A049451 -id:A049451 - cp's OEIS frontend

A306242 Number of ways to write n as x(3x+1) + y(3y-1) + z(3z+2) + w(3w-2), where x,y,z,w are nonnegative integers.

1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 3, 2, 2, 1, 3, 4, 3, 3, 2, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 4, 7, 4, 5, 4, 3, 4, 4, 6, 6, 3, 9, 6, 2, 5, 5, 8, 4, 6, 7, 6, 5, 6, 3, 5, 9, 6, 8, 7, 8, 7, 7, 8, 7, 4, 9, 8, 6, 6, 7, 7, 13, 9, 6, 6, 7, 11, 4, 6, 11, 9, 12

Offset: 0

Zhi-Wei Sun, Jan 31 2019

nonn

Conjecture: a(n) > 0 for all n >= 0, and a(n) = 1 only for n = 0, 1, 2, 3, 4, 9, 13. Moreover, any nonnegative integer n can be written as x*(3x+1) + y*(3y-1) + z*(3z+2) + w*(3w-2), where x,y,z,w are nonnegative integers with x or y even.
The conjecture has been verified for n up to 10^6.
By Theorem 1.3 of the linked 2017 paper of the author, each nonnegative integer can be written as x*(3x+1) + y*(3y-1) + z*(3z+2) + 0*(3*0-2) with x,y,z integers.
We also have some other similar conjectures. For example, we conjecture that every n = 0,1,2,... can be written as x*(5x+1)/2 + y*(5y-1)/2 + z*(5z+3)/2 + w*(5w-3)/2 with x,y,z,w nonnegative integers.

			a(1) = 1 with 1 = 0*(3*0+1) + 0*(3*0-1) + 0*(3*0+2) + 1*(3*1-2).
a(3) = 1 with 3 = 0*(3*0+1) + 1*(3*1-1) + 0*(3*0+2) + 1*(3*1-2).
a(4) = 1 with 4 = 1*(3*1+1) + 0*(3*0-1) + 0*(3*0+2) + 0*(3*0-2).
a(9) = 1 with 9 = 1*(3*1+1) + 0*(3*0-1) + 1*(3*1+2) + 0*(3*0-2).
a(13) = 1 with 13 = 0*(3*0+1) + 0*(3*0-1) + 1*(3*1+2) + 2*(3*2-2).

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, A result similar to Lagrange's theorem, J. Number Theory 162(2016), 190-211.
Zhi-Wei Sun, On x(ax+1)+y(by+1)+z(cz+1) and x(ax+b)+y(ay+c)+z(az+d), J. Number Theory 171(2017), 275-283.

Cf. A000567, A045944, A049450, A049451, A255350.

OctQ[n_]:=OctQ[n]=IntegerQ[Sqrt[3n+1]]&&(n==0||Mod[Sqrt[3n+1]+1,3]==0);
tab={};Do[r=0;Do[If[OctQ[n-x(3x+2)-y(3y+1)-z(3z-1)],r=r+1],{x,0,(Sqrt[3n+1]-1)/3},{y,0,(Sqrt[12(n-x(3x+2))+1]-1)/6},{z,0,(Sqrt[12(n-x(3x+2)-y(3y+1))+1]+1)/6}];tab=Append[tab,r],{n,0,80}];Print[tab]

A142962 Scaled convolution of (n^3)*A000984(n) with A000984(n).

Original entry on oeis.org

4, 26, 81, 184, 350, 594, 931, 1376, 1944, 2650, 3509, 4536, 5746, 7154, 8775, 10624, 12716, 15066, 17689, 20600, 23814, 27346, 31211, 35424, 40000, 44954, 50301, 56056, 62234, 68850, 75919, 83456, 91476, 99994, 109025, 118584, 128686, 139346, 150579

Offset: 1

Wolfdieter Lang, Sep 15 2008

S(3,n) := Sum_{j=0..n} j^3*binomial(2*j,j)*binomial(2*(n-j),n-j).
a(n) = 2^3*S(3,n)/4^n, n >= 1.
O.g.f. for S(3,n) is G(k=3,x). See triangle A142963 for the general G(k,x) formula.
The author was led to compute such sums by a question asked by M. Greiter, Jun 27 2008.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A142961 triangle: row k=3: [3, 5], with the row polynomial 3+5*n.

