A030237 -id:A030237 - cp's OEIS frontend

A278180 Square spiral in which each new term is the sum of its two largest neighbors.

1, 1, 2, 3, 4, 7, 8, 15, 16, 17, 33, 35, 37, 72, 76, 80, 84, 164, 172, 180, 188, 368, 384, 401, 418, 435, 853, 888, 925, 962, 999, 1961, 2037, 2117, 2201, 2285, 2369, 4654, 4826, 5006, 5194, 5382, 5570, 10952, 11336, 11737, 12155, 12590, 13025, 13460, 26485, 27373, 28298, 29260, 30259, 31258, 32257, 63515

Offset: 1

Omar E. Pol, Nov 14 2016

nonn

To evaluate a(n) consider the neighbors of a(n) that are present in the spiral when a(n) should be a new term in the spiral.

For the same idea but for a hexagonal spiral see A278619; and for a right triangle see A278645. It appears that the same idea for an isosceles triangle and also for a square array gives A030237. - Omar E. Pol, Dec 04 2016

			Illustration of initial terms as a square spiral:
.
.          84----80----76-----72----37
.           |                        |
.          164    4-----3-----2     35
.           |     |           |      |
.          172    7     1-----1     33
.           |     |                  |
.          180    8-----15----16----17
.           |
.          188---368---384---401---418
.
a(21) = 188 because the sum of its two largest neighbors is 180 + 8 = 188.
a(22) = 368 because the sum of its two largest neighbors is 180 + 188 = 368.
a(23) = 384 because the sum of its two largest neighbors is 368 + 16 = 384.
a(24) = 401 because the sum of its two largest neighbors is 384 + 17 = 401.
a(25) = 418 because the sum of its two largest neighbors is 401 + 17 = 418.
a(26) = 435 because the sum of its two largest neighbors is 418 + 17 = 435.

Peter Kagey, Table of n, a(n) for n = 1..10000
Peter Kagey, Bitmap illustrating the parity of the first one million terms. (Even and odd numbers are represented with black and white pixels respeectively.)

Cf. A030237, A078510, A141481, A274917, A278181, A278619, A278645.

A096465 Triangle (read by rows) formed by setting all entries in the first column and in the main diagonal ((i,i) entries) to 1 and the rest of the entries by the recursion T(n, k) = T(n-1, k) + T(n, k-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 8, 9, 1, 1, 5, 13, 22, 23, 1, 1, 6, 19, 41, 64, 65, 1, 1, 7, 26, 67, 131, 196, 197, 1, 1, 8, 34, 101, 232, 428, 625, 626, 1, 1, 9, 43, 144, 376, 804, 1429, 2055, 2056, 1, 1, 10, 53, 197, 573, 1377, 2806, 4861, 6917, 6918, 1, 1, 11, 64, 261, 834, 2211, 5017, 9878, 16795, 23713, 23714, 1

Offset: 0

Gerald McGarvey, Aug 12 2004

The third column is A034856 (binomial(n+1, 2) + n-1).
The row sums are A014137 (partial sums of Catalan numbers (A000108)).
The "1st subdiagonal" ((i+1,i) entries) are also A014137.
The "2nd subdiagonal" ((i+2,i) entries) is A014138 ( Partial sums of Catalan numbers (starting 1,2,5,...)).
The "3rd subdiagonal" ((i+3,i) entries) is A001453 (Catalan numbers - 1.)
This is the reverse of A091491 - see A091491 for more information. The sequence of antidiagonal sums gives A124642. - Gerald McGarvey, Dec 09 2006

			Triangle begins as:
  1;
  1, 1;
  1, 2,  1;
  1, 3,  4,  1;
  1, 4,  8,  9,   1;
  1, 5, 13, 22,  23,   1;
  1, 6, 19, 41,  64,  65,   1;
  1, 7, 26, 67, 131, 196, 197, 1;

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened

Cf. A000108, A001453, A014137, A014138, A034856.

Cf. A006134, A024718, A030237, A078478, A091491, A100066, A105848, A124642.

a096465 n k = a096465_tabl !! n !! k
a096465_row n = a096465_tabl !! n
a096465_tabl = map reverse a091491_tabl
-- Reinhard Zumkeller, Jul 12 2012

A096465:= func< n,k | k eq n select 1 else (n-k)*(&+[Binomial(n+k-2*j, n-j)/(n+k-2*j): j in [0..k]]) >;
[A096465(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 30 2021

A096465:= (n,k)-> `if`(k=n, 1, (n-k)*add(binomial(n+k-2*j, n-j)/(n+k-2*j), j=0..k));
seq(seq(A096465(n,k), k=0..n), n=0..12) # G. C. Greubel, Apr 30 2021

