846949229880161, 26255426126284992, 813918209914834753, 25231464507359877344, 782175399728156197665, 24247437391572842127616, 751670559138758105956097

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 31 (A009975).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Shaoshi Chen, Hanqian Fang, Sergey Kitaev, and Candice X.T. Zhang, Patterns in Multi-dimensional Permutations, arXiv:2411.02897 [math.CO], 2024. See pp. 2, 17.
Index entries related to partial sums.
Index entries for linear recurrences with constant coefficients, signature (32,-31).

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-A218733, A132469, A218736-A218753, A133853, A094028, A218723.

[n le 2 select n-1 else 32*Self(n-1)-31*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{32, -31}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218734(n):=(31^n-1)/30$
makelist(A218734(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
a(n)=31^n\30
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1 - x)*(1 - 31*x)).
a(n) = 32*a(n-1) - 31*a(n-2) for n > 1.
a(n) = floor(31^n/30). (End)
E.g.f.: exp(16*x)*sinh(15*x)/15. - Stefano Spezia, Mar 11 2023

A132469 a(n) = (2^(5*n) - 1)/31.

Original entry on oeis.org

0, 1, 33, 1057, 33825, 1082401, 34636833, 1108378657, 35468117025, 1134979744801, 36319351833633, 1162219258676257, 37191016277640225, 1190112520884487201, 38083600668303590433, 1218675221385714893857, 38997607084342876603425, 1247923426698972051309601

Offset: 0

A.K. Devaraj, Aug 22 2007

Partial sums of powers of 32 (A009976), a.k.a. q-numbers for q=32. - M. F. Hasler, Nov 05 2012

A. K. Devaraj, "Minimum Universal Exponent Generalisation of Fermat's Theorem", in ISSN #1550-3747, Proceedings of Hawaii Intl Conference on Statistics, Mathematics & Related Fields, 2004.

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Quynh Nguyen, Jean Pedersen, and Hien T. Vu, New Integer Sequences Arising From 3-Period Folding Numbers, Vol. 19 (2016), Article 16.3.1. See Table 1.
Index entries related to partial sums.
Index entries related to q-numbers.
Index entries for linear recurrences with constant coefficients, signature (33,-32).

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 33*Self(n-1) - 32*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

Table[(2^(5 n) - 1)/31, {n, 16}] (* Robert G. Wilson v *)
LinearRecurrence[{33, -32}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A132469(n):=(32^n-1)/31$
makelist(A132469(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

A132469(n)=32^n\31  \\ M. F. Hasler, Nov 07 2012

[gaussian_binomial(5*n,1,2)/31 for n in range(1,17)] # Zerinvary Lajos, May 28 2009

a(n) = (32^n - 1)/31 = floor(32^n/31) = Sum_{k=0..n} 32^k. - M. F. Hasler, Nov 05 2012
G.f.: x/((1 - x)*(1 - 32*x)). - Bruno Berselli, Nov 06 2012
E.g.f.: exp(x)*(exp(31*x) - 1)/31. - Stefano Spezia, Mar 23 2023

Edited and extended by Robert G. Wilson v, Aug 22 2007

Edited and extended to offset 0 by M. F. Hasler, Nov 05 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

A218736 a(n) = (33^n - 1)/32.

Original entry on oeis.org

0, 1, 34, 1123, 37060, 1222981, 40358374, 1331826343, 43950269320, 1450358887561, 47861843289514, 1579440828553963, 52121547342280780, 1720011062295265741, 56760365055743769454, 1873092046839544391983, 61812037545704964935440, 2039797239008263842869521

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 33 (A009977).

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 (34,-33).

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, A218737-A218753, A133853, A094028, A218723.

[n le 2 select n-1 else 34*Self(n-1)-33*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{34, -33}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218736(n):=(33^n-1)/32$
makelist(A218736(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
A218736(n)=33^n>>5
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1 - x)*(1 - 33*x)).
a(n) = 34*a(n-1) - 33*a(n-2).
a(n) = floor(33^n/32). (End)
E.g.f.: exp(x)*(exp(32*x) - 1)/32. - Stefano Spezia, Mar 24 2023

A055129 Repunits in different bases: table by antidiagonals of numbers written in base k as a string of n 1's.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 4, 7, 4, 1, 5, 13, 15, 5, 1, 6, 21, 40, 31, 6, 1, 7, 31, 85, 121, 63, 7, 1, 8, 43, 156, 341, 364, 127, 8, 1, 9, 57, 259, 781, 1365, 1093, 255, 9, 1, 10, 73, 400, 1555, 3906, 5461, 3280, 511, 10, 1, 11, 91, 585, 2801, 9331, 19531, 21845, 9841, 1023, 11

Offset: 1

Henry Bottomley, Jun 14 2000

			T(3,5)=31 because 111 base 5 represents 25+5+1=31.
      1       1       1       1       1       1       1
      2       3       4       5       6       7       8
      3       7      13      21      31      43      57
      4      15      40      85     156     259     400
      5      31     121     341     781    1555    2801
      6      63     364    1365    3906    9331   19608
      7     127    1093    5461   19531   55987  137257
Starting with the second column, the q-th column list the numbers that are written as 11...1 in base q. - _John Keith_, Apr 12 2021

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (1 <= n <= 150).

