A028552 -id:A028552 - cp's OEIS frontend

A055630 Table T(k,m) = k^2 + m read by antidiagonals.

0, 1, 1, 4, 2, 2, 9, 5, 3, 3, 16, 10, 6, 4, 4, 25, 17, 11, 7, 5, 5, 36, 26, 18, 12, 8, 6, 6, 49, 37, 27, 19, 13, 9, 7, 7, 64, 50, 38, 28, 20, 14, 10, 8, 8, 81, 65, 51, 39, 29, 21, 15, 11, 9, 9, 100, 82, 66, 52, 40, 30, 22, 16, 12, 10, 10, 121, 101, 83, 67, 53, 41, 31, 23, 17, 13

Offset: 0

Henry Bottomley, Jun 05 2000

			Table begins:
..0...1...4...9..16..25..36..49..64..81.100.121.144...
..1...2...5..10..17..26..37..50..65..82.101.122.145...
..2...3...6..11..18..27..38..51..66..83.102.123.146...
..3...4...7..12..19..28..39..52..67..84.103.124.147...
..4...5...8..13..20..29..40..53..68..85.104.125.148...
..5...6...9..14..21..30..41..54..69..86.105.126.149...
..6...7..10..15..22..31..42..55..70..87.106.127.150...
..7...8..11..16..23..32..43..56..71..88.107.128.151...
..8...9..12..17..24..33..44..57..72..89.108.129.152...
..9..10..13..18..25..34..45..58..73..90.109.130.153...
.10..11..14..19..26..35..46..59..74..91.110.131.154...
... - _Philippe Deléham_, Mar 31 2013

First column is A001477, second column is A000027, first row is A000290, second row is A002522, third row (apart from first term) is A010000, main diagonal is A002378, other diagonals include A028387, A028552, A014209, A002061, A014206, A027688-A027694, each row of A055096 (as upper right triangle) is right hand part of some row of this table

Cf. A185910

A192032 Square array read by antidiagonals: W(m,n) (m >= 0, n >= 0) is the Wiener index of the graph G(m,n) obtained in the following way: connect by an edge the center of an m-edge star with the center of an n-edge star. The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.

Original entry on oeis.org

1, 4, 4, 9, 10, 9, 16, 18, 18, 16, 25, 28, 29, 28, 25, 36, 40, 42, 42, 40, 36, 49, 54, 57, 58, 57, 54, 49, 64, 70, 74, 76, 76, 74, 70, 64, 81, 88, 93, 96, 97, 96, 93, 88, 81, 100, 108, 114, 118, 120, 120, 118, 114, 108, 100, 121, 130, 137, 142, 145, 146, 145, 142, 137, 130, 121

Offset: 0

Emeric Deutsch, Jun 30 2011

W(n,0) = W(0,n) = A000290(n+1) = (n+1)^2.
W(n,1) = W(1,n) = A028552(n+1) = (n+1)*(n+4).
W(n,2) = W(2,n) = A028881(n+4) = n^2 + 8*n + 9.
W(n,n) = A079273(n+1) = 5*n^2 + 4*n + 1.
W(n,m) = W(m,n) (trivially).

			W(1,2)=18 because in the graph with vertex set {A,a,B,b,b'} and edge set {AB, Aa, Bb, Bb'} we have 4 pairs of vertices at distance 1 (the edges), 4 pairs at distance 2 (Ab, Ab', Ba, bb') and 2 pairs at distance 3 (ab,ab'); 4*1 + 4*2 + 2*3 = 18.
The square array starts:
   1,  4,  9, 16, 25, ...;
   4, 10, 18, 28, 30, ...;
   9, 18, 29, 42, 57, ...;
  16, 28, 42, 58, 76, ...;

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.

Cf. A000290, A028552, A028881, A079273.

W := proc (m, n) options operator, arrow: m^2+n^2+3*m*n+2*m+2*n+1 end proc: for n from 0 to 10 do seq(W(n-i, i), i = 0 .. n) end do; # yields the antidiagonals in triangular form
W := proc (m, n) options operator, arrow: m^2+n^2+3*m*n+2*m+2*n+1 end proc: for m from 0 to 9 do seq(W(m, n), n = 0 .. 9) end do; # yields the first 10 entries of each of rows 0,1,2,...,9

W(m,n) = m^2 + n^2 + 3*m*n + 2*m + 2*n + 1.

The Wiener polynomial of the graph G(n,m) is P(m,n;t) = (m+n+1)*t + (1/2)*(m^2 + n^2 + m + n)*t^2 + m*n*t^3.

A257936 Decimal expansion of 11/18.

Original entry on oeis.org

6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Bruno Berselli, May 13 2015

Decimal expansion of Sum_{i>=1} 1/A028552(i).

Also, continued fraction expansion of 5+A001622.

			.6111111111111111111111111111111111111111111111111111111111111111...

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A028552, A030880.

Cf. A010716 (decimal expansion of 5/9 = 10/18), A010722 (decimal expansion of 2/3 = 12/18).

Magma
```
[6,1^^90];
```
Mathematica
```
RealDigits[11/18, 10, 90][[1]]
```

11/18. \\ Charles R Greathouse IV, Apr 18 2016

Equals A020773 + A142464.
From Elmo R. Oliveira, Aug 05 2024: (Start)
G.f.: (6-5*x)/(1-x).
E.g.f.: exp(x) + 5.
a(n) = 1, n >= 1. (End)

A303609 a(n) = 2n^3 + 9n^2 + 9*n.

Original entry on oeis.org

0, 20, 70, 162, 308, 520, 810, 1190, 1672, 2268, 2990, 3850, 4860, 6032, 7378, 8910, 10640, 12580, 14742, 17138, 19780, 22680, 25850, 29302, 33048, 37100, 41470, 46170, 51212, 56608, 62370, 68510, 75040, 81972, 89318, 97090, 105300, 113960, 123082, 132678, 142760

Offset: 0

Vincenzo Librandi, Apr 28 2018

y-values solving the Diophantine equation 4*x^3 + 9*x^2 = y^2 for positive x (which are listed in A028552). The equation is also satisfied by y=2 and x=-2.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A014107, A028552 (associated x).

List([0..50],n->n*(2*n^2+9*n+9)); # Muniru A Asiru, Apr 29 2018

Magma
```
[2*n^3+9*n^2+9*n: n in [0..40]];
```

Table[2 n^3 + 9 n^2 + 9 n, {n, 0, 40}] (* or *) CoefficientList[Series[(20 x - 10 x^2 + 2 x^3) / (1 - x)^4, {x, 0, 33}], x]

G.f.: 2*x*(10 - 5*x + x^2)/(1 - x)^4.
a(n) = n*(2*n^2 + 9*n + 9) = n*A014107(n+3).
From Elmo R. Oliveira, Aug 07 2025: (Start)
E.g.f.: exp(x)*x*(20 + 15*x + 2*x^2).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

cp's OEIS Frontend

A055630 Table T(k,m) = k^2 + m read by antidiagonals.

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A257936 Decimal expansion of 11/18.

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A303609 a(n) = 2*n^3 + 9*n^2 + 9*n.

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Formula

A303609 a(n) = 2n^3 + 9n^2 + 9*n.