A014107 -id:A014107 - cp's OEIS frontend

A372025 Maximum second Zagreb index of maximal 3-degenerate graphs with n vertices.

12, 54, 120, 210, 324, 462, 624, 810, 1020, 1254, 1512, 1794, 2100, 2430, 2784, 3162, 3564, 3990, 4440, 4914, 5412, 5934, 6480, 7050, 7644, 8262, 8904, 9570, 10260, 10974, 11712, 12474, 13260, 14070, 14904, 15762, 16644, 17550, 18480, 19434, 20412, 21414, 22440, 23490, 24564, 25662, 26784, 27930

Offset: 3

Allan Bickle, Apr 16 2024

The second Zagreb index of a graph is the sum of the products of the degrees over all edges of the graph.

A maximal 3-degenerate graph can be constructed from a 3-clique by iteratively adding a new 3-leaf (vertex of degree 3) adjacent to three existing vertices. The extremal graphs are 3-stars, so the bound also applies to 3-trees.

			The graph K_3 has 3 degree 2 vertices, so a(3) = 3*4 = 12.

Paolo Xausa, Table of n, a(n) for n = 3..10000
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Allan Bickle, Zagreb Indices of Maximal k-degenerate Graphs, Australas. J. Combin. 89 1 (2024) 167-178.
J. Estes and B. Wei, Sharp bounds of the Zagreb indices of k-trees, J Comb Optim 27 (2014), 271-291.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002378, A152811, A371912 (Zagreb indices of maximal k-degenerate graphs).
Cf. A051624, A372025, A372026 (second Zagreb indices of maximal k-degenerate graphs).
Cf. A372027 (second Zagreb index of MOPs).

LinearRecurrence[{3, -3, 1}, {12, 54, 120}, 50] (* Paolo Xausa, Jan 22 2025 *)

a(n) = 3*(n-1)^2 + 9*(n-3)*(n-1).
From Chai Wah Wu, Apr 16 2024: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 5.
G.f.: x^3*(6*x^2 - 18*x - 12)/(x - 1)^3. (End)
a(n) = 6*A014107(n-1). Sum_{n>=3} 1/a(n) = (1/2+log(2))/9 = 0.1325719... - R. J. Mathar, Apr 22 2024

A131804 Antidiagonal sums of triangular array T: T(j,k) = -(k+1)/2 for odd k, T(j,k) = 0 for k = 0, T(j,k) = j+1-k/2 for even k > 0; 0 <= k <= j.

Original entry on oeis.org

0, 0, -1, -1, 1, 2, 1, 2, 6, 8, 7, 9, 15, 18, 17, 20, 28, 32, 31, 35, 45, 50, 49, 54, 66, 72, 71, 77, 91, 98, 97, 104, 120, 128, 127, 135, 153, 162, 161, 170, 190, 200, 199, 209, 231, 242, 241, 252, 276, 288, 287, 299, 325, 338, 337, 350, 378, 392, 391, 405, 435, 450

Offset: 0

Klaus Brockhaus, Jul 18 2007

sign

T is obtained by replacing the values of the second, fourth, sixth, ... column of the triangular array defined in A129819 by the corresponding negative values.
Interleaving of A000384, A001105, A056220 and A014107 (starting at the second term).
Main diagonal of T is in A001057, row sums are in A131805.

			First seven rows of T are
[ 0 ],
[ 0, -1 ],
[ 0, -1, 2 ],
[ 0, -1, 3, -2 ],
[ 0, -1, 4, -2, 3 ],
[ 0, -1, 5, -2, 4, -3 ],
[ 0, -1, 6, -2, 5, -3, 4 ]

Cf. A129819, A000384 (n*(2*n-1)), A001105 (2*n^2), A056220 (2*n^2-1), A014107 (n*(2*n-3)), A001057, A131805.

m:=62; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do for k:=2 to j do if k mod 2 eq 0 then M[j, k]:=-k div 2; else M[j, k]:=j-(k div 2); end if; end for; end for; [ &+[ M[j-k+1, k]: k in [1..(j+1) div 2] ]: j in [1..m] ];

