A014616 -id:A014616 - cp's OEIS frontend

A193593 Augmentation of the triangle A193592. See Comments.

1, 1, 1, 1, 3, 2, 1, 6, 10, 6, 1, 10, 31, 40, 23, 1, 15, 75, 166, 187, 105, 1, 21, 155, 530, 958, 993, 549, 1, 28, 287, 1415, 3786, 5988, 5865, 3207, 1, 36, 490, 3311, 12441, 28056, 40380, 37947, 20577, 1, 45, 786, 7000, 35469, 109451, 217720, 292092

Offset: 0

Clark Kimberling, Jul 31 2011

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.

Regarding A193592, (column 1)=A014616, (column 2)=A090809, (right edge)=A113227.

			First 5 rows:
1
1...1
1...3...2
1...6...10...6
1...10..31...40...23
Rows reversed as in Callan's n-edge increasing ordered trees with outdegree k:
 1
0      1
0      1      1
0      2      3      1
0      6     10      6      1
0     23     40     31     10      1
0    105    187    166     75     15      1
0    549    993    958    530    155     21     1
0   3207   5865   5988   3786   1415    287    28    1
0  20577  37947  40380  28056  12441   3311   490   36   1
0 143239 265901 292092 217720 109451  35469  7000  786  45 1

D. Callan, A bijection to count (1-23-4)-avoiding permutations, arXiv:1008.2375 (rows reversed)

Cf. A193091, A193592, A113227 (row sums and diagonal), A090809 (3rd col).

p[n_, 0] := 1; p[n_, k_] := n + 1 - k /; k > 0;
Table[p[n, k], {n, 0, 5}, {k, 0, n}] (* A193592 *)
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 12}]]  (* A193593 *)
Flatten[Table[v[n], {n, 0, 10}]]

A234305 Irregular triangle read by rows. Theoretical distribution of electrons based on the Janet's sequence A167268.

Original entry on oeis.org

1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 3, 2, 2, 4, 2, 2, 5, 2, 2, 6, 2, 2, 6, 1, 2, 2, 6, 2, 2, 2, 6, 2, 1, 2, 2, 6, 2, 2, 2, 2, 6, 2, 3, 2, 2, 6, 2, 4, 2, 2, 6, 2, 5, 2, 2, 6, 2, 6, 2, 2, 6, 2, 6, 1, 2, 2, 6, 2, 6, 2, 2, 2, 6, 2, 6, 2, 1, 2, 2, 6, 2, 6, 2, 2, 2, 2, 6, 2, 6, 2, 3, 2, 2, 6, 2, 6, 2, 4

Offset: 1

Paul Curtz, Jan 02 2014

a(n) is not A173642, a compact Bohr-Stoner model (1924), modified by Charles Janet in 1930. The good distribution is A168208.
Only sequences N16(n) in A234398 are used:
N16(1)= 1 followed by 2's              = A040000,
N16(2)= 1, 2, 3, 4, 5, followed by 6's = A101272,
N16(3)= 1 to  9, followed by 10's,
N16(4)= 1 to 13, followed by 14's, etc.
The distribution by rows are in the example.
The N16(n)'s are respectively on columns (hence triangle T)
1, 2, 4,  6,  9, 12, 16, 20, 25, 30, 36,   A002620(n+2)
   3, 5,  8, 11, 15, 19, 24, 29, 35,       A024206(n+2)
      7, 10, 14, 18, 23, 28, 34,           A014616(n+3)
         13, 17, 22, 27, 33,               A004116(n+4)
             21, 26, 32,
                 31, etc.
See A163255.
Antidiagonals give the natural numbers A000027, like rows sums in the example.
A033638=1, 1, 2, 3, 5, 7,... is upon the triangle T.

			1,      H
2,       He
2, 1,    Li
2, 2,    Be
2, 2, 1,
2, 2, 2,
2, 2, 3,
2, 2, 4,
2, 2, 5,
2, 2, 6,
2, 2, 6, 1,
2, 2, 6, 2,
2, 2, 6, 2, 1,
2, 2, 6, 2, 2,
2, 2, 6, 2, 3,
2, 2, 6, 2, 4,
2, 2, 6, 2, 5,
2, 2, 6, 2, 6,
2, 2, 6, 2, 6, 1,
2, 2, 6, 2, 6, 2,
2, 2, 6, 2, 6, 2, 1,
2, 2, 6, 2, 6, 2, 2,
2, 2, 6, 2, 6, 2, 3, etc.

Cf. A002061, A002522 (or A160457), A014206, A059100, diagonals of the triangle T. A004526.

A377802 Triangle read by rows: T(n, k) = (2 * (n+1)^2 + 7 - (-1)^n) / 8 - k.

Original entry on oeis.org

1, 2, 1, 4, 3, 2, 6, 5, 4, 3, 9, 8, 7, 6, 5, 12, 11, 10, 9, 8, 7, 16, 15, 14, 13, 12, 11, 10, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31

Offset: 1

Werner Schulte, Nov 07 2024

The natural numbers, based on quarter-squares (A002620 and A033638); every natural number occurs exactly twice.

