A104249 -id:A104249 - cp's OEIS frontend

A006000 a(n) = (n+1)(n^2+n+2)/2; g.f.: (1 + 2x^2) / (1 - x)^4.

1, 4, 12, 28, 55, 96, 154, 232, 333, 460, 616, 804, 1027, 1288, 1590, 1936, 2329, 2772, 3268, 3820, 4431, 5104, 5842, 6648, 7525, 8476, 9504, 10612, 11803, 13080, 14446, 15904, 17457, 19108, 20860, 22716, 24679, 26752, 28938, 31240, 33661, 36204, 38872, 41668, 44595, 47656, 50854, 54192

Offset: 0

N. J. A. Sloane, Simon Plouffe

Enumerates certain paraffins.
a(n) is the (n+1)st (n+3)-gonal number. - Floor van Lamoen, Oct 20 2001
Sum of n terms of an arithmetic progression with the first term 1 and the common difference n: a(1)=1, a(2) = 1+3, a(3) = 1+4+7, a(4) = 1+5+9+13, etc. - Amarnath Murthy, Mar 25 2004
This is identical to: first triangular number A000217, 2nd square number A000290, 3rd pentagonal number A000326, 4th hexagonal number A000384, 5th heptagonal number A000566, 6th octagonal number A000567, ..., (n+1)-th (n+3)-gonal number = main diagonal of rectangular array T(n,k) of polygonal numbers, by diagonals, referred to in A086271. - Jonathan Vos Post, Dec 19 2007
Also (n + 1)! times the determinant of the n X n matrix given by m(i,j) = (i+1)/i if i=j and otherwise 1. For example, (6 + 1)!*Det[{{2,1,1,1,1,1}, {1,3/2,1,1,1,1},{1,1,4/3,1,1,1}, {1,1,1,5/4,1,1}, {1,1,1,1,6/5,1}, {1,1,1,1,1,7/6}}] = 154 = a(6). - John M. Campbell, May 20 2011
a(n-1) = N_2(n), n>=1, is the number of 2-faces of n planes in generic position in three-dimensional space. See comment under A000125 for general arrangement. Comment to Arnold's problem 1990-11, see the Arnold reference, p. 506. - Wolfdieter Lang, May 27 2011
For n>2, a(n) is 2 * (average cycle weight of primitive Hamiltonian cycles on a simply weighted K_n) (see link). - Jon Perry, Nov 23 2014
a(n) is the partial sums of A104249. - J. M. Bergot, Dec 28 2014
Sum of the numbers in the 1st column of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows. - Wesley Ivan Hurt, May 17 2021
From Enrique Navarrete, Mar 27 2023: (Start)
a(n) is the number of ordered set partitions of an (n+1)-set into 2 sets such that the first set has 0, 1, or 2 elements, the second set has no restrictions, and we choose an element from the second set. For n=4, the a(4) = 55 set partitions of [5] are the following (where the element selected from the second set is in parentheses):
   { }, {(1), 2, 3, 4, 5}  (5 of these);
   {1}, {(2), 3, 4, 5}   (20 of these);
   {1, 2}, {(3), 4, 5}   (30 of these). (End)

V. I. Arnold (ed.), Arnold's Problems, Springer, 2004, comments on Problem 1990-11 (p. 75), pp. 503-510. Numbers N_2.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

William A. Tedeschi, Table of n, a(n) for n = 0..10000
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
P. A. MacMahon, Properties of prime numbers deduced from the calculus of symmetric functions, Proc. London Math. Soc., 23 (1923), 290-316. = Coll. Papers, II, pp. 354-382. [See p. 301].
Jon Perry, Weighted Hamiltonian Cycles
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.
Eric Weisstein's World of Mathematics, Polygonal Number
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000124.

