A016185 -id:A016185 - cp's OEIS frontend

A047969 Square array of nexus numbers a(n,k) = (n+1)^(k+1) - n^(k+1) (n >= 0, k >= 0) read by upwards antidiagonals.

1, 1, 1, 1, 3, 1, 1, 5, 7, 1, 1, 7, 19, 15, 1, 1, 9, 37, 65, 31, 1, 1, 11, 61, 175, 211, 63, 1, 1, 13, 91, 369, 781, 665, 127, 1, 1, 15, 127, 671, 2101, 3367, 2059, 255, 1, 1, 17, 169, 1105, 4651, 11529, 14197, 6305, 511, 1, 1, 19, 217, 1695, 9031

Offset: 0

N. J. A. Sloane

If each row started with an initial 0 (i.e., a(n,k) = (n+1)^k - n^k) then each row would be the binomial transform of the preceding row. - Henry Bottomley, May 31 2001
a(n-1, k-1) is the number of ordered k-tuples of positive integers such that the largest of these integers is n. - Alford Arnold, Sep 07 2005
From Alford Arnold, Jul 21 2006: (Start)
The sequences in A047969 can also be calculated using the Eulerian Array (A008292) and Pascal's Triangle (A007318) as illustrated below: (cf. A101095).
  1       1       1       1       1       1
  1       1       1       1       1       1
  -----------------------------------------
  1       2       3       4       5       6
          1       2       3       4       5
  1       3       5       7       9      11
  -----------------------------------------
  1       3       6      10      15      21
          4      12      24      40      60
                  1       3       6      10
  1       7      19      37      61      91
  -----------------------------------------
  1       4      10      20      35      56
         11      44     110     220     385
                 11      44     110     220
                          1       4      10
  1      15      65     175     369     671
  -----------------------------------------  (End)
From Peter Bala, Oct 26 2008: (Start)
The above remarks of Alford Arnold may be summarized by saying that (the transpose of) this array is the Hilbert transform of the triangle of Eulerian numbers A008292 (see A145905 for the definition of the Hilbert transform). In this context, A008292 is best viewed as the array of h-vectors of permutohedra of type A. See A108553 for the Hilbert transform of the array of h-vectors of type D permutohedra. Compare this array with A009998.
The polynomials n^k - (n-1)^k, k = 1,2,3,..., which give the nonzero entries in the columns of this array, satisfy a Riemann hypothesis: their zeros lie on the vertical line Re s = 1/2 in the complex plane. See A019538 for the connection between the polynomials n^k - (n-1)^k and the Stirling polynomials of the simplicial complexes dual to the type A permutohedra.
(End)
Empirical: (n+1)^(k+1) - n^(k+1) is the number of first differences of length k+1 arrays of numbers in 0..n, k > 0. - R. H. Hardin, Jun 30 2013
a(n-1, k-1) is the number of bargraphs of width k and height n. Examples: a(1,2) = 7 because we have [1,1,2], [1,2,1], [2,1,1], [1,2,2], [2,1,2], [2,2,1], and [2,2,2]; a(2,1) = 5 because we have [1,3], [2,3], [3,1], [3,2], and [3,3] (bargraphs are given as compositions). This comment is equivalent to A. Arnold's Sep 2005 comment. - Emeric Deutsch, Jan 30 2017

			Array a begins:
  [n\k][0  1   2    3    4   5  6  ...
  [0]   1  1   1    1    1   1  1  ...
  [1]   1  3   7   15   31  63  ...
  [2]   1  5  19   65  211  ...
  [3]   1  7  37  175  ...
  ...
Triangle T begins:
  n\m   0   1    2     3     4      5      6      7      8     9  10 ...
  0:    1
  1:    1   1
  2:    1   3    1
  3:    1   5    7     1
  4:    1   7   19    15     1
  5:    1   9   37    65    31      1
  6:    1  11   61   175   211     63      1
  7:    1  13   91   369   781    665    127      1
  8:    1  15  127   671  2101   3367   2059    255      1
  9:    1  17  169  1105  4651  11529  14197   6305    511     1
  10:   1  19  217  1695  9031  31031  61741  58975  19171  1023   1
  ...  - _Wolfdieter Lang_, May 07 2021

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 54.

