A016133 -id:A016133 - cp's OEIS frontend

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

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

A016316 Expansion of 1/((1-2x)(1-8x)(1-9x)).

Original entry on oeis.org

1, 19, 255, 2975, 32231, 333759, 3353335, 32976175, 319155111, 3051352799, 28893830615, 271497720975, 2535105456391, 23548956856639, 217804673719095, 2007154559579375, 18439691005140071, 168959618797797279, 1544655767192730775, 14094055488835543375

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (19,-106,144).

Cf. A016131, A016133.

I:=[1,19,255]; [n le 3 select I[n] else 19*Self(n-1)-106*Self(n-2)+144*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jun 26 2013

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-2*x)*(1-8*x)*(1-9*x)))); // Vincenzo Librandi, Jun 26 2013

CoefficientList[Series[1 / ((1 - 2 x) (1 - 8 x) (1 - 9 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jun 26 2013 *)
LinearRecurrence[{19,-106,144},{1,19,255},30] (* Harvey P. Dale, Dec 29 2021 *)

a(n)=(9^n-2^n)/7-(8^n-2^n)/6 \\ Charles R Greathouse IV, Sep 24 2012

[(9^n - 2^n)/7-(8^n - 2^n)/6 for n in range(2,20)] # Zerinvary Lajos, Jun 05 2009

a(n) = 2^(n+1)/21 - 4*8^(n+1)/3 + 9^(n+2)/7; a(n) = A016133(n+1) - A016131(n+1). - Zerinvary Lajos, Jun 05 2009 [corrected by R. J. Mathar, Mar 14 2011]
From Vincenzo Librandi, Jun 26 2013: (Start)
a(n) = 19*a(n-1) - 106*a(n-2) + 144*a(n-3).
a(n) = 17*a(n-1) - 72*a(n-2) + 2^n. (End)
E.g.f.: exp(2*x)*(2 - 224*exp(6*x) + 243*exp(7*x))/21. - Stefano Spezia, Jul 30 2022

A016321 Expansion of 1/((1-2x)(1-9x)(1-10x)).

Original entry on oeis.org

1, 21, 313, 4065, 49081, 566721, 6350473, 69654225, 751887961, 8016991521, 84652923433, 886876310385, 9231886792441, 95586981129921, 985282830165193, 10117545471478545, 103557909243290521, 1057021183189581921

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 (21,-128,180).

Cf. A016133, A016134.

[(175*10^n +2^n-2*9^(n+2))/14 : n in [0..20]]; // Vincenzo Librandi, Oct 09 2011

CoefficientList[Series[1/((1-2x)(1-9x)(1-10x)),{x,0,20}],x] (* or *) LinearRecurrence[{21,-128,180},{1,21,313},20] (* Harvey P. Dale, Aug 18 2014 *)

a(n) = (175*10^n+2^n-162*9^n)/14 \\ Charles R Greathouse IV, Sep 23 2012

[(10^n - 2^n)/8-(9^n - 2^n)/7 for n in range(2,20)] # Zerinvary Lajos, Jun 05 2009

From Zerinvary Lajos, Jun 05 2009 [corrected by R. J. Mathar, Mar 14 2011]: (Start)
a(n) = 2^(n-1)/7 - 9^(n+2)/7 + 25*10^n/2.
a(n) = A016134(n+1) - A016133(n+1). (End)
From Vincenzo Librandi, Oct 09 2011: (Start)
a(n) = (175*10^n + 2^n - 2*9^(n+2))/14.
a(n) = 19*a(n-1) - 90*a(n-2) + 2^n.
a(n) = 21*a(n-1) - 128*a(n-2) + 180*a(n-3), n >= 3. (End)

A175840 Mirror image of Nicomachus' table: T(n,k) = 3^(n-k)*2^k for n>=0 and 0<=k<=n.

Original entry on oeis.org

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

Offset: 0

Johannes W. Meijer, Sep 21 2010, Jul 13 2011, Jun 03 2012

Lenstra calls these numbers the harmonic numbers of Philippe de Vitry (1291-1361). De Vitry wanted to find pairs of harmonic numbers that differ by one. Levi ben Gerson, also known as Gersonides, proved in 1342 that there are only four pairs with this property of the form 2^n*3^m. See also Peterson’s story ‘Medieval Harmony’.

This triangle is the mirror image of Nicomachus' table A036561. The triangle sums, see the crossrefs, mirror those of A036561. See A180662 for the definitions of these sums.