{for(n=0, 61, r=n%4; k=(n-r)/4; a=if(r==0, k*(2*k-1), if(r==1, 2*k^2, if(r==2, 2*k^2-1, k*(2*k+1)-1))); print1(a, ","))}

a(0) = 0, a(1) = 0, a(2) = -1, a(3) = -1, a(4) = 1, a(5) = 2, a(6) = 1; for n > 6, a(n) = 3*a(n-1) - 5*a(n-2) + 7*a(n-3) - 7*a(n-4) + 5*a(n-5) - 3*a(n-6) + a(n-7);

G.f.: x^2*(-1+2*x-x^2+x^3)/((1-x)^3*(1+x^2)^2).

A200050 a(2) = 1, then (p-1)*(p-4)/2, with p = prime(n), n > 2.

Original entry on oeis.org

1, 2, 9, 35, 54, 104, 135, 209, 350, 405, 594, 740, 819, 989, 1274, 1595, 1710, 2079, 2345, 2484, 2925, 3239, 3740, 4464, 4850, 5049, 5459, 5670, 6104, 7749, 8255, 9044, 9315, 10730, 11025, 11934, 12879, 13529, 14534, 15575, 15930, 17765, 18144, 18914, 19305

Offset: 2

Arkadiusz Wesolowski, Apr 16 2012

nonn

Record values in A192599. The index sequence of this one is 1, 2, 3, 6, 7, 9, 11, 13, 17, 18, 21, 23, 25, 29, 31, 34, 36, 40, 42, 45, 47, 50, 52, 56, 58, 61.

			A192599(13) = 209 since A192599(17) = 350 is the next record value.

Arkadiusz Wesolowski, Table of n, a(n) for n = 2..10000

Cf. A000096, A014107, A192599.

[(p-1)*Abs(p-4)/2: p in [NthPrime(n+1): n in [1..45]]]

Table[p = Prime[n + 1]; (p - 1)*Abs[p - 4]/2, {n, 45}]
Join[{1},((#-1)(#-4))/2&/@Prime[Range[3,50]]] (* Harvey P. Dale, Aug 04 2020 *)

vector(45, n, p=prime(n+1); (p-1)*abs(p-4)/2)

a(2) = 1, a(n) = A006093(n)*A172367(n-2)/2.

A326728 A(n, k) = n(k - 1)k/2 - k, square array for n >= 0 and k >= 0 read by ascending antidiagonals.

Original entry on oeis.org

0, 0, -1, 0, -1, -2, 0, -1, -1, -3, 0, -1, 0, 0, -4, 0, -1, 1, 3, 2, -5, 0, -1, 2, 6, 8, 5, -6, 0, -1, 3, 9, 14, 15, 9, -7, 0, -1, 4, 12, 20, 25, 24, 14, -8, 0, -1, 5, 15, 26, 35, 39, 35, 20, -9, 0, -1, 6, 18, 32, 45, 54, 56, 48, 27, -10

Offset: 0

Peter Luschny, Aug 04 2019

A formal extension of the figurative numbers A139600 to negative n.

			[0] 0, -1, -2, -3, -4, -5, -6,  -7,  -8,  -9, -10, ... A001489
[1] 0, -1, -1,  0,  2,  5,  9,  14,  20,  27,  35, ... A080956
[2] 0, -1,  0,  3,  8, 15, 24,  35,  48,  63,  80, ... A067998
[3] 0, -1,  1,  6, 14, 25, 39,  56,  76,  99, 125, ... A095794
[4] 0, -1,  2,  9, 20, 35, 54,  77, 104, 135, 170, ... A014107
[5] 0, -1,  3, 12, 26, 45, 69,  98, 132, 171, 215, ... A326725
[6] 0, -1,  4, 15, 32, 55, 84, 119, 160, 207, 260, ... A270710
[7] 0, -1,  5, 18, 38, 65, 99, 140, 188, 243, 305, ...

Peter Luschny, Figurate number — a very short introduction. With plots from Stefan Friedrich Birkner.

