A127648 -id:A127648 - cp's OEIS frontend

A127774 Triangle read by rows: row n consists of n-1 zeros followed by A000292(n).

1, 0, 4, 0, 0, 10, 0, 0, 0, 20, 0, 0, 0, 0, 35, 0, 0, 0, 0, 0, 56, 0, 0, 0, 0, 0, 0, 84, 0, 0, 0, 0, 0, 0, 0, 120, 0, 0, 0, 0, 0, 0, 0, 0, 165, 0, 0, 0, 0, 0, 0, 0, 0, 0, 220, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 286, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 364

Offset: 1

Gary W. Adamson, Jan 28 2007

Essentially triangle T(n,k), read by rows, given by (0,0,0,0,0,0,0,...) DELTA (4,-3/2,5/6,-1/3,3/5,-1/10,1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 14 2011

			First few rows of the triangle are:
  1;
  0,  4;
  0,  0, 10;
  0,  0,  0, 20;
  0,  0,  0,  0, 35;
  ...

Cf. A000292, A127648, A127773.

from math import isqrt
from sympy.ntheory.primetest import is_square
def A127774(n): return (a:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(a+1)*(a+2)//6 if is_square((n<<3)+1) else 0 # Chai Wah Wu, Jun 09 2025

G.f.: 1/((x*y-1)^4). - R. J. Mathar, Aug 12 2015

More terms from Michel Marcus, Jun 10 2025

A132921 Triangle read by rows: T(n,k) = n + Fibonacci(k) - 1, 1 <= k <= n.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 4, 5, 6, 5, 5, 6, 7, 9, 6, 6, 7, 8, 10, 13, 7, 7, 8, 9, 11, 14, 19, 8, 8, 9, 10, 12, 15, 20, 28, 9, 9, 10, 11, 13, 16, 21, 29, 42, 10, 10, 11, 12, 14, 17, 22, 30, 43, 64, 11, 11, 12, 13, 15, 18, 23, 31, 44, 65, 99, 12, 12, 13, 14, 16, 19, 24, 32, 45, 66, 100, 155

Offset: 1

Gary W. Adamson, Sep 05 2007

Right border = A081659, row sums = A132922: (1, 4, 10, 19, 32, ...).

			First few rows of the triangle are:
  1;
  2, 2;
  3, 3, 4;
  4, 4, 5, 6;
  5, 5, 6, 7, 9;
  ...
Column 3 = 4, 5, 6, 7, ...; since A081659(2) = 4.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Row sums are A132922.

Cf. A127648, A127647, A081659.

T[n_,k_]:=n+Fibonacci[k]-1;Table[T[n,k],{n,12},{k,n}]//Flatten (* James C. McMahon, Mar 09 2025 *)

T(n,k)=if(k<=n, n + fibonacci(k) - 1, 0) \\ Andrew Howroyd, Sep 01 2018

Equals (A127648 * A000012 + A000012 * A127647) - A000012 as infinite lower triangular matrices.

Name clarified and terms a(56) and beyond from Andrew Howroyd, Sep 01 2018

A188429 L(n) is the minimum of the largest elements of all n-full sets, or 0 if no such set exists.

Original entry on oeis.org

1, 0, 2, 0, 0, 3, 4, 0, 0, 4, 5, 5, 6, 7, 5, 6, 6, 6, 7, 7, 6, 7, 7, 7, 7, 8, 8, 7, 8, 8, 8, 8, 8, 9, 9, 8, 9, 9, 9, 9, 9, 9, 10, 10, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 12, 13, 13

Offset: 1

Madjid Mirzavaziri, Mar 31 2011

Let A be a set of positive integers. We say that A is n-full if (sum A)=[n] for a positive integer n, where (sum A) is the set of all positive integers which are a sum of distinct elements of A and [n]={1,2,...,n}. The number L(n) denotes the minimum of the set {max A: (sum A)=[n] }.

