A081436 -id:A081436 - cp's OEIS frontend

A004068 Number of atoms in a decahedron with n shells.

0, 1, 7, 23, 54, 105, 181, 287, 428, 609, 835, 1111, 1442, 1833, 2289, 2815, 3416, 4097, 4863, 5719, 6670, 7721, 8877, 10143, 11524, 13025, 14651, 16407, 18298, 20329, 22505, 24831, 27312, 29953, 32759, 35735, 38886, 42217, 45733, 49439

Offset: 0

Albert D. Rich (Albert_Rich(AT)msn.com)

Also as a(n)=(n/6)*(5*n^2+1), n>0: structured pentagonal diamond numbers (vertex structure 6) (cf. A081436 = alternate vertex; A000447 = structured diamonds; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Number of atoms in decahedron with n shells, number = 5/6*(n^3) + 1/6*(n) (T. P. Martin, Shells of atoms, eq.(3)). - Brigitte Stepanov, Jul 02 2011
a(n+1) is the number of triples (w,x,y) having all terms in {0,...,n} and x+y >= w. - Clark Kimberling, Jun 14 2012
a(n) = Sum_{k=1..n} A215630(n,k) for n > 0. - Reinhard Zumkeller, Nov 11 2012
a(n) - a(n-2) = A010001(n-1), for n>1. - K. G. Stier, Dec 21 2012
a(n) is also a figurate number representing a cube of side n with a vertex cut off by a tetrahedron of side n-1. As such, a(n) = A000578(n) - A000292(n-1), n > 0. - Jean M. Morales, Aug 11 2013
The sequence starting with 1 is the third partial sum of (1, 4, 5, 5, 5, ...) and the binomial transform of (1, 6, 10, 5, 0, 0, 0, ...). - Gary W. Adamson, Sep 27 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (3).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

(1/12)*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.

Cf. A000292, A005891, A006322 (partial sums), A010001, A081436, A100145, A215630.

[5*n^3/6+n/6: n in [0..50]]; // Vincenzo Librandi, May 15 2011

Table[5*n^3/6+n/6,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)

A004068(n):=5*n^3/6+n/6$ makelist(A004068(n),n,0,20); /* Martin Ettl, Jan 07 2013 */

a(n)=5*n^3/6+n/6 \\ Charles R Greathouse IV, Sep 24 2015

def A004068(n): return n*(5*n**2+1)//6 # Chai Wah Wu, Mar 25 2025

a(n) = 5*binomial(n + 1, 3) + binomial(n, 1).
a(n) = 5*n^3/6 + n/6.
a(n) = Sum_{i=0..n-1} A005891(i). - Xavier Acloque, Oct 08 2003
G.f.: x*(1+3*x+x^2) / (1-x)^4. - R. J. Mathar, Jun 05 2011
E.g.f.: (x/6)*(5x^2 + 15x + 6)*exp(x). - G. C. Greubel, Sep 27 2015
Sum_{n>0} 1/a(n) = 3*(2*gamma + polygamma(0, 1-i/sqrt(5)) + polygamma(0, 1+i/sqrt(5))) = 1.233988011257952852492845364799197179252... where i denotes the imaginary unit. - Stefano Spezia, Aug 31 2023

Typo in definition corrected by Jean M. Morales, Aug 11 2013

A081437 Diagonal in array of n-gonal numbers A081422.

Original entry on oeis.org

1, 10, 33, 76, 145, 246, 385, 568, 801, 1090, 1441, 1860, 2353, 2926, 3585, 4336, 5185, 6138, 7201, 8380, 9681, 11110, 12673, 14376, 16225, 18226, 20385, 22708, 25201, 27870, 30721, 33760, 36993, 40426, 44065, 47916, 51985, 56278, 60801, 65560

Offset: 0

Paul Barry, Mar 21 2003

One of a family of sequences with palindromic generators.
For q a prime power, a(q-1) = q^3 + q^2 - q is the number of pairs of commuting nilpotent 2*2 matrices with coefficients in GF(q). (Proof: the zero matrix commutes with all q^2 nilpotent matrices, there are q^2-1 nonzero nilpotent matrices, all conjugate, each commuting with q nilpotent matrices.) - Mark Wildon, Jun 20 2017
Also the cyclomatic number (= circuit rank) of the n+1 X n+1 rook graph. - Eric W. Weisstein, Jun 20 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Circuit Rank
Eric Weisstein's World of Mathematics, Rook Graph
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A081435, A081436, A081437, A081438.