A006000:=(1+2*z**2)/(z-1)**4; # Simon Plouffe in his 1992 dissertation

a[n_]:=(n^3-n^2)/2+n; Table[a[n],{n,5!}] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2010 *)
CoefficientList[Series[(1 + 2 x^2) / (1 - x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Nov 21 2014 *)

def a(n):
    return n + (n**3 - n**2)//2 # William A. Tedeschi, Aug 22 2010

a(n) = Sum_{j=1..n+1} (binomial(0,0*j) + binomial(n+1,2)). - Zerinvary Lajos, Jul 25 2006
a(n-1) = n + (n^3 - n^2)/2 = n + n*T(n-1) where T(n-1) is a triangular number, n >= 1. - William A. Tedeschi, Aug 22 2010
a(n) = A002817(n)*4/n for n > 0. - Jon Perry, Nov 21 2014
E.g.f.: (1 + x)*(2 + 4*x + x^2)*exp(x)/2. - Robert Israel, Nov 24 2014
a(n) = A057145(n+3,n+1). - R. J. Mathar, Jul 28 2016
a(n) = A000124(n) * (n+1). - Alois P. Heinz, Aug 31 2023

A167560 The ED2 array read by ascending antidiagonals.

Original entry on oeis.org

1, 2, 1, 6, 4, 1, 24, 16, 6, 1, 120, 80, 32, 8, 1, 720, 480, 192, 54, 10, 1, 5040, 3360, 1344, 384, 82, 12, 1, 40320, 26880, 10752, 3072, 680, 116, 14, 1, 362880, 241920, 96768, 27648, 6144, 1104, 156, 16, 1

Offset: 1

Johannes W. Meijer, Nov 10 2009

The coefficients in the upper right triangle of the ED2 array (m>n) were found with the a(n,m) formula while the coefficients in the lower left triangle of the ED2 array (m<=n) were found with the recurrence relation, see below. We use for the array rows the letter n (>=1) and for the array columns the letter m (>=1).
The ED2 array is related to the EG1 matrix, see A162005, because sum(EG1(2*m-1,n) * z^(2*m-1), m=1..infinity) = ((2*n-1)!/(4^(n-1)*(n-1)!^2))*int(sinh(y*(2*z))/cosh(y)^(2*n), y=0..infinity).
For the ED1, ED3 and ED4 arrays see A167546, A167572 and A167584.

			The ED2 array begins with:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1
2, 4, 6, 8, 10, 12, 14, 16, 18, 20
6, 16, 32, 54, 82, 116, 156, 202, 254, 312
24, 80, 192, 384, 680, 1104, 1680, 2432, 3384, 4560
120, 480, 1344, 3072, 6144, 11160, 18840, 30024, 45672, 66864
720, 3360, 10752, 27648, 61440, 122880, 226800, 392832, 646128, 1018080

Johannes W. Meijer, The four Escher-Droste arrays, Mar 08 2013.

A000012, A005843 (n>=1), 2*A104249 (n>=1), A167561, A167562 and A167563 equal the first sixth rows of the array.
A000142 equals the first column of the array.
A047053 equals the a(n, n) diagonal of the array.
2*A034177 equals the a(n+1, n) diagonal of the array.
A167570 equals the a(n+2, n) diagonal of the array,
A167564 equals the row sums of the ED2 array read by antidiagonals.
A167565 is a triangle related to the a(n) formulas of the rows of the ED2 array.
A167568 is a triangle related to the GF(z) formulas of the rows of the ED2 array.
A167569 is the lower left triangle of the ED2 array.
Cf. A162005 (EG1 triangle).
Cf. A167546 (ED1 array), A167572 (ED3 array), A167584 (ED4 array).

nmax:=10; mmax:=10; for n from 1 to nmax do for m from 1 to n do a(n,m) := 4^(m-1)*(m-1)!*(n+m-1)!/(2*m-1)! od; for m from n+1 to mmax do a(n,m):= n! + sum((-1)^(k-1)*binomial(n-1,k)*a(n,m-k),k=1..n-1) od; od: for n from 1 to nmax do for m from 1 to n do d(n,m):=a(n-m+1,m) od: od: T:=1: for n from 1 to nmax do for m from 1 to n do a(T):= d(n,m): T:=T+1: od: od: seq(a(n),n=1..T-1);
# alternative
A167560 := proc(n,m)
    option remember ;
    if m > n then
        n!+add( (-1)^(k-1)*binomial(n-1,k)*procname(n,m-k),k=1..n-1) ;
    else
        4^(m-1)*(m-1)!*(n+m-1)!/(2*m-1)! ;
    end if;
end proc:
seq( seq(A167560(d-m,m),m=1..d-1),d=2..12) ; # R. J. Mathar, Jun 28 2024

