A033431 -id:A033431 - cp's OEIS frontend

A213821 Rectangular array: (row n) = b**c, where b(h) = 3*h-1, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

2, 9, 4, 24, 16, 6, 50, 39, 23, 8, 90, 76, 54, 30, 10, 147, 130, 102, 69, 37, 12, 224, 204, 170, 128, 84, 44, 14, 324, 301, 261, 210, 154, 99, 51, 16, 450, 424, 378, 318, 250, 180, 114, 58, 18, 605, 576, 524, 455, 375, 290, 206

Offset: 1

Clark Kimberling, Jul 04 2012

Principal diagonal: A033431.
Antidiagonal sums: A176060.
Row 1, (2,5,8,11,…)**(1,2,3,4,…): A006002.
Row 2, (2,5,8,11,…)**(2,3,4,5,…): (k^3 + 5*k^2 + 2*k)/2.
Row 3, (1,2,3,4,…)**(8,11,14,17,…): (k^3 + 8*k^2 + 3*k)/2.
For a guide to related arrays, see A212500.

			Northwest corner (the array is read by falling antidiagonals):
2….9….24…50….90
4….16…39…76…130
6….23…54…102…170
8….30…69…128…210
10…37…84…154…250
12…44…99…180…290

Cf. A212500

b[n_]:=3n-1;c[n_]:=n;
t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
r[n_]:=Table[t[n,k],{k,1,60}] (* A213821 *)
Table[t[n,n],{n,1,40}] (* A033431 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A176060 *)

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).

G.f. for row n: f(x)/g(x), where f(x) = x*(2*n - (n-2)*x - (n-1)*x^2) and g(x) = (1-x)^4.

A067389 a(n) = 3n^3 + 2n^2 + n.

Original entry on oeis.org

0, 6, 34, 102, 228, 430, 726, 1134, 1672, 2358, 3210, 4246, 5484, 6942, 8638, 10590, 12816, 15334, 18162, 21318, 24820, 28686, 32934, 37582, 42648, 48150, 54106, 60534, 67452, 74878, 82830, 91326, 100384, 110022, 120258, 131110, 142596

Offset: 0

George E. Antoniou, Jan 21 2002

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

[3*n^3 + 2*n^2 + n: n in [0..60]]; // Vincenzo Librandi, May 08 2011

a:=n->n+2*n^2+3*n^3: seq(a(n), n=0..36); # Zerinvary Lajos, Oct 05 2007

Table[3*n^3+2*n^2+n,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, May 07 2011 *)
LinearRecurrence[{4,-6,4,-1},{0,6,34,102},40] (* Harvey P. Dale, Oct 01 2019 *)

a(n) = n*A056109(n) = A045991(n+1)+A033431(n). - Henry Bottomley, Jan 25 2002
From Chai Wah Wu, Apr 25 2017: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3.
G.f.: 2*x*(x^2 + 5*x + 3)/(x - 1)^4. (End)

More terms from Henry Bottomley, Jan 25 2002

A099834 Maximum number of different determinants that can be produced by permuting the elements of a 3 X 3 integer matrix with nonnegative entries <= n.

Original entry on oeis.org

5, 15, 53, 109, 209, 351, 573, 811, 1193, 1509, 1971, 2501, 3183, 3769, 4511, 5025, 5641, 6165, 6600, 6964, 7354, 7696, 7960, 8110, 8404, 8606, 8704, 8846, 8962, 9125, 9210, 9284, 9362, 9420

Offset: 1

Hugo Pfoertner, Oct 29 2004

For large values of n it is always possible to find a matrix that produces A088021(3)=10080 different determinants. Examples are given in the link. Currently (October 2004) the smallest known n for which a(n)=10080 is 100. The elements of the corresponding matrix are given in A098072.

			a(10)=1509: A corresponding set of matrix elements is {10,9,9,8,7,5,2,1,0}.

Hugo Pfoertner, List of 3 X 3 integer matrices that give 10080 different determinants.
Hugo Pfoertner, Elements of 3 X 3 matrices with maximal number of different determinants.

Cf. A033431, A088021, A088217, A089472, A099815.

Cf. A099815 largest determinant that can be produced by the optimal set of matrix elements.

A333682 Number of nonnegative lattice paths from (0,0) to (4n+3,0) such that slopes of adjacent steps differ by one, assuming zero slope before and after the paths.