Equals A027620(n-1) + 1.

List([0..40], n-> (n+1)^3+n*(n+1)); # G. C. Greubel, Aug 14 2019

[n^3+4*n^2+4*n+1: n in [0..50]]; // Vincenzo Librandi, Aug 08 2013

a:=n->sum(n*k, k=0..n):seq(a(n)+sum(n*k, k=2..n), n=1..40); # Zerinvary Lajos, Jun 10 2008
a:=n->sum(-2+sum(2+sum(2, j=1..n),j=1..n),j=1..n):seq(a(n)/2,n=1..40); # Zerinvary Lajos, Dec 06 2008

Table[n^3 + 4 n^2 + 4n + 1, {n, 0, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {1, 10, 33, 76}, 40] (* Harvey P. Dale, Jan 24 2012 *)
CoefficientList[Series[(1 + 5 x - 7 x^2 + x^3)/(1 - x)^5, {x, 0, 60}], x] (* Vincenzo Librandi, Aug 08 2013 *)

vector(40, n, n--; (n+1)^3+n*(n+1)) \\ G. C. Greubel, Aug 14 2019

[(n+1)^3+n*(n+1) for n in (0..40)] # G. C. Greubel, Aug 14 2019

a(n) = n^3 + 4*n^2 + 4*n + 1.
G.f.: (1 +5*x -7*x^2 +x^3)/(1-x)^5.
a(0)=1, a(1)=10, a(2)=33, a(3)=76; for n>3, a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - Harvey P. Dale, Jan 24 2012
E.g.f.: (1 +9*x +7*x^2 +x^3)*exp(x). - G. C. Greubel, Aug 14 2019

A059722 a(n) = n(2n^2 - 2*n + 1).

Original entry on oeis.org

0, 1, 10, 39, 100, 205, 366, 595, 904, 1305, 1810, 2431, 3180, 4069, 5110, 6315, 7696, 9265, 11034, 13015, 15220, 17661, 20350, 23299, 26520, 30025, 33826, 37935, 42364, 47125, 52230, 57691, 63520, 69729, 76330, 83335, 90756, 98605, 106894, 115635, 124840

Offset: 0

Henry Bottomley, Feb 07 2001

Mean of the first four nonnegative powers of 2n+1, i.e., ((2n+1)^0 + (2n+1)^1 + (2n+1)^2 + (2n+1)^3)/4. E.g., a(2) = (1 + 3 + 9 + 27)/4 = 10.
Equatorial structured meta-diamond numbers, the n-th number from an equatorial structured n-gonal diamond number sequence. There are no 1- or 2-gonal diamonds, so 1 and (n+2) are used as the first and second terms since all the sequences begin as such. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Starting with offset 1 = row sums of triangle A143803. - Gary W. Adamson, Sep 01 2008
Form an array from the antidiagonals containing the terms in A002061 to give antidiagonals 1; 3,3; 7,4,7; 13,8,8,13; 21,14,9,14,21; and so on. The difference between the sum of the terms in n+1 X n+1 matrices and those in n X n matrices is a(n) for n>0. - J. M. Bergot, Jul 08 2013
Sum of the numbers from (n-1)^2 to n^2. - Wesley Ivan Hurt, Sep 08 2014

Harry J. Smith, Table of n, a(n) for n = 0..1000
Bruno Berselli, Table of sequences with the formula m*(m+1)^2 - (k+2)*m^2 (table includes this sequence for k = 2-m, m >= 0).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000330, A005900, A081436, A100178, A100179, A059722: "equatorial" structured diamonds; A000447: "polar" structured meta-diamond; A006484 for other structured meta numbers; and A100145 for more on structured numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004

Cf. A000217, A000290, A001844, A002061, A002414, A053698, A059721, A059723, A136392, A143803.