nmax = 10; mmax = 10; For[n = 1, n <= nmax, n++, For[m = 1, m <= n, m++, a[n, m] = 4^(m - 1)*(m - 1)!*((n + m - 1)!/(2*m - 1)!)]; For[m = n + 1, m <= mmax, m++, a[n, m] = n! + Sum[(-1)^(k - 1)*Binomial[n - 1, k]*a[n, m - k], {k, 1, n - 1}]]; ]; For[n = 1, n <= nmax, n++, For[m = 1, m <= n, m++, d[n, m] = a[n - m + 1, m]]; ]; t = 1; For[n = 1, n <= nmax, n++, For[m = 1, m <= n, m++, a[t] = d[n, m]; t = t + 1]]; Table[a[n], {n, 1, t - 1}] (* Jean-François Alcover, Dec 20 2011, translated from Maple *)

a(n,m) = ((m-1)!/((m-n-1)!))*int(sinh(y*(2*n))/(cosh(y))^(2*m),y=0..infinity) for m>n.
The (n-1)-differences of the n-th array row lead to the recurrence relation
sum((-1)^k*binomial(n-1,k)*a(n-1,m-k),k=0..n-1) = n!
which in its turn leads to, see A167569,
a(n,m) = 4^(m-1)*(m-1)!*(n+m-1)!/(2*m-1)! if m<=n.

A143689 a(n) = (3*n^2 - n + 2)/2.

Original entry on oeis.org

1, 2, 6, 13, 23, 36, 52, 71, 93, 118, 146, 177, 211, 248, 288, 331, 377, 426, 478, 533, 591, 652, 716, 783, 853, 926, 1002, 1081, 1163, 1248, 1336, 1427, 1521, 1618, 1718, 1821, 1927, 2036, 2148, 2263, 2381, 2502, 2626, 2753, 2883, 3016, 3152, 3291

Offset: 0

Gary W. Adamson, Aug 29 2008

Equals left border of triangle A033292.
Equals binomial transform of [1, 1, 3, 0, 0, 0, ...].
A242357(a(n)) = 1. - Reinhard Zumkeller, May 11 2014
These might be called "trisected pentagonal numbers": A figurate pentagonal number is composed of three triangles, of which the central one is the largest, and the removal of the triangular frame (3*n) of the central triangle trisects the figure. This is reflected in the formula a(n) = A000326(n+1) - 3*n. See illustration in links. - John Elias, May 27 2022

Michael De Vlieger, Table of n, a(n) for n = 0..10000
D. Bevan, D. Levin, P. Nugent, J. Pantone, and L. Pudwell, Pattern avoidance in forests of binary shrubs, arXiv preprint arXiv:1510:08036 [math.CO], 2015-2016.
John Elias, Trisected Pentagonal Numbers
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

a(n) = A000326(n+1) - 3n. Third column of A107111.

Cf. A033292, A104249.

a143689 n = n*(3*n-1) `div` 2 + 1 -- Reinhard Zumkeller, May 11 2014

Table[(3n^2-n+2)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,2,6},50] (* Harvey P. Dale, May 05 2014 *)

makelist(binomial(n, 2) + n^2 + 1, n, 0, 100); /* Franck Maminirina Ramaharo, Mar 01 2018 */

a(n)=(3*n^2-n+2)/2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = A000326(n+1) - 3*n. (A000326 are the pentagonal numbers.)
a(n) = (3*n^2 - n + 2)/2 = A027599(n+1)/2. - R. J. Mathar, Sep 03 2008
a(n) = a(n-1) + 3*n - 2 (with a(0)=1). - Vincenzo Librandi, Nov 25 2010
a(n) = 2*a(n-1) - a(n-2) + 3.
O.g.f.: (1-x+3*x^2)/((1-x)^3). - Eric Werley, Jun 27 2011
a(n) = A104249(-n). - Bruno Berselli, Jul 08 2015
a(n) = binomial(n,2) + n^2 + 1 = A152947(n+1) + A000290(n). - Franck Maminirina Ramaharo, Mar 01 2018
E.g.f.: exp(x)*(2 + 2*x + 3*x^2)/2. - Stefano Spezia, Apr 19 2025