T[, 0]= 1; T[n, n_]= 1; T[n_, m_]:= T[n, m]= T[n-1, m] + T[n, m-1]; T[n_, m_] /; n < 0 || m > n = 0; Table[T[n, m], {n, 0, 12}, {m, 0, n}]//Flatten (* Jean-François Alcover, Dec 17 2012 *)

def A096465(n,k): return 1 if (k==n) else (n-k)*sum( binomial(n+k-2*j, n-j)/(n+k-2*j) for j in (0..k))
flatten([[A096465(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 30 2021

From G. C. Greubel, Apr 30 2021: (Start)
T(n, k) = (n-k) * Sum_{j=0..k} binomial(n+k-2*j, n-j)/(n+k-2*j) with T(n,n) = 1.
T(n, k) = A091491(n, n-k).
Sum_{k=0..n} T(n,k) = Sum_{j=0..n} A000108(j) = A014137(n). (End)

Offset changed by Reinhard Zumkeller, Jul 12 2012

A278619 Hexagonal spiral constructed on the nodes of the triangular net in which each new term is the sum of its two largest neighbors in the structure.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 15, 18, 22, 26, 31, 36, 42, 49, 56, 64, 72, 82, 94, 106, 121, 139, 157, 179, 205, 231, 262, 298, 334, 376, 425, 481, 537, 601, 673, 745, 827, 921, 1027, 1133, 1254, 1393, 1550, 1707, 1886, 2091, 2322, 2553, 2815, 3113, 3447, 3781, 4157, 4582, 5063, 5600

Offset: 0

Omar E. Pol, Nov 24 2016

nonn

To evaluate a(n) consider only the two largest neighbors of a(n) that are present in the spiral when a(n) should be a new term in the spiral.
For the same idea but for an right triangle see A278645; for a square spiral see A278180.
It appears that the same idea for an isosceles triangle and also for a square array gives A030237.

			Illustration of initial terms as a spiral:
.
.             18 - 15 - 12
.             /          \
.           22    3 - 2   10
.           /    /     \   \
.         26    4   1 - 1   8
.           \    \         /
.           31    5 - 6 - 7
.             \
.              36 - 42 - 49
.
a(16) = 36 because the sum of its two largest neighbors is 31 + 5 = 36.
a(17) = 42 because the sum of its two largest neighbors is 36 + 6 = 42.
a(18) = 49 because the sum of its two largest neighbors is 42 + 7 = 49.
a(19) = 56 because the sum of its two largest neighbors is 49 + 7 = 56.

Cf. A030237, A274920, A274921, A278180, A278181, A278645.

A355173 The Fuss-Catalan triangle of order 1, read by rows. Related to binary trees.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 3, 5, 0, 1, 4, 9, 14, 0, 1, 5, 14, 28, 42, 0, 1, 6, 20, 48, 90, 132, 0, 1, 7, 27, 75, 165, 297, 429, 0, 1, 8, 35, 110, 275, 572, 1001, 1430, 0, 1, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 0, 1, 10, 54, 208, 637, 1638, 3640, 7072, 11934, 16796

Offset: 0

Peter Luschny, Jun 25 2022

The Fuss-Catalan triangle of order m is a regular, (0, 0)-based table recursively defined as follows: Set row(0) = [1] and row(1) = [0, 1]. Now assume row(n-1) already constructed and duplicate the last element of row(n-1). Next apply the cumulative sum m times to this list to get row(n). Here m = 1. (See the Python program for a reference implementation.)

This definition also includes the classical Fuss-Catalan numbers, since T(n, n) = A000108(n), or row 2 in A355262. For m = 2 see A355172 and for m = 3 A355174. More generally, for n >= 1 all Fuss-Catalan sequences (A355262(n, k), k >= 0) are the main diagonals of the Fuss-Catalan triangles of order n - 1.

			Table T(n, k) begins:
  [0] [1]
  [1] [0, 1]
  [2] [0, 1, 2]
  [3] [0, 1, 3,  5]
  [4] [0, 1, 4,  9,  14]
  [5] [0, 1, 5, 14,  28,  42]
  [6] [0, 1, 6, 20,  48,  90,  132]
  [7] [0, 1, 7, 27,  75, 165,  297, 429]
  [8] [0, 1, 8, 35, 110, 275,  572, 1001, 1430]
  [9] [0, 1, 9, 44, 154, 429, 1001, 2002, 3432, 4862]
Seen as an array reading the diagonals starting from the main diagonal:
  [0] 1, 1, 2,  5,  14,   42,  132,   429,  1430,   4862,   16796, ...  A000108
  [1] 0, 1, 3,  9,  28,   90,  297,  1001,  3432,  11934,   41990, ...  A000245
  [2] 0, 1, 4, 14,  48,  165,  572,  2002,  7072,  25194,   90440, ...  A099376
  [3] 0, 1, 5, 20,  75,  275, 1001,  3640, 13260,  48450,  177650, ...  A115144
  [4] 0, 1, 6, 27, 110,  429, 1638,  6188, 23256,  87210,  326876, ...  A115145
  [5] 0, 1, 7, 35, 154,  637, 2548,  9996, 38760, 149226,  572033, ...  A000588
  [6] 0, 1, 8, 44, 208,  910, 3808, 15504, 62016, 245157,  961400, ...  A115147
  [7] 0, 1, 9, 54, 273, 1260, 5508, 23256, 95931, 389367, 1562275, ...  A115148

