A035312 -id:A035312 - cp's OEIS frontend

A035313 (Largest) diagonal of the Zorach additive triangle A035312.

1, 3, 9, 26, 66, 154, 346, 771, 1726, 3887, 8768, 19700, 43890, 96717, 210665, 453893, 968903, 2053260, 4328489, 9093971, 19068611, 39943689, 83628399, 175018523, 366081209, 765102907, 1597315656, 3330380593, 6933810145

Offset: 0

Alex Zorach

From Philippe Lallouet (philip.lallouet(AT)wanadoo.fr), Apr 22 2007: (Start)
Starting with 1, smallest sequence for which:
all its terms a1(n).............................. 1,3,9,26,66
all terms of first differences a2(n)=a1(n+1)-a1(n) 2,6,17,40
all terms of second differences a3(n)=a2(n+1)-a2(n) 4,11,23
...
all terms of (1+i)th differences ai(n)=ai-1(n+1)-ai-1(n)
are different for any n and any i (End)
Which is to say, this sequence is the lexicographically earliest sequence of positive integers such that the sequence itself and its n-th differences for n >= 1 are pairwise disjoint. - David W. Wilson, Feb 26 2012
Conjecturally, every positive integer occurs in the sequence or one of its n-th differences, which would imply that the sequence and its n-th differences partition the positive integers. - David W. Wilson, Feb 26 2012
Conjecture: lim(n->infinity, a(n+1)/a(n)) = 2. - David W. Wilson, Feb 26 2012
Note that the n-th differences yield the n-th subdiagonals (parallels to the right edge) in the triangle A035312. Therefore Lallouet's statement and Wilson's 1st comment above are just rephrasing the definition of that triangle.  - M. F. Hasler, May 09 2013
Binomial transform of A035311. Hence, from the observed asymptotic equality A035311(n) ~ 2*n, a stronger statement than the one given above follows: a(n) ~ n*2^n. - Andrey Zabolotskiy, Feb 08 2017

			Start with 1; 2 is the next, then add 1+2 to get 3, then 4 is next, then 4+2=6 and 6+3 is 9, then 5 is not next because 5+4=9 and 9 was already used, so 7 is next...which ultimately generates 26 in the final column...

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
A. C. Zorach, Additive triangle
A. C. Zorach, Haskell programs for sequences in connection with Zorach additive triangle

Cf. A035311, A035312, A189713, A035358.

Haskell
```
-- See link for Haskell program.
```

(* Assuming n <= t(n,1) <= 3n *) rows = 29; uniqueQ[t1_, n_] := (t[n, 1] = t1; Do[t[n, k] = t[n, k-1] + t[n-1, k-1], {k, 2, n}]; n*(n+1)/2 == Length[ Union[ Flatten[ Table[t[m, k], {m, 1, n}, {k, 1, m}]]]]); t[n_, 1] := t[n, 1] = Select[ Complement[Range[n, 3 n], Flatten[ Table[t[m, k], {m, 1, n-1}, {k, 1, m}]]], uniqueQ[#, n] &, 1][[1]]; Last /@ Table[t[n, k], {n, 1, rows}, {k, 1, n}] (* Jean-François Alcover, Jun 05 2012 *)

More terms from Christian G. Bower and Dean Hickerson

A035311 First column of the Zorach additive triangle A035312.

Original entry on oeis.org

1, 2, 4, 7, 5, 8, 10, 14, 15, 19, 16, 20, 21, 22, 27, 30, 28, 33, 32, 38, 37, 39, 44, 45, 51, 46, 47, 54, 55, 52, 59, 62, 60, 64, 50, 67, 72, 78, 74, 68, 81, 82, 80, 85, 79, 87, 86, 91, 90, 94, 102, 98, 103, 99, 105, 100, 108, 112, 110, 113, 127, 128, 125, 133, 118, 123

Offset: 0

Alex Zorach

One can see that (least unused number in rows 1 through n of A035312) ~ A035311(n) ~ 2n. (Asymptotic equality, and the first quantity does not exceed either of the two others.) - M. F. Hasler, May 09 2013

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
A. C. Zorach, Additive triangle
Haskell programs for sequences in connection with Zorach additive triangle

Cf. A035312, A035313, A189713, A035358.