Index of A000326 in definition, formula and example corrected by R. J. Mathar, Sep 03 2008

A327622 Number A(n,k) of parts in all k-times partitions of n into distinct parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 1, 0, 1, 1, 5, 3, 1, 0, 1, 1, 7, 8, 5, 1, 0, 1, 1, 9, 16, 15, 8, 1, 0, 1, 1, 11, 27, 35, 28, 10, 1, 0, 1, 1, 13, 41, 69, 73, 49, 13, 1, 0, 1, 1, 15, 58, 121, 160, 170, 86, 18, 1, 0, 1, 1, 17, 78, 195, 311, 460, 357, 156, 25, 1

Offset: 0

Alois P. Heinz, Sep 19 2019

Row n is binomial transform of the n-th row of triangle A327632.

			Square array A(n,k) begins:
  0,  0,  0,   0,    0,    0,    0,     0,     0, ...
  1,  1,  1,   1,    1,    1,    1,     1,     1, ...
  1,  1,  1,   1,    1,    1,    1,     1,     1, ...
  1,  3,  5,   7,    9,   11,   13,    15,    17, ...
  1,  3,  8,  16,   27,   41,   58,    78,   101, ...
  1,  5, 15,  35,   69,  121,  195,   295,   425, ...
  1,  8, 28,  73,  160,  311,  553,   918,  1443, ...
  1, 10, 49, 170,  460, 1047, 2106,  3865,  6611, ...
  1, 13, 86, 357, 1119, 2893, 6507, 13182, 24625, ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened
Wikipedia, Partition (number theory)

Columns k=0-3 give: A057427, A015723, A327605, A327628.
Rows n=0,(1+2),3-5 give: A000004, A000012, A005408, A104249, A005894.
Main diagonal gives: A327623.
Cf. A327618, A327632.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
     `if`(k=0, [1, 1], `if`(i*(i+1)/2 (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
        b(n-i, min(n-i, i-1), k)))(b(i$2, k-1)))))
    end:
A:= (n, k)-> b(n$2, k)[2]:
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, i_, k_] := b[n, i, k] = With[{}, If[n==0, Return@{1, 0}]; If[k == 0, Return@{1, 1}]; If[i(i + 1)/2 < n, Return@{0, 0}]; b[n, i - 1, k] + Function[h, Function[f, f + {0, f[[1]] h[[2]]/h[[1]]}][h[[1]] b[n - i, Min[n - i, i - 1], k]]][b[i, i, k - 1]]];
A[n_, k_] := b[n, n, k][[2]];
Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 03 2020, after Maple *)

A(n,k) = Sum_{i=0..k} binomial(k,i) * A327632(n,i).

A167565 A triangle related to the a(n) formulas for the rows of the ED2 array A167560.

Original entry on oeis.org

1, 2, 0, 3, 1, 2, 4, 4, 16, 0, 5, 10, 67, 14, 24, 6, 20, 202, 124, 368, 0, 7, 35, 497, 601, 2736, 444, 720, 8, 56, 1064, 2120, 13712, 6464, 16896, 0, 9, 84, 2058, 6096, 53121, 48876, 186732, 25584, 40320, 10, 120, 3684, 15168, 171258, 257640, 1350296

Offset: 1

Johannes W. Meijer, Nov 10 2009

The a(n) formulas given below correspond to the first ten rows of the ED2 array A167560.

The recurrence relations for the a(n) formulas for the left hand triangle columns, see the cross-references below, lead to the sequences A003148 and A007318.

			Row 1: a(n) = 1.
Row 2: a(n) = 2*n + 0.
Row 3: a(n) = 3*n^2 + 1*n + 2.
Row 4: a(n) = 4*n^3 + 4*n^2 + 16*n + 0.
Row 5: a(n) = 5*n^4 + 10*n^3 + 67*n^2 + 14*n + 24.
Row 6: a(n) = 6*n^5 + 20*n^4 + 202*n^3 + 124*n^2 + 368*n + 0.
Row 7: a(n) = 7*n^6 + 35*n^5 + 497*n^4 + 601*n^3 + 2736*n^2 + 444*n + 720.
Row 8: a(n) = 8*n^7 + 56*n^6 + 1064*n^5 + 2120*n^4 + 13712*n^3 + 6464*n^2 + 16896*n + 0.
Row 9: a(n) = 9*n^8 + 84*n^7 + 2058*n^6 + 6096*n^5 + 53121*n^4 + 48876*n^3 + 186732*n^2 + 25584*n + 40320.
Row 10: a(n) = 10*n^9 + 120*n^8 + 3684*n^7 + 15168*n^6 + 171258*n^5 + 257640*n^4 + 1350296*n^3 + 533472*n^2 + 1297152*n + 0.