Original entry on oeis.org

1, 3, 16, 119, 1070, 10751, 116287, 1326581, 15756587, 193181910, 2429921124, 31216684816, 408198225495, 5418728779290, 72871393962150, 991102308239835, 13613940451015378, 188650695857473559, 2634681336798911129, 37054660535787380825, 524449965598846642847

Offset: 0

Alois P. Heinz, Apr 01 2020

nonn

The maximal height in all paths of length 4n+3 is (n+1)^2 = A000290(n+1).

The maximal area under all paths of length 4n+3 is 2*(n+1)^3 = A033431(n+1).

Alois P. Heinz, Table of n, a(n) for n = 0..100
Alois P. Heinz, Animation of a(3) = 119 paths
Alois P. Heinz, Plot of a(3) = 119 paths
Alois P. Heinz, Plot of a(4) = 1070 paths
Alois P. Heinz, Plot of a(5) = 10751 paths

Cf. A000290, A004767, A033431, A333647.

b:= proc(x, y, t) option remember; `if`(x=0, 1, add(`if`(j=t, 0,
      b(x-1, y+j, j)), j=max(t-1, -y)..min(x*(x-1)/2-y, t+1)))
    end:
a:= n-> b(4*n+3, 0$2):
seq(a(n), n=0..23);

A184537 a(n) = floor(1/{(2+n^4)^(1/4)}), where {} = fractional part.

Original entry on oeis.org

5, 3, 16, 54, 128, 250, 432, 686, 1024, 1458, 2000, 2662, 3456, 4394, 5488, 6750, 8192, 9826, 11664, 13718, 16000, 18522, 21296, 24334, 27648, 31250, 35152, 39366, 43904, 48778, 54000, 59582, 65536, 71874, 78608, 85750, 93312, 101306, 109744, 118638, 128000, 137842, 148176, 159014, 170368, 182250, 194672, 207646, 221184, 235298, 250000, 265302, 281216, 297754, 314928, 332750, 351232, 370386, 390224, 410758, 432000, 453962, 476656

Offset: 0

Clark Kimberling, Jan 16 2011

Similar to A033431: replacing a(0) by 0 and a(1) by 2 gives A033431, see next comment.
From Bruno Berselli, Feb 04 2011: (Start)
For n >= 1, the value of (n^4+2)^(1/4) is just slightly above n, so the fractional part is (2+n^4)^(1/4)-n. For n > 1, then, 2*n^3 < 1/((2+n^4)^(1/4)-n) < 2*n^3+1.
The proof that 2*n^3*((2+n^4)^(1/4)-n) < 1 follows easily by isolating the quartic root and raising to the 4th power; similarly, 1 < (2*n^3+1)*((2+n^4)^(1/4)-n) needs a sign estimation of a 9th-order polynomial.
In conclusion, a(n)=A033431(n) for n > 1, with g.f. (34*x^2-12*x^3+x^4+x^5+5-17*x) / (x-1)^4. (End)

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. A184536. Essentially the same as A033431.

I:=[5, 3, 16, 54, 128,250]; [n le 6 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..70]]; // Vincenzo Librandi, Jul 04 2012

f[n_] := Floor[1/FractionalPart[(n^4 + 2)^(1/4)]]; Array[f, 40, 0]
CoefficientList[Series[(34*x^2-12*x^3+x^4+x^5+5-17*x)/(x-1)^4,{x,0,50}],x] (* Vincenzo Librandi, Jul 04 2012 *)
LinearRecurrence[{4,-6,4,-1},{5,3,16,54,128,250},70] (* Harvey P. Dale, Apr 06 2018 *)

a(n) = floor( 1 / frac((2+n^4)^(1/4)) ).

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 5.

A269792 a(n) = 5*n^4.

Original entry on oeis.org

0, 5, 80, 405, 1280, 3125, 6480, 12005, 20480, 32805, 50000, 73205, 103680, 142805, 192080, 253125, 327680, 417605, 524880, 651605, 800000, 972405, 1171280, 1399205, 1658880, 1953125, 2284880, 2657205, 3073280, 3536405, 4050000, 4617605, 5242880, 5929605

Offset: 0

Ilya Gutkovskiy, Mar 31 2016

More generally, the ordinary generating function for the sequences of the form k*n^m, is k*Sum_{j>=1}x^j*j^m (when abs(x)<1).