Terms m > 7 occur exactly m times. - Reinhard Zumkeller, Aug 06 2015

			From _Reinhard Zumkeller_, Aug 06 2015: (Start)
Compressed table: no commas and for a and k: 10 replaced by A, 11 by B.
. -----------------------------------------------------------------------------
.   n   1   5   10   15   20   25   30   35   40   45   50   55   60   65   70
. ----  .---.----.----.----.----.----.----.----.----.----.----.----.----.----.-
. t(n)  10100100010000100000100000010000000100000000100000000010000000000100000
. k(n)  1 2  3   4    5     6      7       8        9         A          B
. r(n)  0101201230123401234501234560123456701234567801234567890123456789A012345
. ----  -----------------------------------------------------------------------
. a(n)  102003400455675666776777788788888998999999AA9AAAAAAABBABBBBBBBBCCBCCCCC
. -----------------------------------------------------------------------------
where t(n)=A010054(n), k(n)=A127648(n) zeros blanked, and r(n)=A002262(n). (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Mohammad Saleh Dinparvar, Python program
L. Naranjani and M. Mirzavaziri, Full Subsets of N, Journal of Integer Sequences, 14 (2011), Article 11.5.3.

Cf. A188430, A188431.

Cf. A010054, A127648, A002262.

a188429 n = a188429_list !! (n-1)
a188429_list = [1, 0, 2, 0, 0, 3, 4, 0, 0, 4, 5, 5, 6, 7] ++
               f [15 ..] (drop 15 a010054_list) 0 4
   where f (x:xs) (t:ts) r k | t == 1    = (k + 1) : f xs ts 1 (k + 1)
                             | r < k - 1 = (k + 1) : f xs ts (r + 1) k
                             | otherwise = (k + 2) : f xs ts (r + 1) k
-- Reinhard Zumkeller, Aug 06 2015

kr[n_] := {k, r} /. ToRules[Reduce[0 <= r <= k && n == k*((k+1)/2)+r, {k, r}, Integers]]; L[n_] := Which[{k0, r0} = kr[n]; r0 == 0, k0, 1 <= r0 <= k0-2, k0+1, k0-1 <= r0 <= k0, k0+2]; Join[{1, 0, 2, 0, 0, 3, 4, 0, 0, 4, 5, 5, 6, 7}, Table[L[n], {n, 15, 80}]] (* Jean-François Alcover, Oct 10 2015 *)

for n>= 15. Let n=k(k+1)/2+r, where r=0,1,..., k then
       |k, if r=0
L(n) = |k+1, if 1 <= r <= k-2
       |k+2, if k-1 <= r <= k.

A220698 Indices of triangular numbers generated in A224218.

Original entry on oeis.org

1, 6, 6, 14, 14, 14, 14, 43, 43, 36, 57, 36, 52, 43, 49, 43, 89, 52, 89, 52, 121, 49, 52, 57, 70, 89, 249, 89, 89, 89, 70, 166, 166, 103, 89, 121, 103, 103, 121, 89, 103, 241, 158, 158, 91, 91, 91, 91, 241, 166, 166, 103, 121, 103, 103, 121, 103, 121, 225, 225, 497, 216, 334

Offset: 1

Alex Ratushnyak, Apr 13 2013

nonn

Indices of triangular numbers in A220689. That is, S = triangular(i) XOR triangular(i+1); increment i; if S is a triangular number then index of S is appended to a(n). Initially i=0. XOR is the binary logical exclusive-or operator.

Cf. A000217, A224218, A220689.