Cf. A001489 (n=0), A080956 (n=1), A067998 (n=2), A095794 (n=3), A014107 (n=4), A326725 (n=5), A270710 (n=6).
Columns include A008585, A016933, A017329.
Cf. A139600.

A := (n, k) -> n*(k - 1)*k/2 - k:
seq(seq(A(n - k, k), k=0..n), n=0..11);

def A326728Row(n):
    x, y = 1, 1
    yield 0
    while True:
        yield -x
        x, y = x + y - n, y - n
for n in range(8):
    R = A326728Row(n)
print([next(R) for _ in range(11)])

A386479 a(0) = 0; thereafter a(n) = 2n^2 - 3n + 5.

Original entry on oeis.org

0, 4, 7, 14, 25, 40, 59, 82, 109, 140, 175, 214, 257, 304, 355, 410, 469, 532, 599, 670, 745, 824, 907, 994, 1085, 1180, 1279, 1382, 1489, 1600, 1715, 1834, 1957, 2084, 2215, 2350, 2489, 2632, 2779, 2930, 3085, 3244, 3407, 3574, 3745, 3920, 4099, 4282, 4469, 4660, 4855, 5054, 5257, 5464, 5675, 5890, 6109, 6332, 6559, 6790

Offset: 0

N. J. A. Sloane, Jul 25 2025

For n>0, a(n) is the maximum number of regions the plane can be divided into by drawing two n-chains (both finite and infinite regions are counted). See A386478 for further information.

We do not at present have an explicit construction that will achieve a(n) for n > 5.

N. J. A. Sloane, Two 1-chains (i.e., lines) can divide the plane into at most a(1) = 4 regions; two 2-chains (i.e., V's) can divide the plane into at most a(2) = 7 regions.
N. J. A. Sloane, Illustration for a(3) = 14. Two 3-chains can divide the plane into at most 14 regions.
N. J. A. Sloane, Two 4-chains can divide the plane into at most a(4) = 25 regions. (The two 4-chains are colored respectively black and green.)
N. J. A. Sloane, Two 5-chains can divide the plane into at most a(5) = 40 regions. (The two 5-chains are colored respectively black and red.) It would be nice to have a clearer picture. Region 36 is tiny. Also some of the points where arms of the 5-chains meet are just outside the region shown.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

A column of the array in A386478.

Essentially the same (up to offset, initial terms, and the addition of a small constant) as several other sequences, including A014105, A014107, A084849, A096376, A236257, ....

LinearRecurrence[{3,-3,1},{5,4,7},60] (* or *) a[n_]:=2n^2-3n+5;Array[a,60,0] (* James C. McMahon, Jul 26 2025 *)

From Stefano Spezia, Jul 26 2025: (Start)
G.f.: -x*(4-5*x+5*x^2) / (x-1)^3.
E.g.f.: exp(x)*(5 - x + 2*x^2) - 5. (End)

Changed a(0) so as to match changes to A386478. - N. J. A. Sloane, Jul 26 2025

A002698 Coefficients of Chebyshev polynomials: n(2n-3)2^(2n-5).

Original entry on oeis.org

1, 18, 160, 1120, 6912, 39424, 212992, 1105920, 5570560, 27394048, 132120576, 627048448, 2936012800, 13589544960, 62277025792, 282930970624, 1275605286912, 5712306503680, 25426206392320, 112562502893568, 495879744126976, 2174833999740928, 9499780463984640

Offset: 2

N. J. A. Sloane

nonn

Cornelius Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 516.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Cornelius Lanczos, Applied Analysis. (Annotated scans of selected pages)
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (12,-48,64).
Index entries for sequences related to Chebyshev polynomials.

Cf. A000079, A014107.

A002698:=(-1-6*z+8*z**2)/(4*z-1)**3; # [Simon Plouffe in his 1992 dissertation]

Table[n*(2n-3)*2^(2n-5), {n, 2, 30}] (* Amiram Eldar, Feb 17 2023 *)

From Amiram Eldar, Feb 17 2023: (Start)
a(n) = A014107(n)*A000079(2*n-5).
Sum_{n>=2} 1/a(n) = 12*log(3) - 64*log(2)/3 + 8/3.
Sum_{n>=2} (-1)^n/a(n) = (8/3)*(arctan(1/2) + 4*log(5/4) - 1). (End)

A091435 Array T(n,k) = n*(n+k), read by antidiagonals.