			1;
3, 2;
9, 6, 4;
27, 18, 12, 8;
81, 54, 36, 24, 16;
243, 162, 108, 72, 48, 32;

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
J. O'Connor and E.F. Robertson, Nicomachus of Gerasa, The MacTutor History of Mathematics archive, 2010.
Jay Kappraff, The Arithmetic of Nicomachus of Gerasa and its Applications to Systems of Proportion, Nexus Network Journal, vol. 2, no. 4 (October 2000).
Hendrik Lenstra, Aeternitatem Cogita, Nieuw Archief voor Wiskunde, 5/2, maart 2001, pp. 23-28.
Ivars Peterson, Medieval Harmony, Math Trek, Mathematical Association of America, 1998.

Triangle sums: A001047 (Row1), A015441 (Row2), A016133 (Kn1 & Kn4), A005061 (Kn2 & Kn3), A016153 (Fi1& Fi2), A180844 (Ca1 & Ca4), A016140 (Ca2, Ca3), A180846 (Gi1 & Gi4), A180845 (Gi2 & Gi3), A016185 (Ze1 & Ze4), A180847 (Ze2 & Ze3).

Cf. A000079, A000244, A000400, A003586.

a175840 n k = a175840_tabf !! n !! k
a175840_row n = a175840_tabf !! n
a175840_tabf = iterate (\xs@(x:_) -> x * 3 : map (* 2) xs) [1]
-- Reinhard Zumkeller, Jun 08 2013

A175840 := proc(n,k): 3^(n-k)*2^k end: seq(seq(A175840(n,k),k=0..n),n=0..9);

Flatten[Table[3^(n-k) 2^k,{n,0,10},{k,0,n}]] (* Harvey P. Dale, May 08 2013 *)

T(n,k) = 3^(n-k)*2^k for n>=0 and 0<=k<=n.

T(n,n-k) = T(n,n-k+1) + T(n-1,n-k) for n>=1 and 1<=k<=n with T(n,n) = 2^n for n>=0.

A191465 a(n) = 9^n - 2^n.

Original entry on oeis.org

0, 7, 77, 721, 6545, 59017, 531377, 4782841, 43046465, 387419977, 3486783377, 31381057561, 282429532385, 2541865820137, 22876792438577, 205891132061881, 1853020188786305, 16677181699535497, 150094635296736977, 1350851717672467801, 12157665459055880225, 109418989131510262057

Offset: 0

Vincenzo Librandi, Jun 03 2011

a(n) is the number of words of length n over the alphabet {1,2,...,9} where at least one letter >= 3 appears. - Joerg Arndt, Jan 18 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (11,-18).

Cf. A016133, A016185, A118004.

Table[9^n-2^n,{n,0,20}] (* Harvey P. Dale, Apr 16 2014 *)

a(n)=9^n-1<Charles R Greathouse IV, Jun 08 2011

a(n) = 11*a(n-1) - 18*a(n-2).
G.f.: 7*x/((1-2*x)*(1-9*x)). - Vincenzo Librandi, Oct 04 2014
a(n) = 7*A016133(n-1). - R. J. Mathar, Mar 10 2022
E.g.f.: 2*exp(11*x/2)*sinh(7*x/2). - Elmo R. Oliveira, Mar 31 2025

A096042 Triangle read by rows: T(n,k) = (n+1,k)-th element of (M^8-M)/7, where M is the infinite lower Pascal's triangle matrix, 1<=k<=n.

Original entry on oeis.org

1, 9, 2, 73, 27, 3, 585, 292, 54, 4, 4681, 2925, 730, 90, 5, 37449, 28086, 8775, 1460, 135, 6, 299593, 262143, 98301, 20475, 2555, 189, 7, 2396745, 2396744, 1048572, 262136, 40950, 4088, 252, 8, 19173961, 21570705, 10785348, 3145716, 589806

Offset: 1

Gary W. Adamson, Jun 17 2004

			Triangle begins:
1
9 2
73 27 3
585 292 54 4
4681 2925 730 90 5
37449 28086 8775 1460 135 6

Cf. A007318. First column gives A023001. Row sums give A016133.

P:= proc(n) option remember; local M; M:= Matrix(n, (i, j)-> binomial(i-1, j-1)); (M^8-M)/7 end: T:= (n, k)-> P(n+1)[n+1, k]: seq(seq(T(n, k), k=1..n), n=1..11); # Alois P. Heinz, Oct 07 2009

P[n_] := P[n] = With[{M = Array[Binomial[#1-1, #2-1]&, {n, n}]}, (MatrixPower[M, 8] - M)/7]; T[n_, k_] := P[n+1][[n+1, k]]; Table[ Table[T[n, k], {k, 1, n}], {n, 1, 11}] // Flatten (* Jean-François Alcover, Jan 28 2015, after Alois P. Heinz *)

Edited with more terms by Alois P. Heinz, Oct 07 2009

A109498 Number of closed walks of length 2n on the Heawood graph from a given node.