T. D. Noe, Rows n = 0..100 of triangle, flattened
A. Blecher, C. Brennan, A. Knopfmacher and Helmut Prodinger, The height and width of bargraphs, Discrete Applied Math. 180, (2015), 36-44.
Eric Weisstein's World of Mathematics, Nexus Number

Cf. A047970.
Cf. A009998, A108553 (Hilbert transform of array of h-vectors of type D permutohedra), A145904, A145905.
Row n sequences of array a: A000012, A000225(k+1), A001047(k+1), A005061(k+1), A005060(k+1), A005062(k+1), A016169(k+1), A016177(k+1), A016185(k+1), A016189(k+1), A016195(k+1), A016197(k+1).
Column k sequences of array a: (nexus numbers): A000012, A005408, A003215, A005917(n+1), A022521, A022522, A022523, A022524, A022525, A022526, A022527, A022528.
Cf. A343237 (row reversed triangle).

Flatten[Table[n = d - e; k = e; (n + 1)^(k + 1) - n^(k + 1), {d, 0, 100}, {e, 0, d}]] (* T. D. Noe, Feb 22 2012 *)

T(n,m):=if m=0 then 1 else sum(k!*(-1)^(m+k)*stirling2(m,k)*binomial(n+k-1,n),k,0,m); /* Vladimir Kruchinin, Jan 28 2018 */

From Vladimir Kruchinin: (Start)
O.g.f. of e.g.f of rows of array: ((1-x)*exp(y))/(1-x*exp(y))^2.
T(n,m) = Sum_{k=0..m} k!*(-1)^(m+k)*Stirling2(m,k)*C(n+k-1,n), T(n,0)=1.(End)
From Wolfdieter Lang, May 07 2021: (Start)
T(n,m) = a(n-m,m) = (n-m+1)^(m+1) - (n-m)^(m+1), n >= 0, m = 0, 1,..., n.
O.g.f. column k of the array: polylog(-(k+1), x)*(1-x)/x. See the Peter Bala comment above, and the Eulerian triangle A008292 formula by Vladeta Jovovic, Sep 02 2002.
E.g.f. of e.g.f. of row of the array: exp(y)*(1 + x*(exp(y) - 1))*exp(x*exp(y)).
O.g.f. of triangle's exponential row polynomials R(n, y) = Sum_{m=0} T(n, m)*(y^m)/m!: G(x, y) = exp(x*y)*(1 - x)/(1 - x*exp(x*y))^2. (End)

A036561 Nicomachus triangle read by rows, T(n, k) = 2^(n - k)*3^k, for 0 <= k <= n.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 8, 12, 18, 27, 16, 24, 36, 54, 81, 32, 48, 72, 108, 162, 243, 64, 96, 144, 216, 324, 486, 729, 128, 192, 288, 432, 648, 972, 1458, 2187, 256, 384, 576, 864, 1296, 1944, 2916, 4374, 6561, 512, 768, 1152, 1728, 2592, 3888, 5832, 8748, 13122, 19683

Offset: 0

N. J. A. Sloane

The triangle pertaining to this sequence has the property that every row, every column and every diagonal contains a nontrivial geometric progression. More interestingly every line joining any two elements contains a nontrivial geometric progression. - Amarnath Murthy, Jan 02 2002
Kappraff states (pp. 148-149): "I shall refer to this as Nicomachus' table since an identical table of numbers appeared in the Arithmetic of Nicomachus of Gerasa (circa 150 A.D.)" The table was rediscovered during the Italian Renaissance by Leon Battista Alberti, who incorporated the numbers in dimensions of his buildings and in a system of musical proportions. Kappraff states "Therefore a room could exhibit a 4:6 or 6:9 ratio but not 4:9. This ensured that ratios of these lengths would embody musical ratios". - Gary W. Adamson, Aug 18 2003
After Nichomachus and Alberti several Renaissance authors described this table. See for instance Pierre de la Ramée in 1569 (facsimile of a page of his Arithmetic Treatise in Latin in the links section). - Olivier Gérard, Jul 04 2013
The triangle sums, see A180662 for their definitions, link Nicomachus's table with eleven different sequences, see the crossrefs. It is remarkable that these eleven sequences can be described with simple elegant formulas. The mirror of this triangle is A175840. - Johannes W. Meijer, Sep 22 2010
The diagonal sums Sum_{k} T(n - k, k) give A167762(n + 2). - Michael Somos, May 28 2012
Where d(n) is the divisor count function, then d(T(i,j)) = A003991, the rows of which sum to the tetrahedral numbers A000292(n+1). For example, the sum of the divisors of row 4 of this triangle (i = 4), gives d(16) + d(24) + d(36) + d(54) + d(81) = 5 + 8 + 9 + 8 + 5 = 35 = A000292(5). In fact, where p and q are distinct primes, the aforementioned relationship to the divisor function and tetrahedral numbers can be extended to any triangle of numbers in which the i-th row is of form {p^(i-j)*q^j, 0<=j<=i}; i >= 0 (e.g., A003593, A003595). - Raphie Frank, Nov 18 2012, corrected Dec 07 2012
Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then 2*x and 3*x are in S, and duplicates are deleted as they occur; see A232559. - Clark Kimberling, Nov 28 2013
Partial sums of rows produce Stirling numbers of the 2nd kind: A000392(n+2) = Sum_{m=1..(n^2+n)/2} a(m). - Fred Daniel Kline, Sep 22 2014
A permutation of A003586. - L. Edson Jeffery, Sep 22 2014
Form a word of length i by choosing a (possibly empty) word on alphabet {0,1} then concatenating a word of length j on alphabet {2,3,4}. T(i,j) is the number of such words. - Geoffrey Critzer, Jun 23 2016
Form of Zorach additive triangle (see A035312) where each number is sum of west and northwest numbers, with the additional condition that each number is GCD of the two numbers immediately below it. - Michel Lagneau, Dec 27 2018