Cf. A049451 (scaled k=2 case).

Rest@ CoefficientList[Series[x (4 + 10 x + x^2)/(1 - x)^4, {x, 0, 39}], x] (* Michael De Vlieger, Jul 02 2023 *)

a(n) = n^2*(3+5*n)/2.
a(n) = (2^3)*S(3,n)/4^n with the convolution S(3,n) defined above.
G.f.: x*(4+10*x+x^2)/(1-x)^4. - Joerg Arndt, Jul 02 2023

A300954 Number of Dyck paths whose sequence of ascent lengths is exactly n+1, n+2, ..., 2n.

Original entry on oeis.org

1, 1, 3, 26, 425, 10647, 365512, 16067454, 864721566, 55202528425, 4083666929771, 343854336973368, 32493430569907125, 3406873823160467912, 392619681705581846700, 49342834390595374213214, 6717520607597479710109299, 984991858956314599670220717, 154785386247352261724279606367

Offset: 0

Alois P. Heinz, Mar 16 2018

nonn

Dyck paths counted by a(n) have semilength (3*n^2 + n)/2 = A005449(n) and length A049451(n).

			a(0) = 1: the empty path.
a(1) = 1: uudd.
a(2) = 3: uuuduuuudddddd, uuudduuuuddddd, uuuddduuuudddd.

Alois P. Heinz, Table of n, a(n) for n = 0..150
Wikipedia, Counting lattice paths

Main diagonal of A107876.

Cf. A005449, A049451.

a:= proc(m) option remember; local b; b:=
      proc(n, i) option remember; `if`(i>=2*m, 1,
        add(b(n+i-j, i+1), j=1..n+i))
      end; b(0, m+1)
    end:
seq(a(n), n=0..20);

a[m_] := a[m] = Module[{b}, b[n_, i_] := b[n, i] = If[i >= 2m, 1, Sum[b[n + i - j, i + 1], {j, 1, n + i}]]; b[0, m + 1]];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 02 2018, from Maple *)

a(n) = A107876(2n,n).

A306250 Number of ways to write n as x(3x+1) + y(3y-1) + z(3z+2) + w(3w-2), where x,y,z,w are nonnegative integers with xyz = 0.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Feb 01 2019

nonn

Conjecture: a(n) > 0 for any nonnegative integer n.

Clearly, a(n) <= A306242(n). We have verified a(n) > 0 for all n = 0..10^6.

			a(12) = 1 with 12 = 1*(3*1+1) + 0*(3*0-1) + 0*(3*0+2) + 2*(3*2-2).
a(42) = 1 with 42 = 0*(3*0+1) + 1*(3*1-1) + 0*(3*0+2) + 4*(3*4-2).
a(62) = 3 with 62 = 3*(3*3+1) + 3*(3*3-1) + 0*(3*0+2) + 2*(3*2-2)
= 4*(3*4+1) + 2*(3*2-1) + 0*(3*0+2) + 0*(3*0-2) = 4*(3*4+1) + 1*(3*1-1) + 0*(3*0+2) + 2*(3*2-2).
a(99) = 1 with 99 = 2*(3*2+1) + 0*(3*0-1) + 5*(3*5+2) + 0*(3*0-2).
a(118) = 1 with 118 = 0*(3*0+1) + 6*(3*6-1) + 2*(3*2+2) + 0*(3*0-2).

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, A result similar to Lagrange's theorem, J. Number Theory 162(2016), 190-211.
Zhi-Wei Sun, On x(ax+1)+y(by+1)+z(cz+1) and x(ax+b)+y(ay+c)+z(az+d), J. Number Theory 171(2017), 275-283.

Cf. A000567, A045944, A049450, A049451, A255350, A306242.

OctQ[n_]:=OctQ[n]=IntegerQ[Sqrt[3n+1]]&&(n==0||Mod[Sqrt[3n+1]+1,3]==0);
tab={};Do[r=0;Do[If[OctQ[n-x(3x+2)-y(3y+1)-z(3z-1)],r=r+1],{x,0,(Sqrt[3n+1]-1)/3},{y,0,(Sqrt[12(n-x(3x+2))+1]-1)/6},{z,0,If[x>0&&y>0,0,(Sqrt[12(n-x(3x+2)-y(3y+1))+1]+1)/6]}];tab=Append[tab,r],{n,0,100}];Print[tab]

A027903 a(n) = n(n + 1)(3*n + 1).