			Triangle T(n, k) for 1 <= k <= n starts:
n\ k :   1   2   3   4   5   6   7   8   9  10  11  12  13
==========================================================
   1 :   1
   2 :   2   1
   3 :   4   3   2
   4 :   6   5   4   3
   5 :   9   8   7   6   5
   6 :  12  11  10   9   8   7
   7 :  16  15  14  13  12  11  10
   8 :  20  19  18  17  16  15  14  13
   9 :  25  24  23  22  21  20  19  18  17
  10 :  30  29  28  27  26  25  24  23  22  21
  11 :  36  35  34  33  32  31  30  29  28  27  26
  12 :  42  41  40  39  38  37  36  35  34  33  32  31
  13 :  49  48  47  46  45  44  43  42  41  40  39  38  37
  etc.

A002620 (column 1), A024206 (column 2), A014616 (column 3), A004116 (column 4), A033638 (main diagonal), A290743 (1st subdiagonal).

Cf. A006003, A026741, A002061, A000290, A002378, A005563, A246694.

PARI
```
T(n,k)=(2*(n+1)^2+7-(-1)^n)/8-k
```

T(n, k) = A002620(n+1) + 1 - k.
T(2*n-1, n) = n^2 - n + 1 = A002061(n); T(2*n-2, n) = (n-1)^2 = A000290(n-1) for n > 1; T(2*n-3, n) = (n-1) * (n-2) = A002378(n-2) for n > 2; T(2*n-4, n) = (n-1) * (n-3) = A005563(n-3) for n > 3.
Row sums are (2 * n^3 + 5 * n - n * (-1)^n) / 8 = (A006003(n) + A026741(n)) / 2.
G.f.: x*y*(1 - x*y + x^2*y + x^4*y^2 - x^5*y^3 + x^6*y^3 - x^3*y*(1 + 2*y - y^2))/((1 - x)^3*(1 + x)*(1 - x*y)^3*(1 + x*y)). - Stefano Spezia, Nov 08 2024

A193591 Augmentation of the Euler partition triangle A026820. See Comments.

Original entry on oeis.org

1, 1, 2, 1, 4, 7, 1, 7, 19, 31, 1, 10, 45, 103, 161, 1, 14, 82, 297, 617, 937, 1, 18, 146, 652, 2057, 4005, 5953, 1, 23, 228, 1395, 5251, 15004, 27836, 40668, 1, 28, 355, 2555, 13023, 43470, 115110, 205516, 295922, 1, 34, 509, 4689, 27327, 122006, 371942

Offset: 0

Clark Kimberling, Jul 31 2011

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.

			First 5 rows:
  1
  1...2
  1...4...7
  1...7...19...31
  1...10..45...103...161

Cf. A014616 (column 1), A026820, A193091.

p[n_, k_] := Length@IntegerPartitions[n + 1,
   k + 1] (* A026820, Euler partition triangle *)
Table[p[n, k], {n, 0, 5}, {k, 0, n}]
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 12}]]  (* A193591 *)
Flatten[Table[v[n], {n, 0, 9}]]

A257566 Triangle read by rows, T(n,k) = Sum_{j=0..n-k+1} P(n,j)*T(n-j,k-1) if k>0 else 0^n where P(n,j) is the number of j-partitions of n; for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 5, 13, 7, 1, 0, 7, 30, 30, 10, 1, 0, 11, 76, 119, 65, 14, 1, 0, 15, 152, 357, 306, 113, 18, 1, 0, 22, 330, 1119, 1375, 746, 193, 23, 1, 0, 30, 633, 2973, 5059, 3888, 1497, 295, 28, 1, 0, 42, 1245, 8036, 18605, 19423, 10298, 2930, 447, 34, 1

Offset: 0

Peter Luschny, Apr 30 2015

			1;
0,  1;
0,  2,  1;
0,  3,  4,   1;
0,  5, 13,   7,  1;
0,  7, 30,  30, 10,  1;
0, 11, 76, 119, 65, 14, 1;

Cf. A000041, A014616.

T := proc(n,k) option remember; `if`(k=0, 0^n,
add(combinat:-numbpart(n,j)*T(n-j, k-1), j=0..n-k+1)) end:
for n from 0 to 12 do seq(T(n, k), k=0..n) od;

T(n,1) = A000041(n) for n>=1.

T(n,n-1) = A014616(n-1) for n>=2.

A114139 Changes in United States postal rates per ounce since 1863.