[n*(2*n^2 - 2*n + 1) : n in [0..50]]; // Wesley Ivan Hurt, Sep 08 2014

A059722:=n->n*(2*n^2 - 2*n + 1): seq(A059722(n), n=0..50); # Wesley Ivan Hurt, Sep 08 2014

Table[n (2 n^2 - 2 n + 1), {n, 0, 50}] (* Wesley Ivan Hurt, Sep 08 2014 *)

a(n) = { n*(2*n^2 - 2*n + 1) } \\ Harry J. Smith, Jun 28 2009

a(n) = A053698(2*n-1)/4.
a(n) = Sum_{j=1..n} ((n+j-1)^2-j^2+1). - Zerinvary Lajos, Sep 13 2006
From R. J. Mathar, Sep 02 2008: (Start)
G.f.: x*(1 + x)*(1 + 5*x)/(1 - x)^4.
a(n) = A002414(n-1) + A002414(n).
a(n+1) - a(n) = A136392(n+1). (End)
a(n) = (A000290(n) + A000290(n+1)) * (A000217(n+1) - A000217(n)). - J. M. Bergot, Nov 02 2012
a(n) = n * A001844(n-1). - Doug Bell, Aug 18 2015
a(n) = A000217(n^2) - A000217(n^2-2*n). - Bruno Berselli, Jun 26 2018
E.g.f.: exp(x)*x*(1 + 4*x + 2*x^2). - Stefano Spezia, Jun 20 2021

Edited with new definition by N. J. A. Sloane, Aug 29 2008

A110449 Triangle read by rows: T(n,k) = n((2k+1)*n+1)/2, 0<=k<=n.

Original entry on oeis.org

0, 1, 2, 3, 7, 11, 6, 15, 24, 33, 10, 26, 42, 58, 74, 15, 40, 65, 90, 115, 140, 21, 57, 93, 129, 165, 201, 237, 28, 77, 126, 175, 224, 273, 322, 371, 36, 100, 164, 228, 292, 356, 420, 484, 548, 45, 126, 207, 288, 369, 450, 531, 612, 693, 774, 55, 155, 255, 355, 455, 555, 655, 755, 855, 955, 1055

Offset: 0

Reinhard Zumkeller, Jul 21 2005

Row sums give A110450; central terms give A110451;
T(n,0) = A000217(n);
T(n,1) = A005449(n) for n>0;
T(n,2) = A005475(n) for n>1;
T(n,3) = A022265(n) for n>2;
T(n,4) = A022267(n) for n>3;
T(n,5) = A022269(n) for n>4;
T(n,6) = A022271(n) for n>5;
T(n,7) = A022263(n) for n>6;
T(n+1,n-1) = A059270(n) for n>1;
T(n,n-1) = A081436(n) for n>1;
T(n,n) = A085786(n).

			Triangle starts:
0;
1, 2;
3, 7, 11;
6, 15, 24, 33;
10, 26, 42, 58, 74;
...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A126890.

Table[n*((2*k + 1)*n + 1)/2, {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 23 2017 *)

tabl(nn) = {for (n=0, nn, for (k=0, n, print1(n*((2*k+1)*n+1)/2, ", ");); print(););} \\ Michel Marcus, Jun 22 2015

T(n,k) = n*((2*k + 1)*n + 1)/2, 0 <= k <= n.

A267471 T(n,k)=Number of length-n 0..k arrays with no following elements larger than the first repeated value.

Original entry on oeis.org

2, 3, 4, 4, 9, 7, 5, 16, 24, 12, 6, 25, 58, 62, 21, 7, 36, 115, 204, 160, 38, 8, 49, 201, 515, 712, 418, 71, 9, 64, 322, 1096, 2285, 2490, 1112, 136, 10, 81, 484, 2072, 5921, 10119, 8770, 3018, 265, 11, 100, 693, 3592, 13216, 31880, 44901, 31200, 8352, 522, 12, 121

Offset: 1

R. H. Hardin, Jan 15 2016

Table starts
...2.....3......4.......5........6.........7.........8..........9.........10
...4.....9.....16......25.......36........49........64.........81........100
...7....24.....58.....115......201.......322.......484........693........955
..12....62....204.....515.....1096......2072......3592.......5829.......8980
..21...160....712....2285.....5921.....13216.....26440......48657......83845
..38...418...2490...10119....31880.....83972....193852.....404589.....779938
..71..1112...8770...44901...171601....532840...1418740....3357537....7240267
.136..3018..31200..200119...925176...3381860..10378144...27838701...67140808
.265..8352.112300..897301..5002641..21491464..75944464..230790033..622347697
.522.23522.409254.4052183.27155800.136856180.556295860.1914051597.5768860606

			Some solutions for n=6 k=4
..0....1....1....4....0....1....0....4....1....2....3....4....4....1....0....3
..4....0....4....0....4....0....3....0....2....3....1....4....3....4....1....0
..4....2....2....4....3....4....2....4....0....0....2....4....2....0....3....2
..0....3....4....0....3....4....3....3....1....1....4....2....4....4....4....3
..2....4....1....1....0....0....1....1....2....1....3....1....3....2....1....3
..4....4....1....1....3....2....0....2....0....0....0....1....4....3....1....1

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A005126(n-1).
Row 1 is A000027(n+1).
Row 2 is A000290(n+1).
Row 3 is A081436.