More generally, the ordinary generating function for the values of quartic polynomial p*n^4 + q*n^3 + k*n^2 + m*n + r, is (r + (p + q + k + m - 4*r)*x + (11*p + 3*q - k - 3*m + 6*r)*x^2 + (11*p - 3*q - k + 3*m - 4*r)*x^3 + (p - q + k - m + r)*x^4)/(1 - x)^5.

Ilya Gutkovskiy, Examples of the ordinary generating function for the values of quartic polynomial
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Cf. similar sequences of the form k*n^m, for k = 1...5, m = 1...10: A001477(k = 1, m = 1), A005843 (k = 2, m = 1), A008585 (k = 3, m = 1), A008586 (k = 4, m = 1), A008587 (k = 5, m = 1), A000290 (k = 1, m = 2), A001105 (k = 2, m = 2), A033428 (k = 3, m = 2), A016742 (k = 4, m = 2), A033429 (k = 5, m = 2), A000578 (k = 1, m = 3), A033431 (k = 2, m = 3), A117642 (k = 3, m = 3), A033430 (k = 4, m = 3), A244725 (k = 5, m = 3), A000583 (k = 1, m = 4), A244730 (k = 2, m = 4), A219056 (k = 3, m = 4), A141046 (k = 4, m = 4), this sequence(k = 5, m = 4), A000584 (k = 1, m = 5), A001014 (k = 1, m = 6), A106318 (k = 2, m = 6), A001015 (k = 1, m = 7), A001016 (k = 1, m = 8), A001017 (k = 1, m = 9), A008454 (k = 1, m = 10).

A269792:=n->5*n^4: seq(A269792(n), n=0..50); # Wesley Ivan Hurt, Apr 28 2017

Table[5 n^4, {n, 0, 33}]
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 5, 80, 405, 1280}, 34]

x='x+O('x^99); concat(0, Vec(5*x*(1+11*x+11*x^2+x^3)/(1-x)^5)) \\ Altug Alkan, Mar 31 2016

G.f.: 5*x*(1 + 11*x + 11*x^2 + x^3)/(1 - x)^5.
E.g.f.: 5*exp(x)^x*x*(1 + 7*x + 6*x^2 + x^3).
a(n) = 5*a(n-1) - 10*(9n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = 5*A000583(n) = A008587(n)*A000578(n).
Sum_{n>=1} 1/a(n) = Pi^4/450 = (1/450)*A092425 = 0.216464646742...

A275709 a(n) = 2n^3 + 3n^2.

Original entry on oeis.org

0, 5, 28, 81, 176, 325, 540, 833, 1216, 1701, 2300, 3025, 3888, 4901, 6076, 7425, 8960, 10693, 12636, 14801, 17200, 19845, 22748, 25921, 29376, 33125, 37180, 41553, 46256, 51301, 56700, 62465, 68608, 75141, 82076, 89425, 97200, 105413, 114076, 123201, 132800, 142885

Offset: 0

Joshua Giambalvo, Aug 06 2016

Apart from the initial zero this sequence gives the 2nd pentagonal number, the 4th hexagonal number, the 6th heptagonal number, the 8th octagonal number, the 10th nonagonal number, etc. as well as the 5th nonnegative number, the 7th triangular number, the 9th square, the 11th pentagonal number, the 13th hexagonal number, etc. This is a reliable pattern that does not seem to appear on any other pairs of polygonal numbers (see link).

a(n) is the maximal determinant of a 3 X 3 matrix with integer elements from {1, ..., n+1}, so (for example) the maximum determinant of a 3 X 3 matrix with integer elements from {1, ..., 5} = det(1, 5, 5; 5, 1, 5; 5, 5, 1) = a(4) = 176. - Matthew Scroggs, Dec 31 2022

Carauleanu Marc and Colin Barker, Table of n, a(n) for n = 0..3030 (first 1000 terms from Colin Barker)
Joshua Giambalvo, Illustration of initial terms in a square array, Imgur, (2016).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000290, A005408, A005563, A033428, A033431, A057145, A139600.