Haskell
```
-- See link for Haskell program.
```

(* Assuming n <= t(n,1) <= 3n *) uniqueQ[t1_, n_] := (t[n, 1] = t1; Do[t[n, k] = t[n, k-1] + t[n-1, k-1], {k, 2, n}]; n*(n+1)/2 == Length[ Union[ Flatten[ Table[ t[m, k], {m, 1, n}, {k, 1, m}]]]]); t[n_ , 1] := t[n, 1] = Select[ Complement[ Range[n, 3 n], Flatten[ Table[t[m, k], {m, 1, n - 1}, {k, 1, m}]]], uniqueQ[#, n] & , 1][[1]]; Table[t[n, 1], {n, 1, 66}](* Jean-François Alcover, Dec 02 2011 *)

More terms from Christian G. Bower

A035358 k such that A035312(k-1) = n or 0 if there is none.

Original entry on oeis.org

1, 2, 3, 4, 11, 5, 7, 16, 6, 22, 8, 12, 17, 29, 37, 56, 9, 23, 46, 67, 79, 92, 13, 30, 18, 10, 106, 137, 38, 121, 24, 172, 154, 47, 57, 68, 211, 191, 232, 14, 80, 31, 93, 254, 277, 326, 352, 19, 107, 596, 301, 436, 39, 379, 407, 25, 122, 138, 466, 529, 155, 497, 48

Offset: 1

Christian G. Bower, Nov 15 1998

nonn

At least up through the first 19 terms (ending at 46), this appears to be identical to the inverse of sequence A035312 considered as a permutation of the positive integers. - Howard A. Landman, Sep 23 2001

This sequence is *by definition* the inverse of A035312 (upon shifting its offset), provided that A035312 indeed is surjective on the positive integers. - M. F. Hasler, May 09 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
A. C. Zorach, Additive triangle
Reinhard Zumkeller, Haskell programs for sequences in connection with Zorach additive triangle
Index entries for sequences that are permutations of the natural numbers

Cf. A035311, A035312, A035313.

Haskell
```
-- See link for Haskell program.
```

a(n) = A000217(A072039(n) - 1) + A072038(n) if the sequence is in fact the inverse permutation of the flattened Zorach additive triangle. - Reinhard Zumkeller, Apr 30 2011

A189713 Central terms of Zorach additive triangle (cf. A035312).

Original entry on oeis.org

1, 6, 23, 56, 168, 459, 1080, 2400, 5461, 12678, 28728, 63336, 138717, 303346, 663179, 1438098, 3062730, 6450121, 13583957, 28627773, 59954459, 124413466, 258510987, 543867101, 1152248068, 2428235533, 5069886692, 10511339217, 21696371682, 44710463120

Offset: 0

Reinhard Zumkeller, Apr 30 2011

nonn

a(n) = A035312(2*n,n).

Reinhard Zumkeller, Table of n, a(n) for n = 0..500
A. C. Zorach, Additive triangle
Haskell programs for sequences in connection with Zorach additive triangle

Cf. A035311, A035313.

Haskell
```
-- See link.
```

A189714 Sums of rows of Zorach additive triangle (cf. A035312).

Original entry on oeis.org

1, 5, 19, 61, 146, 336, 757, 1711, 3868, 8752, 19680, 43869, 96695, 210638, 453863, 968875, 2053227, 4328457, 9093933, 19068574, 39943650, 83628355, 175018478, 366081158, 765102861, 1597315609, 3330380539, 6933810090, 14414812994, 29924339675, 62038821363

Offset: 0

Reinhard Zumkeller, Apr 30 2011

nonn

a(n) = Sum(A035312(n,k): 0 <= k <= n).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
A. C. Zorach, Additive triangle
Haskell programs for sequences in connection with Zorach additive triangle