A167560 is the ED2 array.
A000012, A005843 (n=>1), 2*A104249 (n=>1), A167561, A167562 and A167563 equal the first sixth rows of the array.
A005359 equals the first right hand triangle column.
A000027, A000292, A167566, A167567 and A168304 equal the first five left hand triangle columns.
A000142 equals the row sums.
Cf. A003148 and A007318.

Comment and links added by Johannes W. Meijer, Nov 23 2009

A242357 Crescendo trapezoidal.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, May 11 2014

nonn

a(A000326(n)) = a(A143689(n-1)) = 1;

for all n: a(k) = n, A104249(n-1) <= k <= A005448(n).

			. Initial values
. seen trapezoidal:                                  666666
.                                    55555          5......5
.                       4444        4.....4        4........4
.             333      3....3      3.......3      3..........3
.      22    2...2    2......2    2.........2    2............2
.  1  1..1  1.....1  1........1  1...........1  1..............1 .

Reinhard Zumkeller, Table of n, a(n) for n = 1..1001

Cf. A004737.

import Data.List (inits)
a242357 n = a242357_list !! (n-1)
a242357_list = concatMap f $ tail $ inits [1..] where
   f us = (init us) ++ (take v [v, v ..]) ++ vs
          where (v:vs) = reverse us

Table[Join[Range[n-1],PadRight[{},n,n],Range[n-1,1,-1]],{n,9}]//Flatten (* Harvey P. Dale, Oct 23 2020 *)

A164845 a(n) = (6 + 10n + 5n^2 + n^3)/2.

Original entry on oeis.org

3, 11, 27, 54, 95, 153, 231, 332, 459, 615, 803, 1026, 1287, 1589, 1935, 2328, 2771, 3267, 3819, 4430, 5103, 5841, 6647, 7524, 8475, 9503, 10611, 11802, 13079, 14445, 15903, 17456, 19107, 20859, 22715, 24678, 26751, 28937, 31239, 33660, 36203, 38871

Offset: 0

Paul Curtz, Aug 28 2009

Row sums of the triangle defined by non-interrupted runs in A080036.
If the sequence of integers is split at positions defined by A000124 we obtain A080036. Its runs of consecutive integers can be placed into rows of a triangle:
    3;
    5,  6;
    8,  9, 10;
   12, 13, 14, 15;
   17, 18, 19, 20, 21;
   ...
The a(n) are the row sums of this triangle.
The a(n) are also the binomial transform of the quasi-finite sequence 3, 8, 8, 3, 0 (0 continued).
An associated integer sequence could be defined by a(n)/A026741(n+1) = 3, 11, 9, 27, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A135278.

[3+5*n+5*n^2/2+n^3/2: n in [0..50]]; // Vincenzo Librandi, Aug 07 2011

Table[(6 + 10*n + 5*n^2 + n^3)/2, {n,0,50}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {3, 11, 27, 54}, 50] (* G. C. Greubel, Apr 21 2018 *)

for(n=0, 50, print1((6+10*n+5*n^2+n^3)/2, ", ")) \\ G. C. Greubel, Apr 21 2018

a(n) = A162607(n+3) + n.
First differences: a(n+1) - a(n) = A104249(n+2), i.e., a(n) = a(n-1) + 3*n^2/2 + 7*n/2 +3.
Second differences: a(n+2) - 2*a(n+1) + a(n) = A016789(n+2).
a(n) = 2*a(n-1) - a(n-2) + 3*n + 5, n>1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 3, n>2.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n>3.
G.f.: (3-x+x^2)/(x-1)^4.
E.g.f.: (6 + 16*x + 8*x^2 + x^3)*exp(x)/2. - G. C. Greubel, Apr 21 2018

Edited and extended by R. J. Mathar, Aug 31 2009

Corrected typo in recurrence, observed by Paul Curtz - R. J. Mathar, Sep 25 2009

A238410 a(n) = floor((3(n-1)^2 + 1)/2).