Empirical for column k:
k=1: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3)
k=2: a(n) = 8*a(n-1) -23*a(n-2) +28*a(n-3) -12*a(n-4)
k=3: a(n) = 13*a(n-1) -65*a(n-2) +155*a(n-3) -174*a(n-4) +72*a(n-5)
k=4: a(n) = 19*a(n-1) -145*a(n-2) +565*a(n-3) -1174*a(n-4) +1216*a(n-5) -480*a(n-6)
k=5: [order 7]
k=6: [order 8]
k=7: [order 9]
Empirical for row n:
n=1: a(n) = n + 1
n=2: a(n) = n^2 + 2*n + 1
n=3: a(n) = n^3 + (5/2)*n^2 + (5/2)*n + 1
n=4: a(n) = n^4 + (17/6)*n^3 + 4*n^2 + (19/6)*n + 1
n=5: a(n) = n^5 + (37/12)*n^4 + (11/2)*n^3 + (77/12)*n^2 + 4*n + 1
n=6: a(n) = n^6 + (197/60)*n^5 + 7*n^4 + (43/4)*n^3 + 10*n^2 + (149/30)*n + 1
n=7: a(n) = n^7 + (69/20)*n^6 + (17/2)*n^5 + (97/6)*n^4 + 20*n^3 + (893/60)*n^2 + 6*n + 1

A355576 Number A(n,k) of n-tuples (p_1, p_2, ..., p_n) of positive integers such that p_{i-1} <= p_i <= k^(i-1); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 3, 7, 1, 0, 1, 1, 4, 24, 44, 1, 0, 1, 1, 5, 58, 541, 516, 1, 0, 1, 1, 6, 115, 3236, 35649, 11622, 1, 0, 1, 1, 7, 201, 12885, 713727, 6979689, 512022, 1, 0, 1, 1, 8, 322, 39656, 7173370, 627642640, 4085743032, 44588536, 1, 0

Offset: 0

Alois P. Heinz, Jul 07 2022

			A(2,3) = 3: (1,1), (1,2), (1,3).
A(3,2) = 7: (1,1,1), (1,1,2), (1,1,3), (1,1,4), (1,2,2), (1,2,3), (1,2,4).
A(3,3) = 24: (1,1,1), (1,1,2), (1,1,3), (1,1,4), (1,1,5), (1,1,6), (1,1,7), (1,1,8), (1,1,9), (1,2,2), (1,2,3), (1,2,4), (1,2,5), (1,2,6), (1,2,7), (1,2,8), (1,2,9), (1,3,3), (1,3,4), (1,3,5), (1,3,6), (1,3,7), (1,3,8), (1,3,9).
Square array A(n,k) begins:
  1, 1,     1,       1,         1,           1,            1, ...
  1, 1,     1,       1,         1,           1,            1, ...
  0, 1,     2,       3,         4,           5,            6, ...
  0, 1,     7,      24,        58,         115,          201, ...
  0, 1,    44,     541,      3236,       12885,        39656, ...
  0, 1,   516,   35649,    713727,     7173370,     46769781, ...
  0, 1, 11622, 6979689, 627642640, 19940684251, 330736663032, ...

Alois P. Heinz, Antidiagonals n = 0..43, flattened

Columns k=1-9 give: A000012, A107354, A109055, A109056, A109057, A109058, A109059, A109060, A109061.
Rows n=1-4 give: A000012, A001477, A081436(k-1) for k>0, A354608.
Main diagonal gives A355561.

A:= proc(n, k) option remember; `if`(n=0, 1, -add(
      A(j, k)*(-1)^(n-j)*binomial(k^j, n-j), j=0..n-1))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[n_, k_] := A[n, k] = If[n==0, 1, -Sum[A[j, k]*(-1)^(n-j)*Binomial[If[j==0, 1, k^j], n-j], {j, 0, n-1}]];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Sep 21 2022, after Alois P. Heinz *)

A005945 Number of n-step mappings with 4 inputs.