Original entry on oeis.org

0, 8, 42, 120, 260, 480, 798, 1232, 1800, 2520, 3410, 4488, 5772, 7280, 9030, 11040, 13328, 15912, 18810, 22040, 25620, 29568, 33902, 38640, 43800, 49400, 55458, 61992, 69020, 76560, 84630, 93248, 102432, 112200, 122570, 133560, 145188, 157472, 170430

Offset: 0

N. J. A. Sloane

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

Cf. A000567, A001477, A002378, A016742, A016777, A049451, A117642.

A027903:=n->n*(n + 1)*(3*n + 1); seq(A027903(n), n=0..100); # Wesley Ivan Hurt, Dec 02 2013

Table[n (n + 1) (3*n + 1), {n, 0, 100}] (* Wesley Ivan Hurt, Dec 02 2013 *)

From Wesley Ivan Hurt, Dec 02 2013: (Start)
a(n) = n*(n + 1)*(3*n + 1).
a(n) = 3*n^3 + 4*n^2 + n.
a(n) = A002378(n) * A016777(n).
a(n) = A049451(n) * A001477(n+1).
a(n) = A001477(n) * A000567(n-1).
a(n) = A001477(n) * A001477(n+1) * A016777(n).
a(n) = A117642(n) + A016742(n) + A001477(n). (End)
From Amiram Eldar, Aug 15 2025: (Start)
Sum_{n>=1} 1/a(n) = 4 - sqrt(3)*Pi/4 - 9*log(3)/4.
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/2 + 2*log(2) - 4. (End)
From Elmo R. Oliveira, Aug 29 2025: (Start)
G.f.: 2*x*(4 + 5*x)/(1 - x)^4.
E.g.f.: x*(8 + 13*x + 3*x^2)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

A157481 Number of primes between n^3-n^2 and (n+1)^3-(n+1)^2.

Original entry on oeis.org

0, 2, 5, 8, 10, 16, 21, 24, 32, 36, 43, 53, 57, 65, 74, 86, 92, 104, 114, 123, 133, 150, 151, 175, 180, 194, 207, 224, 238, 251, 271, 275, 306, 305, 332, 349, 359, 383, 408, 410, 434, 458, 473, 497, 502, 549, 553, 570, 590, 630, 641, 668, 685, 718, 726, 748, 780

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 01 2009

nonn

Cf. A049451, A060199, A014085, A144391

Table[PrimePi[(n+1)^3-(n+1)^2]-PrimePi[n^3-n^2],{n,0,5!}]

A165410 Hankel transform of the transform of 2^n given by A165409.

Original entry on oeis.org

1, 0, -4, -16, 0, 1024, 16384, 0, -16777216, -1073741824, 0, 17592186044416, 4503599627370496, 0, -1180591620717411303424, -1208925819614629174706176, 0, 5070602400912917605986812821504, 20769187434139310514121985316880384

Offset: 0

Paul Barry, Sep 17 2009

sign

Powers of two occurring in this sequence are based on the hexagonal spiral pattern of A049450 and A049451 (see A152749):
.
        16--15--14
        /         \
      17   5---4  13
      /   /     \   \
    18   6   0   3  12
    /   /   /   /   /
  19   7   1---2  11  26
    \   \         /   /
    20   8---9--10  25
      \             /
      21--22--23--24
.
The powers, (0,-oo,2,4,-oo,10,14,-oo,24,30,-oo,...) correspond to vertically joining pairs on the (0,4) and (0,2) radial lines, with -oo corresponding to the jump to the next pair.
The Hankel transforms of transforms of r^n behave similarly -- we get 1, 0, -r^2, -r^4, 0, r^10, r^14, ....
Note the Somos-4 property: a(3n) = 4*a(3n-1)*a(3n-3)/e(3n-4). Related to elliptic curve y^2 = 1 - 8x^3 in g.f. of A165409.

Cf. A165409.

A157482 Number of primes between n^3-n^2-n^1 and (n+1)^3-(n+1)^2-(n+1)^1.