Haskell
```
-- See link.
```

A323013 Form of Zorach additive triangle T(n,k) (see A035312) where each number is sum of west and northwest numbers, with the additional condition that the first element T(n,1) is a Fibonacci number.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 8, 13, 20, 30, 21, 29, 42, 62, 92, 34, 55, 84, 126, 188, 280, 89, 123, 178, 262, 388, 576, 856, 144, 233, 356, 534, 796, 1184, 1760, 2616, 377, 521, 754, 1110, 1644, 2440, 3624, 5384, 8000, 610, 987, 1508, 2262, 3372, 5016, 7456, 11080, 16464, 24464

Offset: 1

Michel Lagneau, Jan 02 2019

Conjecture: Let F(i) be the i-th Fibonacci number. Each number of T(n, k), k = 1, 2, 3 is the difference between two Fibonacci numbers F(i) - F(j) for some i, j, where F(i) is the smallest Fibonacci number greater than T(n, k). The case T(n, 1) is trivial. Examples: 10 = 13 - 3, 29 = 34 - 5, 20 = 21 - 1, 42 = 55 - 13, 84 = 89 - 5, ...
We observe interesting properties:
T(n,1) = A117647(n) = 1, 2, 5, 8, 21, ... where n = 1, 2, ...
T(2n,2) = A033887(n) = 3, 13, 55, ... (Fibonacci(3n+1)), and T(2n+1,2) = A048876(n) = 7, 29, 123, ... (Generalized Pell equation with second term of 7) where n = 1, 2, ...
T(3n,3) = 10, 84, 754, 6388,... If n = 2m - 1, T(6m - 3, 3) = F(9m - 2) - F(9m - 5) and if n = 2m, T(6m, 3) =  F(9m + 2) - F(9m - 4).
T(3n+1,3) = 20, 178, 1508, 13530, ... If n = 2m - 1, T(6m - 2, 3) = F(9m - 1) - F(9m - 7) and if n = 2m, T(6m+1, 3) =  F(9m + 4) - F(9m + 1).
T(3n+2,3) = 42, 356, 3194, 27060, ... If n = 2m - 1, T(6m - 1, 3) = F(9m + 1) - F(9m - 2) and if n = 2m, T(6m + 2, 3) =  F(9m + 5) - F(9m - 1).
Other property:
T(2m, 1) + T(2m, 2) = T(2m +1, 1) with T(2m, 1)= F(3m), T(2m, 2) = F(3m + 1) and T(2m + 1, 1) = F(3m + 2).
T(2m + 1, 1) + T(2m + 1, 2) = F(3m + 4) - F(3m - 1).

			The start of the sequence as a triangular array T(n, k) read by rows:
   1;
   2,   3;
   5,   7,  10;
   8,  13,  20,   30;
  21,  29,  42,   62,   92;
  34,  55,  84,  126,  188,  280;
  ...

Cf. A000045, A002878, A033887, A035312, A036561, A117647.

with(combinat,fibonacci):
lst:={1}:lst2:=lst:
for n from 2 to 15 do :
lst1:={}:ii:=0:
  for j from 1 to 1000 while(ii=0) do:
     i:=fibonacci(j):
     if {i} intersect lst2 = {} and {i+lst[1]} intersect lst2 = {}
      then
      lst1:=lst1 union {i}:ii:=1:
      else
     fi:
   od:
    for k from 1 to n-1 do:
      lst1:=lst1 union {lst1[k]+lst[k]}:
    od:
    lst:=lst1:lst2:=lst2 union lst:
    print(lst1):
   od:

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

A072038 Column of Zorach additive triangle in which n appears (or 0 in the unlikely event that it does not).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 2, 2, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 3, 2, 3, 4, 1, 1, 2, 1, 3, 1, 1, 2, 2, 2, 1, 1, 1, 4, 2, 3, 2, 1, 1, 1, 1, 4, 2, 1, 1, 1, 3, 1, 1, 4, 2, 2, 1, 1, 2, 1, 3, 1, 2, 5, 1, 1, 3, 2, 3, 1, 4, 1, 2, 2, 3, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 5, 2, 1, 1, 3, 2, 1, 4, 2, 2, 1, 1, 1, 2, 1, 1, 5, 1