Original entry on oeis.org

0, 1, 15, 60, 154, 315, 561, 910, 1380, 1989, 2755, 3696, 4830, 6175, 7749, 9570, 11656, 14025, 16695, 19684, 23010, 26691, 30745, 35190, 40044, 45325, 51051, 57240, 63910, 71079, 78765, 86986, 95760, 105105, 115039, 125580, 136746

Offset: 0

N. J. A. Sloane

a(n) is the coefficient of x^4/4! in n-th iteration of exp(x)-1.

			G.f. = x + 15*x^2 + 60*x^3 + 154*x^4 + 315*x^5 + 561*x^6 + 910*x^7 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
T. Hogg and B. A. Huberman, Attractors on finite sets: the dissipative dynamics of computing structures, Phys. Review A 32 (1985), 2338-2346.
T. Hogg and B. A. Huberman, Attractors on finite sets: the dissipative dynamics of computing structures, Phys. Review A 32 (1985), 2338-2346. (Annotated scanned copy)
B. A. Huberman, T. H. Hogg, & N. J. A. Sloane, Correspondence, 1985
Pierpaolo Natalini, Paolo E. Ricci, Integer Sequences Connected with Extensions of the Bell Polynomials, Journal of Integer Sequences, 2017, Vol. 20, #17.10.2.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. for recursive method [Ar(m) is the m-th term of a sequence in the OEIS] a(n) = n*Ar(n) - A000217(n-1) or a(n) = (n+1)*Ar(n+1) - A000217(n) or similar: A081436, A005920, A006003 and the terms T(2, n) or T(3, n) in the sequence A125860. [Bruno Berselli, Apr 25 2010]

Cf. A094952.

I:=[0, 1, 15, 60]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..45]]; // Vincenzo Librandi Jun 18 2012

LinearRecurrence[{4,-6,4,-1},{0,1,15,60},50] (* Vincenzo Librandi, Jun 18 2012 *)
a[ n_] := 3 n^3 - 5/2 n^2 + 1/2 n; (* Michael Somos, Jun 10 2015 *)

{a(n) = 3*n^3 - 5/2*n^2 + 1/2*n}; /* Michael Somos, Jan 23 2014 */

G.f.: x*(1+11*x+6*x^2)/(1-x)^4. a(n)=n*(3*n-1)*(2*n-1)/2.
For n>0, a(n) = n*A000567(n) - A000217(n-1). - Bruno Berselli, Apr 25 2010; Feb 01 2011
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 18 2012
a(n) = -A094952(-n) for all n in Z. - Michael Somos, Jan 23 2014

Edited by Michael Somos, Oct 29 2002

A081422 Triangle read by rows in which row n consists of the first n+1 n-gonal numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 6, 10, 1, 4, 9, 16, 25, 1, 5, 12, 22, 35, 51, 1, 6, 15, 28, 45, 66, 91, 1, 7, 18, 34, 55, 81, 112, 148, 1, 8, 21, 40, 65, 96, 133, 176, 225, 1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451

Offset: 0

Paul Barry, Mar 21 2003

			The array starts
  1  1  3 10 ...
  1  2  6 16 ...
  1  3  9 22 ...
  1  4 12 28 ...
The triangle starts
  1;
  1,  1;
  1,  2,  3;
  1,  3,  6, 10;
  1,  4,  9, 16, 25;
  ...

T. D. Noe, Rows n = 0..100 of triangle, flattened
Eric Weisstein's World of Mathematics, Polygonal Number

Rows include A060354, A064808, A006000, A006003, A002411.
Diagonals include A001093, A053698, A069778, A000578, A002414, A081423, A081435, A081436, A081437, A081438, A081441.
Antidiagonals are composed of n-gonal numbers.