A220698 := proc(n)
    A127648(A220689(n)-1) ;
end proc: # R. J. Mathar, Apr 23 2013

nmax = 100;
pmax = 2 nmax^2; (* increase coeff 2 if A224218 is too short *)
A224218 = Join[{0}, Flatten[Position[Partition[Accumulate[Range[pmax]], 2, 1], _?(OddQ[Sqrt[1 + 8 BitXor[#[[1]], #[[2]]]]]&), {1}, Heads -> False]]];
a[n_] := Module[{i, t}, i = A224218[[n]]; t = BitXor[PolygonalNumber[i], PolygonalNumber[i + 1]]; (Sqrt[8 t + 1] - 1)/2];
Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Aug 07 2023, after Harvey P. Dale in A224218 *)

def rootTriangular(a):
    sr = 1<<33
    while a < sr*(sr+1)//2:
      sr>>=1
    b = sr>>1
    while b:
        s = sr+b
        if a >= s*(s+1)//2:
          sr = s
        b>>=1
    return sr
for i in range(1<<12):
        s = (i*(i+1)//2) ^ ((i+1)*(i+2)//2)
        t = rootTriangular(s)
        if s == t*(t+1)//2:
            print(str(t), end=',')

a(n) = i where A000217(i) = A220689(n).

A132898 Triangle read by rows: T(n,k) = (-1)^(n-1)n + (-1)^(k-1)k - 1, 1 <= k <= n.

Original entry on oeis.org

1, -2, -5, 3, 0, 5, -4, -7, -2, -9, 5, 2, 7, 0, 9, -6, -9, -4, -11, -2, -13, 7, 4, 9, 2, 11, 0, 13, -8, -11, -6, -13, -4, -15, -2, -17, 9, 6, 11, 4, 13, 2, 15, 0, 17, -10, -13, -8, -15, -6, -17, -4, -19, -2, -21, 11, 8, 13, 6, 15, 4, 17, 2, 19, 0, 21

Offset: 1

Gary W. Adamson, Sep 03 2007

Row sums = A132899: (1, -7, 8, -22, 23, -45, ...).

			First few rows of the triangle:
   1;
  -2,  -5;
   3,   0,   5;
  -4,  -7,  -2,  -9;
   5,   2,   7,   0,   9;
  -6,  -9,  -4, -11,  -2, -13;
   7,   4,   9,   2,  11,   0,  13;
  -8, -11,  -6, -13,  -4, -15,  -2, -17;
   9,   6,  11,   4,  13,   2,  15,   0,   7;
  ...
T(5,3) = 7 = S(5) + S(3) = 5 + 3 - 1.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Row sums are A132899.

Cf. A127648.

T(n,k) = if(k<=n, (-1)^(n-1)*n + (-1)^(k-1)*k - 1, 0); \\ Andrew Howroyd, Sep 01 2018

Name clarified and terms a(56) and beyond from Andrew Howroyd, Sep 01 2018

A132919 Triangle read by rows: T(n,k) = Fibonacci(n) + k - 1.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 3, 4, 5, 6, 5, 6, 7, 8, 9, 8, 9, 10, 11, 12, 13, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 34, 35, 36, 37, 38, 39, 40, 41, 42, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99

Offset: 1

Gary W. Adamson, Sep 05 2007

Left border = Fibonacci numbers, right border = A081659.

Infinite lower triangular matrix by rows: n-th row = n terms of: F(n) followed by (F(n) + 1), (F(n) + 2), (F(n) + 3), ...

			First few rows of the triangle:
  1;
  1,  2;
  2,  3,  4;
  3,  4,  5,  6;
  5,  6,  7,  8,  9;
  8,  9, 10, 11, 12, 13;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Row sums are A132920.
Cf. A127647, A127648, A081659, A000045.
Cf. A132921, A132923.

T[n_,k_]:=Fibonacci[n]+k-1;Table[T[n,k],{n,11},{k,n}]//Flatten (* James C. McMahon, Mar 09 2025 *)

T(n,k) = if(k<=n, fibonacci(n) + k - 1, 0); \\ Andrew Howroyd, Aug 10 2018

Name changed and terms a(56) and beyond from Andrew Howroyd, Aug 10 2018

A127651 Triangle T(n,k) = n*k if k|n, 0 otherwise; 1<=k<=n.