A000108 (main diagonal), A000245 (subdiagonal), A002057 (diagonal 2), A000344 (diagonal 3), A000027 (column 2), A000096 (column 3), A071724 (row sums), A000958 (alternating row sums), A262394 (main diagonal of array).
Variants: A009766 (main variant), A030237, A130020.
Cf. A123110 (triangle of order 0), A355172 (triangle of order 2), A355174 (triangle of order 3), A355262 (Fuss-Catalan array).
Cf. A115144, A115145, A000588, A115147, A115148.

from functools import cache
from itertools import accumulate
@cache
def Trow(n: int) -> list[int]:
    if n == 0: return [1]
    if n == 1: return [0, 1]
    row = Trow(n - 1) + [Trow(n - 1)[n - 1]]
    return list(accumulate(row))
for n in range(11): print(Trow(n))

The general formula for the Fuss-Catalan triangles is, for m >= 0 and 0 <= k <= n:
FCT(n, k, m) = (m*(n - k) + m + 1)*(m*n + k - 1)!/((m*n + 1)!*(k - 1)!) for k > 0 and FCT(n, 0, m) = 0^n. The case considered here is T(n, k) = FCT(n, k, 1).
T(n, k) = (n - k + 2)*(n + k - 1)!/((n + 1)!*(k - 1)!) for k > 0; T(n, 0) = 0^n.
The g.f. of row n of the FC-triangle of order m is 0^n + (x - (m + 1)*x^2) / (1 - x)^(m*n + 2), thus:
T(n, k) = [x^k] (0^n + (x - 2*x^2)/(1 - x)^(n + 2)).

A278645 Triangle read by rows in which each new term is the sum of its two largest neighbors in the structure.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 8, 15, 22, 29, 23, 45, 74, 103, 132, 68, 142, 245, 377, 509, 641, 210, 455, 832, 1341, 1982, 2623, 3264, 665, 1497, 2838, 4820, 7443, 10707, 13971, 17235, 2162, 5000, 9820, 17263, 27970, 41941, 59176, 76411, 93646, 7162, 16982, 34245, 62215, 104156, 163332, 239743, 333389, 427035, 520681

Offset: 1

Omar E. Pol, Nov 24 2016

To evaluate T(n,k) consider only the two largest neighbors of T(n,k) that are present in the triangle when T(n,k) should be a new term in the triangle.
For the same idea but for a square spiral see A278180; and for a hexagonal spiral see A278619.
It appears that the same idea for an isosceles triangle and also for a square array gives A030237.

			Triangle begins:
1;
1,    2;
3,    5,     7;
8,    15,    22,    29;
23,   45,    74,    103,   132;
68,   142,   245,   377,   509,    641;
210,  455,   832,   1341,  1982,   2623,   3264;
665,  1497,  2838,  4820,  7443,   10707,  13971,  17235;
2162, 5000,  9820,  17263, 27970,  41941,  59176,  76411,  93646;
7162, 16982, 34245, 62215, 104156, 163332, 239743, 333389, 427035, 520681;
...

Cf. A030237, A278180, A278317, A278619.

cp's OEIS Frontend

A278180 Square spiral in which each new term is the sum of its two largest neighbors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A096465 Triangle (read by rows) formed by setting all entries in the first column and in the main diagonal ((i,i) entries) to 1 and the rest of the entries by the recursion T(n, k) = T(n-1, k) + T(n, k-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

Sage

Formula

Extensions

A278619 Hexagonal spiral constructed on the nodes of the triangular net in which each new term is the sum of its two largest neighbors in the structure.

Views

Author

Keywords

Comments

Examples

Crossrefs

A355173 The Fuss-Catalan triangle of order 1, read by rows. Related to binary trees.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Python

Formula

A278645 Triangle read by rows in which each new term is the sum of its two largest neighbors in the structure.

Views

Author

Keywords

Comments

Examples

Crossrefs