Original entry on oeis.org

0, 1, 0, 4, 2, 0, 9, 6, 3, 0, 16, 12, 8, 4, 0, 25, 20, 15, 10, 5, 0, 36, 30, 24, 18, 12, 6, 0, 49, 42, 35, 28, 21, 14, 7, 0, 64, 56, 48, 40, 32, 24, 16, 8, 0, 81, 72, 63, 54, 45, 36, 27, 18, 9, 0, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, 0, 121, 110, 99, 88, 77, 66, 55, 44, 33, 22, 11, 0

Offset: 0

Ross La Haye, Mar 02 2004

			Table begins
   0;
   1,  0;
   4,  2,  0;
   9,  6,  3,  0;
  16, 12,  8,  4,  0;
  25, 20, 15, 10,  5,  0;
  36, 30, 24, 18, 12,  6,  0;
  ...
a(5,3) = 40 because 5 * (5 + 3) = 5 * 8 = 40.

Muniru A Asiru, Table of n, a(n) for n = 0..5150 (rows n = 0..100,flattened)
P. De Geest, Palindromic Quasipronics of the form n(n+x)

Columns: a(n, 0) = A000290(n), a(n, 1) = A002378(n), a(n, 2) = A005563(n), a(n, 3) = A028552(n), a(n, 4) = A028347(n+2), a(n, 5) = A028557(n), a(n, 6) = A028560(n), a(n, 7) = A028563(n), a(n, 8) = A028566(n). Diagonals: a(n, n-4) = A054000(n-1), a(n, n-3) = A014107(n), a(n, n-2) = A046092(n-1), a(n, n-1) = A000384(n), a(n, n) = A001105(n), a(n, n+1) = A014105(n), a(n, n+2) = A046092(n), a(n, n+3) = A014106(n), a(n, n+4) = A054000(n+1), a(n, n+5) = A033537(n). Also note that the sums of the antidiagonals = A002411.

Cf. A056536, A056537, A082156.

Flat(List([0..11],j->List([0..j],i->j*(j-i)))); # Muniru A Asiru, Sep 11 2018

Maple
```
seq(seq((j-i)*j,i=0..j),j=0..14);
```

Table[# (# + k) &[m - k], {m, 0, 11}, {k, 0, m}] // Flatten (* Michael De Vlieger, Oct 15 2018 *)

G.f.: x*(1+x-2*x^2*y)/((1-x*y)^2*(1-x)^3). - Vladeta Jovovic, Mar 05 2004

More terms from Emeric Deutsch, Mar 15 2004

A109094 Length of the longest path (repeated edges not allowed) between two arbitrary, distinct nodes in K_n, the complete graph on n vertices.

Original entry on oeis.org

0, 1, 2, 5, 9, 13, 20, 25, 35, 41, 54, 61, 77, 85, 104, 113, 135, 145, 170, 181, 209, 221, 252, 265, 299, 313, 350, 365, 405, 421, 464, 481, 527, 545, 594, 613, 665, 685, 740, 761, 819, 841, 902, 925, 989, 1013, 1080, 1105, 1175, 1201, 1274, 1301, 1377, 1405

Offset: 1

Ryan Propper, Jun 18 2005

			a(4) = 5 because the length of the longest path between any two distinct vertices in K_4 is 5.

R. J. Mathar, Comments on this sequence
Eric Weisstein's World of Mathematics, Complete Graph.
Eric Weisstein's World of Mathematics, Euler Path.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A057979, A128223.

a(1)=0; a(2n+1) = n*(n-1)/2-1 = A014107(n+1), n>0; a(2n)=n*(n-2)/2+1= A001844(n-1). - Martin Fuller, R. J. Mathar and Mitch Harris, Dec 06 2007

O.g.f.: x^2*(x^4-2*x^3-x^2-x-1)/((-1+x)^3 *(x+1)^2) . - R. J. Mathar, Jan 17 2008

A177732 The sums of two or more consecutive positive numbers, the largest being even.