			The start of the sequence as a triangular array read by rows:
   1
   2   3
   4   6   9
   8  12  18  27
  16  24  36  54  81
  32  48  72 108 162 243
  ...
The start of the sequence as a table T(n,k) n, k > 0:
    1    2    4    8   16   32 ...
    3    6   12   24   48   96 ...
    9   18   36   72  144  288 ...
   27   54  108  216  432  864 ...
   81  162  324  648 1296 2592 ...
  243  486  972 1944 3888 7776 ...
  ...
- _Boris Putievskiy_, Jan 08 2013

Jay Kappraff, Beyond Measure, World Scientific, 2002, p. 148.
Flora R. Levin, The Manual of Harmonics of Nicomachus the Pythagorean, Phanes Press, 1994, p. 114.

Reinhard Zumkeller and Matthew House, Rows n = 0..300 of triangle, flattened [Rows 0 through 120 were computed by Reinhard Zumkeller; rows 121 through 300 by Matthew House, Jul 09 2015]
Fred Daniel Kline, How do I convert this Nicomachus' Triangle to one with edges?
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Pierre de la Ramée (Petrus Ramus), P. Rami Arithmeticae (anno 1569) Liber 2, Cap. XVI "De inventione continue proportionalium" p.46 (leaf 0055) describes this integer triangle in a layout close to the current OEIS 'tabl' layout.
Marko Riedel, Proof of identity by Egorychev method.
Thomas Scheuerle, Version of this triangle from Boethius (480-524), Anicius Manlius Severinus Boethius, De institutione arithmetica, Medeltidshandskrift 1 (Mh 1), Lund University Library, early 10th century, page 4r.
Robert Sedgewick, Analysis of shellsort and related algorithms, Fourth European Symposium on Algorithms, Barcelona, September, 1996.

Cf. A001047 (row sums), A000400 (central terms), A013620, A007318.
Cf. A003586, A000079, A000244, A007283, A025197, A005010, A003946, A005051.
Triangle sums (see the comments): A001047 (Row1); A015441 (Row2); A005061 (Kn1, Kn4); A016133 (Kn2, Kn3); A016153 (Fi1, Fi2); A016140 (Ca1, Ca4); A180844 (Ca2, Ca3); A180845 (Gi1, Gi4); A180846 (Gi2, Gi3); A180847 (Ze1, Ze4); A016185 (Ze2, Ze3). - Johannes W. Meijer, Sep 22 2010, Sep 10 2011
Antidiagonal cumulative sum: A000392; square arrays cumulative sum: A160869. Antidiagonal products: 6^A000217; antidiagonal cumulative products: 6^A000292; square arrays products: 6^A005449; square array cumulative products: 6^A006002.

a036561 n k = a036561_tabf !! n !! k
a036561_row n = a036561_tabf !! n
a036561_tabf = iterate (\xs@(x:_) -> x * 2 : map (* 3) xs) [1]
-- Reinhard Zumkeller, Jun 08 2013

/* As triangle: */ [[(2^(i-j)*3^j)/3: j in [1..i]]: i in [1..10]]; // Vincenzo Librandi, Oct 17 2014