Flat(List([0..10], n-> List([1..n+1], k-> k*((n-2)*k-(n-4))/2 ))); # G. C. Greubel, Aug 14 2019

[[k*((n-2)*k-(n-4))/2: k in [1..n+1]]: n in [0..10]]; // G. C. Greubel, Oct 13 2018

Table[PolygonalNumber[n,i],{n,0,10},{i,n+1}]//Flatten (* Requires Mathematica version 10.4 or later *) (* Harvey P. Dale, Aug 27 2016 *)

tabl(nn) = {for (n=0, nn, for (k=1, n+1, print1(k*((n-2)*k-(n-4))/2, ", ");); print(););} \\ Michel Marcus, Jun 22 2015

[[k*((n-2)*k -(n-4))/2 for k in (1..n+1)] for n in (0..10)] # G. C. Greubel, Aug 14 2019

Array of coefficients of x in the expansions of T(k, x) = (1 + k*x -(k-2)*x^2)/(1-x)^4, k > -4.

T(n, k) = k*((n-2)*k -(n-4))/2 (see MathWorld link). - Michel Marcus, Jun 22 2015

A081435 Diagonal in array of n-gonal numbers A081422.

Original entry on oeis.org

1, 5, 18, 46, 95, 171, 280, 428, 621, 865, 1166, 1530, 1963, 2471, 3060, 3736, 4505, 5373, 6346, 7430, 8631, 9955, 11408, 12996, 14725, 16601, 18630, 20818, 23171, 25695, 28396, 31280, 34353, 37621, 41090, 44766, 48655, 52763, 57096, 61660

Offset: 0

Paul Barry, Mar 21 2003

One of a family of sequences with palindromic generators.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A081434, A081436, A081437.

List([0..40], n-> (n+1)*(2*(n+1)^2-3*n)/2); # G. C. Greubel, Aug 14 2019

[(2*n^3+3*n^2+3*n+2)/2: n in [0..40]]; // Vincenzo Librandi, Aug 08 2013

a := n-> (n+1)*(2*(n+1)^2-3*n)/2; seq(a(n), n = 0..40); # G. C. Greubel, Aug 14 2019

Table[(n^3 +(n+1)^3 -1)/2 +1, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, May 04 2011 *)
CoefficientList[Series[(1 +3x^2 -4x^3)/(1-x)^5, {x,0,40}], x] (* Vincenzo Librandi, Aug 08 2013 *)
LinearRecurrence[{4,-6,4,-1},{1,5,18,46},40] (* Harvey P. Dale, Dec 28 2024 *)

vector(40, n, n--; (n+1)*(2*(n+1)^2-3*n)/2) \\ G. C. Greubel, Aug 14 2019

[(n+1)*(2*(n+1)^2-3*n)/2 for n in (0..40)] # G. C. Greubel, Aug 14 2019

a(n) = (2*n^3 +3*n^2 +3*n +2)/2.
G.f.: (1 +3*x^2 -4*x^3)/(1-x)^5.
E.g.f.: (2 +8*x +9*x^2 +2*x^3)*exp(x)/2. - G. C. Greubel, Aug 14 2019

A125860 Rectangular table where column k equals row sums of matrix power A097712^k, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 17, 12, 4, 1, 1, 86, 69, 22, 5, 1, 1, 698, 612, 178, 35, 6, 1, 1, 9551, 8853, 2251, 365, 51, 7, 1, 1, 226592, 217041, 46663, 5990, 651, 70, 8, 1, 1, 9471845, 9245253, 1640572, 161525, 13131, 1057, 92, 9, 1, 1, 705154187

Offset: 0

Paul D. Hanna, Dec 13 2006

Triangle A097712 satisfies: A097712(n,k) = A097712(n-1,k) + [A097712^2](n-1,k-1) for n > 0, k > 0, with A097712(n,0)=A097712(n,n)=1 for n >= 0. Column 1 equals A016121, which counts the sequences (a_1, a_2, ..., a_n) of length n with a_1 = 1 satisfying a_i <= a_{i+1} <= 2*a_i.

T(2, n) = (n+1)*A005408(n) - Sum_{i=0..n} A001477(i) = (n+1)*(2*n+1) - A000217(n) = (n+1)*(3*n+2)/2; T(3, n) = (n+1)*A001106(n+1) - Sum_{i=0..n} A001477(i) = (n+1)*((n+1)*(7*n+2)/2) - A000217(n) = (n+1)*(7*n^2 + 8*n + 2)/2. - Bruno Berselli, Apr 25 2010