Offset: 1

Henry Bottomley, Jun 07 2002

nonn

			a(8)=1 since 8 appears in the first (i.e. left hand) column.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
A. C. Zorach, Additive triangle
Haskell programs for sequences in connection with Zorach additive triangle

If a(n)=1 then n is in A035311, if a(n)=A072039(n) then n is in A035313. Cf. A035312, A035358, A072039.

Haskell
```
-- See link.
```

If a(n) > 0: n = A035312(A072039(n)-1, a(n)-1).

A072039 Row of Zorach additive triangle in which n appears (or 0 in the unlikely event that it does not).

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 6, 3, 7, 4, 5, 6, 8, 9, 11, 4, 7, 10, 12, 13, 14, 5, 8, 6, 4, 15, 17, 9, 16, 7, 19, 18, 10, 11, 12, 21, 20, 22, 5, 13, 8, 14, 23, 24, 26, 27, 6, 15, 35, 25, 30, 9, 28, 29, 7, 16, 17, 31, 33, 18, 32, 10, 34, 19, 5, 36, 40, 11, 20, 12, 37, 8, 39, 21, 22, 13, 38, 45, 43

Offset: 1

Henry Bottomley, Jun 07 2002

nonn

			a(8)=6 since 8 appears in the sixth row.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
A. C. Zorach, Additive triangle
Haskell programs for sequences in connection with Zorach additive triangle

Cf. A035312, A035311, A035313, A035358, A072038.

Haskell
```
-- See link.
```

If a(n) > 0: n = A035312(a(n)-1, A072038(n)-1).

A136561 Triangle read by rows: n-th diagonal (from the right) is the sequence of (signed) differences between pairs of consecutive terms in the (n-1)th diagonal. The rightmost diagonal (A136562) is defined: A136562(1)=1; A136562(n) is the smallest integer > A136562(n-1) such that any (signed) integer occurs at most once in the triangle A136561.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, -5, -1, 5, 14, 13, 8, 7, 12, 26, -30, -17, -9, -2, 10, 36, 75, 45, 28, 19, 17, 27, 63, -200, -125, -80, -52, -33, -16, 11, 74, 524, 324, 199, 119, 67, 34, 18, 29, 103, -1299, -775, -451, -252, -133, -66, -32, -14, 15, 118

Offset: 1

Leroy Quet, Jan 06 2008

Requiring that the absolute values of the differences in the difference triangle only occur at most once each leads to the Zorach additive triangle. (See A035312.)

			The triangle begins:
1,
2,3,
4,6,9,
-5,-1,5,14,
13,8,7,12,26,
-30,-17,-9,-2,10,36.
Example:
Considering the rightmost value of the 4th row: Writing a 10 here instead, the first 4 rows of the triangle become:
1
2,3
4,6,9
-9,-5,1,10
But 1 already occurs earlier in the triangle. So 10 is not the rightmost element of row 4.
Checking 11,12,13,14; 14 is the smallest value that can be the rightmost element of row 4 and not have any elements of row 4 occur earlier in the triangle.

Cf. A035312, A136562, A136563.

Rows 7-10 from Andrey Zabolotskiy, May 29 2017

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A035313 (Largest) diagonal of the Zorach additive triangle A035312.

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A035311 First column of the Zorach additive triangle A035312.

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A035358 k such that A035312(k-1) = n or 0 if there is none.

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A189713 Central terms of Zorach additive triangle (cf. A035312).

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A189714 Sums of rows of Zorach additive triangle (cf. A035312).

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A323013 Form of Zorach additive triangle T(n,k) (see A035312) where each number is sum of west and northwest numbers, with the additional condition that the first element T(n,1) is a Fibonacci number.

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

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A072038 Column of Zorach additive triangle in which n appears (or 0 in the unlikely event that it does not).

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