A036561 := proc(n,k): 2^(n-k)*3^k end:
seq(seq(A036561(n,k),k=0..n),n=0..9);
T := proc(n,k) option remember: if k=0 then 2^n elif k>=1 then procname(n,k-1) + procname(n-1,k-1) fi: end: seq(seq(T(n,k),k=0..n),n=0..9);
# Johannes W. Meijer, Sep 22 2010, Sep 10 2011

Flatten[Table[ 2^(i-j) 3^j, {i, 0, 12}, {j, 0, i} ]] (* Flatten added by Harvey P. Dale, Jun 07 2011 *)

for(i=0,9,for(j=0,i,print1(3^j<<(i-j)", "))) \\ Charles R Greathouse IV, Dec 22 2011

{T(n, k) = if( k<0 || k>n, 0, 2^(n - k) * 3^k)} /* Michael Somos, May 28 2012 */

T(n,k) = A013620(n,k)/A007318(n,k). - Reinhard Zumkeller, May 14 2006
T(n,k) = T(n,k-1) + T(n-1,k-1) for n>=1 and 1<=k<=n with T(n,0) = 2^n for n>=0. - Johannes W. Meijer, Sep 22 2010
T(n,k) = 2^(k-1)*3^(n-1), n, k > 0 read by antidiagonals. - Boris Putievskiy, Jan 08 2013
a(n) = 2^(A004736(n)-1)*3^(A002260(n)-1), n > 0, or a(n) = 2^(j-1)*3^(i-1) n > 0, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor[(-1+sqrt(8*n-7))/2]. - Boris Putievskiy, Jan 08 2013
G.f.: 1/((1-2x)(1-3yx)). - Geoffrey Critzer, Jun 23 2016
T(n,k) = (-1)^n * Sum_{q=0..n} (-1)^q * C(k+3*q, q) * C(n+2*q, n-q). - Marko Riedel, Jul 01 2024

A016189 a(n) = 10^n - 9^n.

Original entry on oeis.org

0, 1, 19, 271, 3439, 40951, 468559, 5217031, 56953279, 612579511, 6513215599, 68618940391, 717570463519, 7458134171671, 77123207545039, 794108867905351, 8146979811148159, 83322818300333431, 849905364703000879, 8649148282327007911, 87842334540943071199, 890581010868487640791

Offset: 0

N. J. A. Sloane

Almost all numbers contain any given sequence of digits (in any base) [Theorem 143 of Hardy and Wright]. a(7) = 5217031, more than 52% of the numbers < 10^7 contain any given nonzero decimal digit. - Frank Ellermann, May 30 2001
a(n) gives the number of integers from 0 to 10^n-1 which contain (at least) any one given decimal digit except 0. - Michael Taktikos, Aug 24 2004
These are the numerators of a(n)=(integral{x=0 to 0.2} (1-0.5*x)^n dx). E.g., a(3)=3439/20000. The denominators are b(n)=5*(n+1)*10^n. E.g., b(3)=20000. - Al Hakanson (hawkuu(AT)excite.com), Feb 22 2004
Binomial transforms of sequences defined by a(n)=(C+1)^n-C^n are the sequences (C+2)^n-(C+1)^n. The binomial transform of this here is in A016195, for example. - R. J. Mathar, Nov 27 2008
First differences are given in A088924. - M. F. Hasler, May 04 2015

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 143

Vincenzo Librandi, Table of n, a(n) for n = 0..130
Alexander Bogomolny, Almost every integer has a digit 3 in it
John Elias, Illustration of Initial Terms
James Grime, 3 is everywhere, Numberphile video
Index entries for linear recurrences with constant coefficients, signature (19, -90).

Base 2: A000225, 3: A001047, 4: A005061, 5: A005060, 6: A005062, base 7: A016169, 8: A016177, 9: A016185 11: A016195 12: A016197.
Equals A155671 - 1.
Cf. A011557, A011533.

a016189 n = 10 ^ n - 9 ^ n
a016189_list = 0 : zipWith (+) (map (* 9) a016189_list) a011557_list
-- Reinhard Zumkeller, Apr 03 2015

[10^n - 9^n: n in [0..20]]; // Vincenzo Librandi, Apr 26 2011

f[n_]:=10^n-9^n;f[Range[0,40]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011 *)

a(n)=10^n-9^n \\ M. F. Hasler, May 04 2015

G.f.: x/((1-9x)(1-10x)).
a(0) = 0, a(1) = 1, then a(n+1) = 9*a(n) + 10^n.
a(n) = 19*a(n-1) - 90*a(n-2), n > 1; a(0)=0, a(1)=1. - Philippe Deléham, Jan 01 2009
E.g.f.: e^(10*x) - e^(9*x). - Mohammad K. Azarian, Jan 14 2009

A016140 Expansion of 1/((1-3x)(1-8*x)).