Original entry on oeis.org

3, 7, 9, 10, 11, 15, 18, 19, 20, 21, 23, 26, 27, 30, 31, 33, 34, 35, 36, 39, 40, 42, 43, 45, 47, 49, 50, 51, 52, 54, 55, 57, 58, 59, 60, 63, 66, 67, 68, 69, 70, 71, 72, 74, 75, 77, 78, 79, 80, 81, 82, 83, 84, 87, 90, 91, 93, 95, 98, 99, 100, 102, 103, 104, 105, 106, 107, 108

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 12 2010

nonn

Numbers of the form (j+2l)*(2l-j+1)/2 with j>=1 and 2l>j. Subsequences are A014105 where >=3, (j=1), A014107 where >=9 (j=2). - R. J. Mathar, Jul 14 2012

			3=1+2, 7=3+4, 9=2+3+4, 10=1+2+3+4, 11=5+6,..

Cf. A177712, A177713, A177731.

z=200;lst2={};Do[c=a;Do[c+=b;If[c<=2*z,AppendTo[lst2,c]],{b,a-1,1,-1}],{a,2,z,2}];Union@lst2
With[{upto=108},Select[Union[Flatten[Table[Accumulate[Range[2n-1,1,-1]]+ 2n,{n,upto/4}]]],#<=upto&]] (* Harvey P. Dale, May 19 2019 *)

A187287 Number of 2-step one or two space at a time rook's tours on an n X n board summed over all starting positions.

Original entry on oeis.org

0, 8, 36, 80, 140, 216, 308, 416, 540, 680, 836, 1008, 1196, 1400, 1620, 1856, 2108, 2376, 2660, 2960, 3276, 3608, 3956, 4320, 4700, 5096, 5508, 5936, 6380, 6840, 7316, 7808, 8316, 8840, 9380, 9936, 10508, 11096, 11700, 12320, 12956, 13608, 14276, 14960, 15660

Offset: 1

R. H. Hardin, Mar 08 2011

nonn

			Some solutions for 4 X 4:
..0..0..0..0....0..0..0..0....0..0..0..0....0..0..0..0....0..2..0..1
..1..0..0..0....0..1..0..0....0..1..2..0....2..0..0..0....0..0..0..0
..0..0..0..0....0..0..0..0....0..0..0..0....0..0..0..0....0..0..0..0
..2..0..0..0....0..2..0..0....0..0..0..0....1..0..0..0....0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..50

Row 2 of A187286.

Cf. A014107.

A187287:=n->`if`(n=1, 0, 8*n^2 - 12*n); seq(A187287(n), n=1..50); # Wesley Ivan Hurt, Feb 28 2014

Empirical: a(n) = 8*n^2 - 12*n for n>1.

Empirical g.f.: 4*x^2*(2+3*x-x^2)/(1-x)^3. - Colin Barker, Jan 22 2012

cp's OEIS Frontend

A372025 Maximum second Zagreb index of maximal 3-degenerate graphs with n vertices.

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A131804 Antidiagonal sums of triangular array T: T(j,k) = -(k+1)/2 for odd k, T(j,k) = 0 for k = 0, T(j,k) = j+1-k/2 for even k > 0; 0 <= k <= j.

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A200050 a(2) = 1, then (p-1)*(p-4)/2, with p = prime(n), n > 2.

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A326728 A(n, k) = n*(k - 1)*k/2 - k, square array for n >= 0 and k >= 0 read by ascending antidiagonals.

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A386479 a(0) = 0; thereafter a(n) = 2*n^2 - 3*n + 5.

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A002698 Coefficients of Chebyshev polynomials: n*(2*n-3)*2^(2*n-5).

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A091435 Array T(n,k) = n*(n+k), read by antidiagonals.

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A326728 A(n, k) = n(k - 1)k/2 - k, square array for n >= 0 and k >= 0 read by ascending antidiagonals.

A386479 a(0) = 0; thereafter a(n) = 2n^2 - 3n + 5.

A002698 Coefficients of Chebyshev polynomials: n(2n-3)2^(2n-5).