Original entry on oeis.org

1, 2, 4, 3, 0, 9, 4, 8, 0, 16, 5, 0, 0, 0, 25, 6, 12, 18, 0, 0, 36, 7, 0, 0, 0, 0, 0, 49, 8, 16, 0, 32, 0, 0, 0, 64, 9, 0, 27, 0, 0, 0, 0, 0, 81, 10, 20, 0, 0, 50, 0, 0, 0, 0, 100, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 121, 12, 24, 36, 48, 0, 72, 0, 0, 0, 0, 0, 144

Offset: 1

Gary W. Adamson, Jan 22 2007

Equals the matrix product A127648 * A127093 as infinite lower triangular matrices.

			First few rows of the triangle are:
1;
2, 4;
3, 0, 9;
4, 8, 0, 16;
5, 0, 0, 0, 25;
6, 12, 18, 0, 0, 36;
7, 0, 0, 0, 0, 0, 49;
8, 16, 0, 32, 0, 0, 0, 64;
...

Cf. A127648, A127093, A064987 (row sums).

A127651 := proc(n,k)
        if n mod k =0 then
                n*k;
        else
                0 ;
        end if;
end proc: # R. J. Mathar, Oct 01 2011

T(n,k) = n*A127093(n,k). - R. J. Mathar, Oct 01 2011

A127740 Natural number transform of Aitken's triangle.

Original entry on oeis.org

1, 2, 4, 6, 9, 15, 20, 28, 40, 60, 75, 100, 135, 185, 260, 312, 402, 522, 684, 906, 1218, 1421, 1785, 2254, 2863, 3661, 4718, 6139, 7016, 8640, 10680, 13256, 16528, 20712, 26104, 33120, 37260, 45153, 54873, 66888, 81801, 100395, 123696, 153063, 190323

Offset: 0

Gary W. Adamson, Jan 27 2007

Left column (1, 2, 6, 20, ...) = A052889.

Row sums give A127741.

			First few rows of the triangle:
   1;
   2,   4;
   6,   9,  15;
  20,  28,  40,  60;
  75, 100, 135, 185, 260;
  ...

Cf. A005493, A052889, A011971, A127741.

A127648 * A011971 as infinite lower triangular matrices.

A188023 Triangle read by rows, T(n,k) = k*A115361(n-1,k-1).

Original entry on oeis.org

1, 1, 2, 0, 0, 3, 1, 2, 0, 4, 0, 0, 0, 0, 5, 0, 0, 3, 0, 0, 6, 0, 0, 0, 0, 0, 0, 7, 1, 2, 0, 4, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 5, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 3, 0, 0, 6, 0, 0, 0, 0, 0, 12

Offset: 1

Gary W. Adamson, Mar 19 2011

Triangle obtained by multiplying the lower triangular matrices A115361 and A127648.

			First few rows of the triangle =
1
1, 2
0, 0, 3
1, 2, 0, 4
0, 0, 0, 0, 5
0, 0, 3, 0, 0, 6
0, 0, 0, 0, 0, 0, 7
1, 2, 0, 4, 0, 0, 0, 8
0, 0, 0, 0, 0, 0, 0, 0, 9
0, 0, 0, 0, 5, 0, 0, 0, 0, 10
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11
0, 0, 3, 0, 0, 6, 0, 0, 0, 0, 0, 12
...

Cf. A115361, A127648, A129527 (row sums)

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A127774 Triangle read by rows: row n consists of n-1 zeros followed by A000292(n).

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A132921 Triangle read by rows: T(n,k) = n + Fibonacci(k) - 1, 1 <= k <= n.

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A188429 L(n) is the minimum of the largest elements of all n-full sets, or 0 if no such set exists.

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A220698 Indices of triangular numbers generated in A224218.

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A132898 Triangle read by rows: T(n,k) = (-1)^(n-1)*n + (-1)^(k-1)*k - 1, 1 <= k <= n.

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A132919 Triangle read by rows: T(n,k) = Fibonacci(n) + k - 1.

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A127651 Triangle T(n,k) = n*k if k|n, 0 otherwise; 1<=k<=n.

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A132898 Triangle read by rows: T(n,k) = (-1)^(n-1)n + (-1)^(k-1)k - 1, 1 <= k <= n.