Original entry on oeis.org

1, 11, 97, 803, 6505, 52283, 418993, 3354131, 26839609, 214736555, 1717951489, 13743789059, 109950843913, 879608345627, 7036871547985, 56294986732787, 450359936909017, 3602879624412299, 28823037382718881, 230584300224012515, 1844674405278884521, 14757395252691429371, 118059162052912494577, 944473296517443135443

Offset: 0

N. J. A. Sloane

In general, for expansion of 1/((1-b*x)*(1-c*x)): a(n) = (c^(n+1) - b^(n+1))/(c-b) = (b+c)*a(n-1) - b*c*a(n-2) = b*a(n-1) + c^n = c*a(n-1) + b^n = Sum_{i=0..n} b^i*c^(n-i). - Henry Bottomley, Jul 20 2000

8*a(n) gives the number of edges in the n-th-order Sierpiński carpet graph. - Eric W. Weisstein, Aug 19 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (11,-24).

Sequences with g.f. 1/((1-n*x)*(1-8*x)): A001018 (n=0), A023001 (n=1), A016131 (n=2), this sequence (n=3), A016152 (n=4), A016162 (n=5), A016170 (n=6), A016177 (n=7), A053539 (n=8), A016185 (n=9), A016186 (n=10), A016187 (n=11), A016188 (n=12), A060195 (n=16).

Cf. A190543.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1-8*x)))); // Vincenzo Librandi, Jun 24 2013

Table[(8^(n+1)-3^(n+1))/5, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2011 *)
CoefficientList[Series[1/((1-3 x)(1-8 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 24 2013 *)
LinearRecurrence[{11,-24},{1,11},30] (* Harvey P. Dale, Feb 03 2022 *)

Vec(1/((1-3*x)*(1-8*x))+O(x^30)) \\ Charles R Greathouse IV, Sep 23 2012

[lucas_number1(n,11,24) for n in range(1, 30)] # Zerinvary Lajos, Apr 27 2009

a(n) = (8^(n+1) - 3^(n+1))/5.
a(n) = 11*a(n-1) - 24*a(n-2).
a(n) = 3*a(n-1) + 8^n.
a(n) = 8*a(n-1) + 3^n.
a(n) = Sum_{i=0..n} 3^i*8^(n-i).
E.g.f.: (1/5)*(8*exp(8*x) - 3*exp(3*x)). - G. C. Greubel, Nov 14 2024

A180845 a(n) = (16^n-3^n)/13.

Original entry on oeis.org

0, 1, 19, 313, 5035, 80641, 1290499, 20648713, 330381595, 5286112081, 84577812979, 1353245066713, 21651921244555, 346430740444321, 5542891848703459, 88686269584038313, 1418980313358961915

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+1) appear in several triangle sums of Nicomachus' table A036561, i.e Gi1(4*n), Gi1(4*n+1)/2, Gi1(4*n+2)/4, Gi1(4*n+3)/8 and Gi4(n). See A180662 for information about these giraffe and other chess sums.

Index entries for linear recurrences with constant coefficients, signature (19, -48).

Cf. A016153, A016140, A180844, A180845, A180846, A180847, A016185.

Table[(16^n-3^n)/13,{n,0,30}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011*)

a(n) = (16^n-3^n)/13

G.f.: x/((16*x-1)*(3*x-1))

A180844 a(n) = (27^n - 2^n)/25.

Original entry on oeis.org

0, 1, 29, 787, 21257, 573955, 15496817, 418414123, 11297181449, 305023899379, 8235645283745, 222362422662139, 6003785411879801, 162102206120758723, 4376759565260493713, 118172508262033346635, 3190657723074900391913

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+1) appear in several triangle sums of Nicomachus's table A036561, i.e., Ca2(3*n), Ca2(3*n+1)/3, Ca2(3*n+2)/9 and Ca3(n). See A180662 for information about these camel sums and other chess sums.

Index entries for linear recurrences with constant coefficients, signature (29, -54).