Original entry on oeis.org

-2, -2, 1, -1, 1, 1, 1, 1, 2, 2, 0, 3, 2, 3, 2, 2, 3, 4, 3, 3, 1, 1, 3, 2

Offset: 1

Jonathan Vos Post, Feb 03 2006

Benjamin Franklin, first Postmaster General of the United States, applied computational complexity theory to Economics by changing the business plan for American mail by changing from payment by distance to payment by weight. "Before stamps were used a person had to collect his mail at the post office and pay for it. Franklin stopped the money loss on unclaimed mail in Philadelphia by printing in his paper the names of persons who had mail awaiting them. He also developed a simple, accurate way of keeping post-office accounts. In 1753 Franklin was made deputy postmaster general for all the colonies." [Encyclopedia Britannica]

			a(1) = -2 because the rate per half ounce was lowered effective 3 March 1863 from 3 cents to 2 cents; thereafter rates were per ounce.

R. K. Guy, Unsolved Problems in Number Theory, C12.

R. Alter and J. A. Barnett, A postage stamp problem, Amer. Math. Monthly, Vol. 87, No. 3 (1980), 206-210.
A. K. Dart, The History of Postage Rates in the United States
W. F. Lunnon, A postage stamp problem, Comput. J., Vol. 12, No. 4 (1969) 377-380.
USPS, Benjamin Franklin Commemorative Stamps
1847usa.com, United States Postage Stamps By Year.

First differences of A114062.

Cf. A001212, A001213, A014616, A001208, A001209, A001210, A001211, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A014616, A053346, A053348, A075060, A084192, A084193.

A140571 Decimal expansion of the universal constant in E(h), the maximum number of essential elements of order h.

Original entry on oeis.org

2, 0, 5, 7, 2, 8, 4, 1, 2, 8, 4, 7, 8, 7, 9, 3, 4, 1, 2, 8, 5, 8, 2, 2, 3, 9, 6, 4, 4, 8, 3, 7, 6, 9, 0, 9, 1, 0, 0, 4, 3, 4, 7, 8, 2, 7, 4, 9, 4, 2, 1, 2, 6, 8, 0, 7, 4, 1, 5, 3, 8, 1, 9, 6, 6, 2, 4, 2, 3, 6, 9, 2, 9, 5, 4, 2, 7, 6, 3, 5, 1, 3, 3, 4, 9, 8, 5, 1, 9, 0, 8, 0, 7, 8, 9, 0, 1, 6, 5, 3, 6, 5, 5, 9, 7, 7

Offset: 1

Jonathan Vos Post, Jul 05 2008

A fundamental result of Erdos and Graham is that every integer basis possesses only finitely many essential elements. Grekos refined this, showing that the number of essential elements in a basis or order h is bounded by a function of h only. Deschamps and Farhi (2007) proved a best possible upper bound on this function, which contains a constant whose digits are this sequence.
Abstract: Plagne recently determined the asymptotic behavior of the function E(h), which counts the maximum possible number of essential elements in an additive basis for N of order h. Here we extend his investigations by studying asymptotic behavior of the function E(h,k), which counts the maximum possible number of essential subsets of size k, in a basis of order h. For a fixed k and with h going to infinity, we show that
E(h,k) = Theta_{k} ([h^{k}/log h]^{1/(k+1)}). The determination of a more precise asymptotic formula is shown to depend on the solution of the well-known "postage stamp problem" in finite cyclic groups. On the other hand, with h fixed and k going to infinity, we show that E(h,k) ~ (h-1) (log k)/(log log k).

			2.0572841284787934...

Bruno Deschamps, Bakir Farhi, Essentialité dans les bases additives, J. Number Theory, 123 (2007), p. 170-192.
P. Erdos and R. L. Graham, On Bases with an Exact Order, Acta Arith. 37(1980)201-207.
G. Grekos, Sur l'ordre d'une base additive, (French) Séminaire de Théorie des nombres de Bordeaux, 1987/1988, exposé 31.
Peter Hegarty, The Postage Stamp Problem and Essential Subsets in Integer Bases, arXiv:0807.0463 [math.NT], 2008.

Cf. postage stamp sequences: A001208, A001209, A001210, A001211, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A014616, A053346, A053348, A075060, A084192, A084193.

RealDigits[(30*Sqrt[Log[1564]/1564]),10,120][[1]] (* Harvey P. Dale, Sep 27 2023 *)

30*sqrt(log(1564)/1564) \\ Michel Marcus, Oct 18 2018

Equals 30*sqrt(log(1564)/1564).

a(100) corrected by Georg Fischer, Jul 12 2021

cp's OEIS Frontend

A193593 Augmentation of the triangle A193592. See Comments.

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A234305 Irregular triangle read by rows. Theoretical distribution of electrons based on the Janet's sequence A167268.

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A377802 Triangle read by rows: T(n, k) = (2 * (n+1)^2 + 7 - (-1)^n) / 8 - k.

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A193591 Augmentation of the Euler partition triangle A026820. See Comments.

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A257566 Triangle read by rows, T(n,k) = Sum_{j=0..n-k+1} P(n,j)*T(n-j,k-1) if k>0 else 0^n where P(n,j) is the number of j-partitions of n; for n>=0 and 0<=k<=n.

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A114139 Changes in United States postal rates per ounce since 1863.

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A140571 Decimal expansion of the universal constant in E(h), the maximum number of essential elements of order h.

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