Original entry on oeis.org

1, 3, 15, 111, 951, 8463, 75975, 683391, 6149751, 55346223, 498112935, 4483010271, 40347080151, 363123696783, 3268113221895, 29413018898751, 264717169892151, 2382454528636143, 21442090756938855

Offset: 0

Mitch Harris, Jun 30 2005

G. C. Greubel, Table of n, a(n) for n = 0..1000
Wikipedia, Heawood graph
Index entries for linear recurrences with constant coefficients, signature (11,-18).

I:=[1,3]; [n le 2 select I[n] else 11*Self(n-1) - 18*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 30 2017

CoefficientList[Series[(1 - 8*x)/((9*x - 1)*(2*x - 1)), {x,0,50}], x] (* or *) LinearRecurrence[{11,-18}, {1,3}, 30] (* G. C. Greubel, Dec 30 2017 *)

x='x+O('x^30); Vec((1 - 8*x)/((9*x - 1)*(2*x - 1))) \\ G. C. Greubel, Dec 30 2017

O.g.f.: ( 1-8*x ) / ( (9*x-1)*(2*x-1) ).
a(n) = (9^n + 6*2^n)/7.
a(n) = A106133(n) - 8*A016133(n-1).

A135247 a(n) = 3a(n-1) + 2a(n-2) - 6*a(n-3).

Original entry on oeis.org

1, 3, 11, 33, 103, 309, 935, 2805, 8431, 25293, 75911, 227733, 683263, 2049789, 6149495, 18448485, 55345711, 166037133, 498111911, 1494335733, 4483008223, 13449024669, 40347076055, 121041228165, 363123688591, 1089371065773, 3268113205511, 9804339616533

Offset: 0

Paul Curtz, Feb 15 2008

This sequence interleaves A016133 and 3*A016133, see formulas. - Mathew Englander, Jan 08 2024

a(n) is the number of partitions of n into parts 1 (in three colors) and 2 (in two colors) where the order of colors matters. For example, the a(2)=11 such partitions (using parts 1, 1', 1'', 2, and 2') are 2, 2', 1+1, 1+1', 1+1'', 1'+1, 1'+1', 1'+1'', 1''+1, 1''+1', 1''+1''. For such partitions where the order of colors does not matter see A002624. - Joerg Arndt, Jan 18 2024

Sean A. Irvine, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,2,-6).

Cf. A016133.

a:=[1,3,11];; for n in [4..30] do a[n]:=3*a[n-1]+2*a[n-2]-6*a[n-3]; od; a; # G. C. Greubel, Nov 20 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1-3*x-2*x^2+6*x^3) )); // G. C. Greubel, Nov 20 2019

seq(coeff(series(1/(1-3*x-2*x^2+6*x^3), x, n+1), x, n), n = 0..30); # G. C. Greubel, Nov 20 2019

LinearRecurrence[{3,2,-6},{1,3,11},30] (* Harvey P. Dale, Jun 27 2015 *)

my(x='x+O('x^30)); Vec(1/(1-3*x-2*x^2+6*x^3)) \\ G. C. Greubel, Nov 20 2019

def A135247_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-3*x-2*x^2+6*x^3) ).list()
A135247_list(30) # G. C. Greubel, Nov 20 2019

G.f.: 1/((1-3*x)*(1-2*x^2)). - G. C. Greubel, Oct 04 2016
From Mathew Englander, Jan 08 2024: (Start)
a(n) = A010684(n) * A016133(floor(n/2)).
a(n) = 3*a(n-1) + A077957(n) for n >= 1.
a(n) = (A000244(n+2) - A164073(n+3))/7.
(End)

More terms from Harvey P. Dale, Jun 27 2015

Dropped two leading terms = 0. - Joerg Arndt, Jan 18 2024

cp's OEIS Frontend

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

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A016316 Expansion of 1/((1-2x)*(1-8x)*(1-9x)).

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A016321 Expansion of 1/((1-2x)(1-9x)(1-10x)).

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A175840 Mirror image of Nicomachus' table: T(n,k) = 3^(n-k)*2^k for n>=0 and 0<=k<=n.

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A191465 a(n) = 9^n - 2^n.

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A096042 Triangle read by rows: T(n,k) = (n+1,k)-th element of (M^8-M)/7, where M is the infinite lower Pascal's triangle matrix, 1<=k<=n.

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A109498 Number of closed walks of length 2n on the Heawood graph from a given node.

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A016316 Expansion of 1/((1-2x)(1-8x)(1-9x)).

A135247 a(n) = 3a(n-1) + 2a(n-2) - 6*a(n-3).