Cf. A016153, A016140, A180844, A180845, A180846, A180847, A016185.

(#[[1]]-#[[2]])/25&/@Partition[Riffle[27^Range[0,20],2^Range[0,20]],2]  (* Harvey P. Dale, Jan 22 2011 *)

a(n) = (27^n - 2^n)/25 \\ Iain Fox, Dec 12 2017

first(n) = Vec(x/((27*x-1)*(2*x-1)) + O(x^n), -n) \\ Iain Fox, Dec 12 2017

a(n) = (27^n - 2^n)/25.

G.f.: x/((27*x-1)*(2*x-1)).

A180846 a(n) = (81^n - 2^n)/79.

Original entry on oeis.org

0, 1, 83, 6727, 544895, 44136511, 3575057423, 289579651327, 23455951757615, 1899932092367071, 153894499481733263, 12465454458020395327, 1009701811099652023535, 81785846699071813910431, 6624653582624816926753103

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+1) appear in several triangle sums of Nicomachus's table A036561, i.e., Gi2(4*n), Gi2(4*n+1)/2, Gi2(4*n+2)/4, Gi2(4*n+3)/8 and Gi3(n). See A180662 for information about these giraffe and other chess sums.

Nathaniel Johnston, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (83,-162).

Cf. A016153, A016140, A180844, A180845, A180846, A180847, A016185.

[(81^n-2^n)/79: n in [0..50]]; // Vincenzo Librandi, Apr 15 2011

Table[(81^n-2^n)/79,{n,0,15}] (* Vladimir Joseph Stephan Orlovsky, Apr 13 2011 *)

a(n) = (81^n - 2^n)/79.

G.f.: x/((81*x-1)*(2*x-1)).

A180847 a(n) = (27^n-4^n)/23.

Original entry on oeis.org

0, 1, 31, 853, 23095, 623821, 16844191, 454797253, 12279542215, 331547705341, 8951788306351, 241698285320053, 6525853707835735, 176198050128342061, 4757347353532344511, 128448378545641737253, 3468106220733400647655

Offset: 0

Johannes W. Meijer, Sep 21 2010

For n>0, a(n) appears in several triangle sums of Nicomachus' table A036561, i.e. Ze1(2*n), Ze1(2*n+1)/2; Ze4(3*n), Ze4(3*n+1)/3 and Ze4(3*n+2)/9. See A180662 for information about these zebra and other chess sums.

Index entries for linear recurrences with constant coefficients, signature (31,-108).

Cf. A016153, A016140, A180844, A180845, A180846, A180847, A016185.

Table[(27^n-4^n)/23,{n,0,20}] (* or *) LinearRecurrence[{31,-108},{0,1},20]  (* Harvey P. Dale, Sep 01 2011 *)

a(n)=(27^n-4^n)/23 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (27^n-4^n)/23.
G.f.: x/((27*x-1)*(4*x-1)).
a(0)=0, a(1)=1, a(n) = 31*a(n-1)-108*a(n-2). - Harvey P. Dale, Sep 01 2011

A016197 a(n) = 12^n - 11^n.

Original entry on oeis.org

0, 1, 23, 397, 6095, 87781, 1214423, 16344637, 215622815, 2801832661, 35979939623, 457696700077, 5777672071535, 72470493235141, 904168630965623, 11229773405170717, 138934529031464255, 1713164078241143221

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (23, -132).

Cf. k^n-(k-1)^n: A000225 (k=2), A001047 (k=3), A005061 (k=4), A005060 (k=5), A005062 (k=6), A016169 (k=7), A016177 (k=8), A016185 (k=9), A016189 (k=10), A016195 (k=11), this sequence (k=12).

[12^n-11^n: n in [0..40]]; // Vincenzo Librandi, Aug 03 2014

f[n_]:= 12^n - 11^n; f[Range[0, 40]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011 *)
CoefficientList[Series[x/((1 - 11 x) (1 - 12 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 03 2014 *)

a(n)=12^n-11^n \\ Charles R Greathouse IV, Aug 03 2014

G.f.: x/((1-11x)(1-12x)).
E.g.f.: e^(12*x)-e^(11*x). - Mohammad K. Azarian, Jan 14 2009
a(0)=0, a(n)=12*a(n-1)+11^(n-1). - _Vincenzo Librandi-, Feb 09 2011
a(0)=0, a(1)=1, a(n)=23*a(n-1)-132*a(n-2). - Vincenzo Librandi, Feb 09 2011

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