			Recurrence is illustrated by:
  T(4,1) = T(3,1) + T(3,2) = 17 + 69 = 86;
  T(4,2) = T(3,2) + T(3,3) + T(3,4) = 69 + 178 + 365 = 612;
  T(4,3) = T(3,3) + T(3,4) + T(3,5) + T(3,6) = 178 + 365 + 651 + 1057 = 2251.
Rows of this table begin:
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,...;
  1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, ...;
  1, 17, 69, 178, 365, 651, 1057, 1604, 2313, 3205, 4301, 5622, 7189,..;
  1, 86, 612, 2251, 5990, 13131, 25291, 44402, 72711, 112780, 167486,..;
  1, 698, 8853, 46663, 161525, 435801, 996583, 2025458, 3768273, ...;
  1, 9551, 217041, 1640572, 7387640, 24530016, 66593821, 156664796, ...;
  1, 226592, 9245253, 100152049, 586285040, 2394413286, 7713533212, ...;
  1, 9471845, 695682342, 10794383587, 82090572095, 412135908606, ...;
  1, 705154187, 93580638024, 2079805452133, 20540291522675, ...;
  1, 94285792211, 22713677612832, 723492192295786, 9278896006526795,...;
  1, 22807963405043, 10025101876435413, 458149292979837523, ...;
  ...
where column k equals the row sums of matrix power A097712^k for k >= 0.
Triangle A097712 begins:
  1;
  1,      1;
  1,      3,       1;
  1,      8,       7,       1;
  1,     25,      44,      15,       1;
  1,    111,     346,     208,      31,      1;
  1,    809,    4045,    3720,     912,     63,     1;
  1,  10360,   77351,   99776,   35136,   3840,   127,   1;
  1, 236952, 2535715, 4341249, 2032888, 308976, 15808, 255; ...
where A097712(n,k) = A097712(n-1,k) + [A097712^2](n-1,k-1);
e.g., A097712(5,2) = A097712(4,2) + [A097712^2](4,1) = 44 + 302 = 346.
Matrix square A097712^2 begins:
     1;
     2,     1;
     5,     6,     1;
    17,    37,    14,     1;
    86,   302,   193,    30,    1;
   698,  3699,  3512,   881,   62,   1;
  9551, 73306, 96056, 34224, 3777, 126, 1; ...
Matrix cube A097712^3 begins:
       1;
       3,      1;
      12,      9,      1;
      69,     87,     21,      1;
     612,   1146,    447,     45,    1;
    8853,  22944,  12753,   2019,   93,   1;
  217041, 744486, 549453, 120807, 8595, 189, 1; ...

Olivier Danvy, Summa Summarum: Moessner's Theorem without Dynamic Programming, arXiv:2412.03127 [cs.DM], 2024. See p. 16.

Cf. A097712; columns: A016121, A125862, A125863, A125864, A125865; A125861 (diagonal), A125859 (antidiagonal sums). Variants: A125790, A125800.

Cf. for recursive method [Ar(m) is the m-th term of a sequence in the OEIS] a(n) = n*Ar(n) - A000217(n-1) or a(n) = (n+1)*Ar(n+1) - A000217(n) and similar: A081436, A005920, A005945, A006003. - Bruno Berselli, Apr 25 2010

T[n_, k_] := T[n, k] = If[Or[n == 0, k == 0], 1, Sum[T[n - 1, j + k], {j, 0, k}]];
Table[T[#, k] &[n - k + 1], {n, 0, 9}, {k, 0, n + 1}] (* Michael De Vlieger, Dec 10 2024, after PARI *)

T(n,k)=if(n==0 || k==0,1,sum(j=0,k,T(n-1,j+k)))

T(n,k) = Sum_{j=0..k} T(n-1, j+k) for n > 0, with T(0,n)=T(n,0)=1 for n >= 0.

cp's OEIS Frontend

A004068 Number of atoms in a decahedron with n shells.

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A081437 Diagonal in array of n-gonal numbers A081422.

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A059722 a(n) = n*(2*n^2 - 2*n + 1).

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A110449 Triangle read by rows: T(n,k) = n*((2*k+1)*n+1)/2, 0<=k<=n.

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A267471 T(n,k)=Number of length-n 0..k arrays with no following elements larger than the first repeated value.

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A355576 Number A(n,k) of n-tuples (p_1, p_2, ..., p_n) of positive integers such that p_{i-1} <= p_i <= k^(i-1); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A005945 Number of n-step mappings with 4 inputs.

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A059722 a(n) = n(2n^2 - 2*n + 1).

A110449 Triangle read by rows: T(n,k) = n((2k+1)*n+1)/2, 0<=k<=n.