[n^2*(2*n + 3): n in [0..30]]; // G. C. Greubel, Oct 19 2018

seq(2*n^3+3*n^2, n=0..30); # Robert Israel, Aug 09 2016

Table[2 n^3 + 3 n^2, {n, 0, 41}] (* or *)
CoefficientList[Series[x (5 + 8 x - x^2)/(1 - x)^4, {x, 0, 41}], x] (* Michael De Vlieger, Aug 11 2016 *)

concat(0, Vec(x*(5+8*x-x^2)/(1-x)^4 + O(x^50))) \\ Colin Barker, Aug 28 2016

a(n)=n^2*(2*n+3) \\ Charles R Greathouse IV, Aug 28 2016

for n in range(0,50): print(n**2*(2*n+3), end=' ') # Stefano Spezia, Oct 19 2018

From Colin Barker, Aug 06 2016: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4.
G.f.: x*(5+8*x-x^2) / (1-x)^4. (End)
a(n) = A033431(n) + A033428(n). - Omar E. Pol, Aug 09 2016
a(n) = A000290(n) * A005408(n+1). - Robert Israel, Aug 09 2016
a(n) = A320047(1, n, 0). - Kolosov Petro, Oct 04 2018
E.g.f.: x*(5 + 9*x + 2*x^2)*exp(x). - G. C. Greubel, Oct 19 2018
From Amiram Eldar, Jan 21 2023: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/18 + 4*log(2)/9 - 16/27.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/36 + Pi/9 -2*log(2)/9 - 8/27. (End)

A282093 Larger member of a pair (x,y) which solves x^2+y^2 = z^3 for positive x, y and z.

Original entry on oeis.org

2, 10, 11, 16, 26, 30, 39, 46, 52, 54, 68, 80, 88, 100, 110, 117, 120, 128, 130, 142, 145, 170, 198, 205, 208, 222, 236, 240, 250, 270, 286, 297, 310, 312, 322, 350, 366, 368, 371, 377, 406, 414, 415, 416, 432, 455, 481, 488, 505, 518, 520, 524, 544, 549, 584

Offset: 1

R. J. Mathar, Feb 06 2017

nonn

Values y such that x^2+y^2 = z^3 has a solution 1<=x<=y with integer x, y and z.
The positive values of A033431 are a subsequence, induced by solutions where x=y.
There are entries which have more than one representation, e.g., 10^2 + 198^2 = 34^3 and 107^2 + 198^2 = 37^3 both with y=198. 234^2 + 415^2 = 61^3 and 320^2 + 415^2 = 65^3 both with y=415.
The ordered sequence of x can apparently be constructed by retrieving the perfect squares in A106265 and printing their square roots: 1, 2, 5, 7, 8, 9, 10, 11, 16, 17, 18 , 26, 27, 30,...

			2^2+2^2=2^3, so 2 is in. 5^2+10^2=5^3, so 10 is in. 2^2+11^2 = 5^3, so 11 is in. 16^2+16^2=8^3, so 16 is in.

Cf. A000404 (values of z), A033431, A106265.

isA282093 := proc(y)
    local x,z3 ;
    for x from 1 to y do
        z3 := x^2+y^2 ;
        if isA000578(z3) then
            return true ;
        end if;
    end do:
    return false ;
end proc:
for y from 1 to 800 do
    if isA282093(y) then
        printf("%d,\n",y) ;
    end if;
end do:

isA282093[y_] := Module[{x, z3},
For[x = 1, x <= y, x++, z3 = x^2+y^2; If[IntegerQ[z3^(1/3)], Return[True]]]; Return[False]];
Reap[For[y = 1, y <= 800, y++, If[isA282093[y], Print[y]; Sow[y]]]][[2, 1]] (* Jean-François Alcover, May 29 2023, after R. J. Mathar *)

A304161 a(n) = 2n^3 - 4n^2 + 10*n - 2 (n>=1).

Original entry on oeis.org

6, 18, 46, 102, 198, 346, 558, 846, 1222, 1698, 2286, 2998, 3846, 4842, 5998, 7326, 8838, 10546, 12462, 14598, 16966, 19578, 22446, 25582, 28998, 32706, 36718, 41046, 45702, 50698, 56046, 61758, 67846, 74322, 81198, 88486, 96198, 104346, 112942, 121998

Offset: 1

Emeric Deutsch, May 09 2018

For n>=2, a(n) is the first Zagreb index of the graph KK_n, defined as 2 copies of the complete graph K_n, with one vertex from one copy joined to two vertices of the other copy (see the Stevanovic et al. reference, p. 396).
The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternatively, it is the sum of the degree sums d(i) + d(j) over all edges ij of the graph.
The M-polynomial of KK_n is M(KK_n; x,y) = (n-2)^2*x^{n-1}*y^{n-1}+2*(n-2)*x^{n-1}*y^n + (n-1)*x^{n-1}*y^{n+1} + x^n*y^n +2*x^n*y^{n+1}.