Original entry on oeis.org

0, 2, 6, 14, 24, 38, 54, 74, 96, 122, 150, 182, 216, 254, 294, 338, 384, 434, 486, 542, 600, 662, 726, 794, 864, 938, 1014, 1094, 1176, 1262, 1350, 1442, 1536, 1634, 1734, 1838, 1944, 2054, 2166, 2282, 2400, 2522, 2646, 2774, 2904, 3038, 3174, 3314, 3456, 3602, 3750, 3902, 4056, 4214, 4374, 4538, 4704

Offset: 1

Emeric Deutsch, Feb 27 2014

a(n) = the eccentric connectivity index of the path P[n] on n vertices. The eccentric connectivity index of a simple connected graph G is defined to be the sum over all vertices i of G of the product E(i)D(i), where E(i) is the eccentricity and D(i) is the degree of vertex i. For example, a(4)=14 because the vertices of P[4] have degrees 1,2,2,1 and eccentricities 3,2,2,3; we have 1*3 + 2*2 + 2*2 + 1*3 = 14.
From Paul Curtz, Feb 23 2023: (Start)
East spoke of the hexagonal spiral using A004526 with a single 0:
.
            43  42  42  41  41  40
          43  28  28  27  27  26  40
        44  29  17  16  16  15  26  39
      44  29  17   8   8   7  15  25  39
    45  30  18   9   3   2   7  14  25  38
  45  30  18   9   3   0---2---6--14--24--38-->
    31  19  10   4   1   1   6  13  24  37
      31  19  10   4   5   5  13  23  37
        32  20  11  11  12  12  23  36
          32  20  21  21  22  22  36
            33  33  34  34  35  35
.
Other spokes are A032528, A005449, A227017, A045943, A005448, A212959, A000326, A282513, A143689, and A104249. (End)

Matthew House, Table of n, a(n) for n = 1..10000
M. J. Morgan, S. Mukwembi and H. C. Swart, On the eccentric connectivity index of a graph, Discrete Math., 311, 2011, 1229-1234.
B. Zhou and Zh. Du, On eccentric connectivity index, Comm. Math. Comp. Chem. (MATCH), 63, 2010, 181-198.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A004526, A077043.

(Cf. A056105, A056107.)

a := proc (n) options operator, arrow: floor((3/2)*(n-1)^2+1/2) end proc: seq(a(n), n = 1 .. 70);

Table[Floor[(3(n-1)^2+1)/2],{n,80}]  (* or *) LinearRecurrence[{2,0,-2,1},{0,2,6,14},80] (* Harvey P. Dale, Apr 30 2022 *)

a(n)=(3*(n-1)^2 + 1)\2 \\ Charles R Greathouse IV, Feb 15 2017

a(n) = (3*n)^2/6 for n even and a(n) = ((3*n)^2 + 3)/6 for n odd. - Miquel Cerda, Jun 17 2016
From Ilya Gutkovskiy, Jun 17 2016: (Start)
G.f.: 2*x^2*(1 + x + x^2)/((1 - x)^3*(1 + x)).
a(n) = (6*n^2 - 12*n + 7 + (-1)^n)/4.
a(n) = 2* A077043(n-1). (End)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Matthew House, Feb 15 2017
Sum_{n>=2} 1/a(n) = Pi^2/36 + tanh(Pi/(2*sqrt(3)))*Pi/(2*sqrt(3)). - Amiram Eldar, Mar 12 2023

A116445 Array read by antidiagonals: the binomial transform of the sequence (1,2,..n,0,0,0..) in row n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 3, 5, 1, 1, 3, 8, 7, 1, 1, 3, 8, 16, 9, 1, 1, 3, 8, 20, 27, 11, 1, 1, 3, 8, 20, 43, 41, 13, 1, 1, 3, 8, 20, 48, 81, 58, 15, 1, 1, 3, 8, 20, 48, 106, 138, 78, 17, 1, 1, 3, 8, 20, 48, 112, 213, 218, 101, 19, 1

Offset: 1

Gary W. Adamson, Feb 15 2006

Create an array by rows: (binomial transforms of 1,0,0,0,...; 1,2,0,0,0,...; 1,2,3,0,0,0,...; etc.). Antidiagonals of the array become rows of the triangle.