Rows include A000012, A000027, A002061, A053698, A053699, A053700. Columns (see recurrence) include A000027, A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002275, A016123, A016125. Diagonals include A023037, A031973. Numbers in the table (apart from the first column and first two rows) are ordered in A053696.

A055129 := proc(n,k)
    add(k^j,j=0..n-1) ;
end proc: # R. J. Mathar, Dec 09 2015

Table[FromDigits[ConstantArray[1, #], k] &[n - k + 1], {n, 11}, {k, n, 1, -1}] // Flatten (* or *)
Table[If[k == 1, n, (k^# - 1)/(k - 1) &[n - k + 1]], {n, 11}, {k, n, 1, -1}] // Flatten (* Michael De Vlieger, Dec 11 2016 *)

T(n, k) = (k^n-1)/(k-1) [with T(n, 1) = n] = T(n-1, k)+k^(n-1) = (k+1)*T(n-1, k)-k*T(n-2, k) [with T(0, k) = 0 and T(1, k) = 1].
From Werner Schulte, Aug 29 2021 and Sep 18 2021: (Start)
T(n,k) = 1 + k * T(n-1,k) for k > 0 and n > 1.
Sum_{m=2..n} T(m-1,k)/Product_{i=2..m} T(i,k) = (1 - 1/Product_{i=2..n} T(i,k))/k for k > 0 and n > 1.
Sum_{n > 1} T(n-1,k)/Product_{i=2..n} T(i,k) = 1/k for k > 0.
Sum_{i=1..n} k^(i-1) / (T(i,k) * T(i+1,k)) = T(n,k) / T(n+1,k) for k > 0 and n > 0. (End)

A022172 Triangle of Gaussian binomial coefficients [ n,k ] for q = 8.

Original entry on oeis.org

1, 1, 1, 1, 9, 1, 1, 73, 73, 1, 1, 585, 4745, 585, 1, 1, 4681, 304265, 304265, 4681, 1, 1, 37449, 19477641, 156087945, 19477641, 37449, 1, 1, 299593, 1246606473, 79936505481, 79936505481, 1246606473, 299593, 1

Offset: 0

N. J. A. Sloane

			Triangle begins:
  1;
  1,      1;
  1,      9,          1;
  1,     73,         73,           1;
  1,    585,       4745,         585,           1;
  1,   4681,     304265,      304265,        4681,          1;
  1,  37449,   19477641,   156087945,    19477641,      37449,      1;
  1, 299593, 1246606473, 79936505481, 79936505481, 1246606473, 299593, 1;

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.

G. C. Greubel, Rows n=0..50 of triangle, flattened
R. Mestrovic, Lucas' theorem: its generalizations, extensions and applications (1878--2014), arXiv preprint arXiv:1409.3820 [math.NT], 2014.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
Index entries for sequences related to Gaussian binomial coefficients

Cf. A023001 (k=1), A022242 (k=2).

A027876 := proc(n)
    mul(8^i-1,i=1..n) ;
end proc:
A022172 := proc(n,m)
    A027876(n)/A027876(m)/A027876(n-m) ;
end proc: # R. J. Mathar, Jul 19 2017

a027878[n_]:=Times@@ Table[8^i - 1, {i, n}]; T[n_, m_]:=a027878[n]/( a027878[m] a027878[n - m]); Table[T[n, m], {n, 0, 10}, {m, 0, n}]//Flatten (* Indranil Ghosh, Jul 20 2017 *)
Table[QBinomial[n,k,8], {n,0,10}, {k,0,n}]//Flatten (* or *) q:= 8; T[n_, 0]:= 1; T[n_,n_]:= 1; T[n_,k_]:= T[n,k] = If[k < 0 || n < k, 0, T[n-1, k -1] +q^k*T[n-1,k]]; Table[T[n,k], {n,0,10}, {k,0,n}] // Flatten  (* G. C. Greubel, May 27 2018 *)

{q=8; T(n,k) = if(k==0,1, if (k==n, 1, if (k<0 || nG. C. Greubel, May 27 2018

from operator import mul
def a027878(n): return 1 if n==0 else reduce(mul, [8**i - 1 for i in range(1, n + 1)])
def T(n, m): return a027878(n)//(a027878(m)*a027878(n - m))
for n in range(11): print([T(n, m) for m in range(n + 1)]) # Indranil Ghosh, Jul 20 2017

T(n,k) = T(n-1,k-1) + q^k * T(n-1,k). - Peter A. Lawrence, Jul 13 2017

G.f. of column k: x^k * exp( Sum_{j>=1} f((k+1)*j)/f(j) * x^j/j ), where f(j) = 8^j - 1. - Seiichi Manyama, May 09 2025

cp's OEIS Frontend

A135519 Generalized repunits in base 14.

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A190958 a(n) = 2*a(n-1) - 10*a(n-2), with a(0) = 0, a(1) = 1.

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A218724 a(n) = (21^n - 1)/20.

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A218734 a(n) = (31^n - 1)/30.

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A132469 a(n) = (2^(5*n) - 1)/31.

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

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A190958 a(n) = 2a(n-1) - 10a(n-2), with a(0) = 0, a(1) = 1.