Colin Barker, Table of n, a(n) for n = 1..1000
E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
D. Stevanovic, I. Stankovic, and M. Milosevic, More on the relation between energy and Laplacian energy of graphs, MATCH Commun. Math. Comput. Chem. 61, 2009, 395-401.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A033431, A054000.

Table[2n^3-4n^2+10n-2 ,{n,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{6,18,46,102},50] (* Harvey P. Dale, Oct 17 2022 *)

Vec(2*x*(3 - 3*x + 5*x^2 + x^3) / (1 - x)^4 + O(x^60)) \\ Colin Barker, May 09 2018

a(n) = A033431(n-1) + A054000(n+1). - Omar E. Pol, May 09 2018
From Colin Barker, May 09 2018: (Start)
G.f.: 2*x*(3 - 3*x + 5*x^2 + x^3) / (1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4.
(End)

A305290 Numbers k such that 4*k + 1 is a perfect cube, sorted by absolute values.

Original entry on oeis.org

0, -7, 31, -86, 182, -333, 549, -844, 1228, -1715, 2315, -3042, 3906, -4921, 6097, -7448, 8984, -10719, 12663, -14830, 17230, -19877, 22781, -25956, 29412, -33163, 37219, -41594, 46298, -51345, 56745, -62512, 68656, -75191, 82127, -89478, 97254, -105469, 114133, -123260, 132860

Offset: 1

Bruno Berselli, May 29 2018

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (-3,-2,2,3,1).

Cf. A016755.
Cf. A000290: k such that 4*k is a square.
Cf. A002378: k such that 4*k+1 is a square.
Cf. A033431: k such that 4*k is a nonnegative cube.
Cf. A305291: k such that 4*k+3 is a cube.
Cf. A141046: k such that 4*k is a fourth power.
Cf. 4*A219086: k such that 4*k+1 is a fourth power.

seq(coeff(series(x^2*(-7+10*x-7*x^2)/((1-x)*(1+x)^4), x,50),x,n),n=1..45); # Muniru A Asiru, May 31 2018

LinearRecurrence[{-3, -2, 2, 3, 1}, {0, -7, 31, -86, 182}, 45] (* Jean-François Alcover, Jun 04 2018 *)

concat(0, Vec(-x^2*(7 - 10*x + 7*x^2) / ((1 - x)*(1 + x)^4) + O(x^40))) \\ Colin Barker, Jun 04 2018

G.f.: x^2*(-7 + 10*x - 7*x^2)/((1 - x)*(1 + x)^4).
a(n) = -3*a(n-1) - 2*a(n-2) + 2*a(n-3) + 3*a(n-4) + a(n-5).
a(n) = (-1 - A016755(n-1)*(-1)^n)/4.
a(n) + a(-n) = (-1)^n*2^((1-(-1)^n)/2).
(n - 2)*(4*n^2 - 16*n + 19)*a(n) + (12*n^2 - 36*n + 31)*a(n-1) - (n - 1)*(4*n^2 - 8*n + 7)*a(n-2) = 0.
From Colin Barker, May 30 2018: (Start)
a(n) = n*(4*n^2 + 6*n + 3)/2 for n even.
a(n) = -(n + 1)*(4*n^2 + 2*n + 1)/2 for n odd.
(End)

cp's OEIS Frontend

A213821 Rectangular array: (row n) = b**c, where b(h) = 3*h-1, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

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A067389 a(n) = 3*n^3 + 2*n^2 + n.

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A099834 Maximum number of different determinants that can be produced by permuting the elements of a 3 X 3 integer matrix with nonnegative entries <= n.

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A333682 Number of nonnegative lattice paths from (0,0) to (4n+3,0) such that slopes of adjacent steps differ by one, assuming zero slope before and after the paths.

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Maple

A184537 a(n) = floor(1/{(2+n^4)^(1/4)}), where {} = fractional part.

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Magma

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A269792 a(n) = 5*n^4.

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A275709 a(n) = 2*n^3 + 3*n^2.

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A282093 Larger member of a pair (x,y) which solves x^2+y^2 = z^3 for positive x, y and z.

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A067389 a(n) = 3n^3 + 2n^2 + n.

A275709 a(n) = 2n^3 + 3n^2.

A304161 a(n) = 2n^3 - 4n^2 + 10*n - 2 (n>=1).