			First few rows of the array:
  1, 1, 1,  1,  1,   1,   1,   1,   1, 1, ...
  1, 3, 5,  7,  9,  11,  13,  15,  17, ...
  1, 3, 8, 16, 27,  41,  58,  78, 101, ...  A104249
  1, 3, 8, 20, 43,  81, 138, 218, ...       A139488
  1, 3, 8, 20, 48, 106, 213, ...
  1, 3, 8, 20, 48, 112, 249, ...
  ...
Diagonals converge to A001792, binomial transform of (1,2,3,...); and the first few rows of the triangle created by reading upwards antidiagonals are:
  1
  1, 1;
  1, 3, 1;
  1, 3, 5,  1;
  1, 3, 8,  7,  1;
  1, 3, 8, 16,  9,  1;
  1, 3, 8, 20, 27, 22, 1;
  ...
a(4), a(5), a(6) = 1, 3, 1 = antidiagonals of the array becoming row three of the triangle.

Cf. A001629 (antidiagonal sums), A104249.

A116445 := proc(n,k)
    local a,i ;
    a := 0 ;
    for i from 0 to n do
        a := a+binomial(k,i)*(i+1) ;
    end do:
    a ;
end proc:
seq(seq(A116445(d-k,k),k=0..d),d=0..12) ; # R. J. Mathar, Aug 17 2022

Detailed NAME by R. J. Mathar, Aug 17 2022

A139488 Binomial transform of [1, 2, 3, 4, 0, 0, 0, ...].

Original entry on oeis.org

1, 3, 8, 20, 43, 81, 138, 218, 325, 463, 636, 848, 1103, 1405, 1758, 2166, 2633, 3163, 3760, 4428, 5171, 5993, 6898, 7890, 8973, 10151, 11428, 12808, 14295, 15893, 17606, 19438, 21393, 23475, 25688, 28036, 30523, 33153, 35930, 38858, 41941, 45183

Offset: 0

Gary W. Adamson, Apr 23 2008

			a(5) = 43 = (1, 4, 6, 4, 1) dot (1, 2, 3, 4, 0) = (1 + 8, + 18 + 16 + 0).

Cf. A005408, A007318, A104249, A116445.

a:=proc(n) options operator, arrow: (2/3)*n^3-(1/2)*n^2+(11/6)*n+1 end proc: seq(a(n),n=0..35); # Emeric Deutsch, Apr 30 2008

f[n_] := Plus @@ (Table[ Binomial[n - 1, i], {i, 0, n - 1}] PadRight[{1, 2, 3, 4}, n]); Array[f, 43] (* Robert G. Wilson v, Apr 24 2008 *)

Equals A007318 * [1, 2, 3, 4, 0, 0, 0, ...].
a(n) = (4n^3 - 3n^2 + 11n + 6)/6. - Emeric Deutsch, Apr 30 2008
G.f.: (1 - x + 2*x^2 + 2*x^3)/(1-x)^4. - Colin Barker, Feb 01 2012

More terms from Robert G. Wilson v and Emeric Deutsch, Apr 24 2008

cp's OEIS Frontend

A006000 a(n) = (n+1)*(n^2+n+2)/2; g.f.: (1 + 2*x^2) / (1 - x)^4.

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A167560 The ED2 array read by ascending antidiagonals.

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A143689 a(n) = (3*n^2 - n + 2)/2.

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A327622 Number A(n,k) of parts in all k-times partitions of n into distinct parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A167565 A triangle related to the a(n) formulas for the rows of the ED2 array A167560.

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A242357 Crescendo trapezoidal.

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A164845 a(n) = (6 + 10*n + 5*n^2 + n^3)/2.

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A006000 a(n) = (n+1)(n^2+n+2)/2; g.f.: (1 + 2x^2) / (1 - x)^4.

A164845 a(n) = (6 + 10n + 5n^2 + n^3)/2.