Original entry on oeis.org

0, 1, 5, 8, 10, 16, 21, 24, 30, 39, 42, 52, 57, 65, 75, 86, 92, 102, 115, 122, 133, 150, 151, 176, 181, 192, 209, 221, 239, 252, 270, 273, 307, 308, 328, 350, 359, 383, 407, 414, 430, 460, 472, 494, 504, 548, 554, 571, 590, 629, 642, 669, 681, 722, 724, 749, 776

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 01 2009

nonn

Cf. A049451, A060199, A014085, A144391, A157481

Table[PrimePi[(n+1)^3-(n+1)^2-(n+1)^1]-PrimePi[n^3-n^2-n^1],{n,0,5!}]

A257144 Numbers n not of the form x+y*x^2 for x>1 and y>0.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 37, 40, 41, 43, 44, 45, 47, 49, 51, 53, 59, 60, 61, 63, 64, 65, 67, 69, 71, 73, 76, 77, 79, 81, 83, 85, 87, 88, 89, 91, 92, 95, 96, 97, 99, 101, 103, 104, 107, 108, 109, 112, 113, 115, 117, 119, 121

Offset: 1

Gionata Neri, Apr 16 2015

nonn

Number n such that (d*k+1) /= (n/d), for k>0 and each value of d, where d is a divisor >1 of n.

For numbers of the form x+y*x^2 with 0A002378 (y=1), A014105 (y=2), A049451 (y=3), A007742 (y=4), A202803 (y=5), A049453 (y=6), A092277 (y=7), A139275 (y=8), A154517 (y=9), A055437 (y=10). - Danny Rorabaugh, Apr 20 2015

n = 71; Take[Complement[Range[n^2], DeleteDuplicates@ Sort@ Flatten@ Table[x + y x^2, {x, 2, n}, {y, 1, n}]], n] (* Michael De Vlieger, Apr 17 2015 *)

A295089 a(n) = 3*n^2 + n + 3.

Original entry on oeis.org

3, 7, 17, 33, 55, 83, 117, 157, 203, 255, 313, 377, 447, 523, 605, 693, 787, 887, 993, 1105, 1223, 1347, 1477, 1613, 1755, 1903, 2057, 2217, 2383, 2555, 2733, 2917, 3107, 3303, 3505, 3713, 3927, 4147, 4373, 4605, 4843, 5087, 5337, 5593, 5855, 6123, 6397, 6677, 6963, 7255, 7553, 7857

Offset: 0

Ron Knott, Nov 14 2017

Numbers represented as the palindrome 313 in number base n including base n=1, base 2 (binary) and base 3 with 'illegal' digit 3: 313_1=7, 313_2=17, 313_3=33, ... 313_9=255, 313_10=313, ...

			313 in base 7 is 3*7^2 + 1*7 + 3 = 157.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A049451, A056108, A131649, A144391.

Array[3 #^2 + # + 3 &, 52, 0] (* Michael De Vlieger, Nov 15 2017 *)
LinearRecurrence[{3, -3, 1}, {3, 7, 17}, 52] (* or *)
CoefficientList[Series[-(5 x^2 - 2 x + 3)/(x - 1)^3, {x, 0, 51}], x] (* Robert G. Wilson v, Nov 29 2017 *)

a(n) = 3*n^2 + n + 3; \\ Michel Marcus, Dec 15 2017

a(n) = A131649(n+3) + 1, n >= 2 (conjectured).
a(n) = A056108(n) + 2 = A049451(n) + 3 = A144391(n) + 4.
From Elmo R. Oliveira, Sep 02 2025: (Start)
G.f.: (3 - 2*x + 5*x^2)/(1-x)^3.
E.g.f.: (3 + 4*x + 3*x^2)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)

cp's OEIS Frontend

A306242 Number of ways to write n as x*(3x+1) + y*(3y-1) + z*(3z+2) + w*(3w-2), where x,y,z,w are nonnegative integers.

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A142962 Scaled convolution of (n^3)*A000984(n) with A000984(n).

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A300954 Number of Dyck paths whose sequence of ascent lengths is exactly n+1, n+2, ..., 2n.

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A306250 Number of ways to write n as x*(3x+1) + y*(3y-1) + z*(3z+2) + w*(3w-2), where x,y,z,w are nonnegative integers with x*y*z = 0.

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Mathematica

A027903 a(n) = n*(n + 1)*(3*n + 1).

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Maple

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Formula

A157481 Number of primes between n^3-n^2 and (n+1)^3-(n+1)^2.

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Mathematica

A165410 Hankel transform of the transform of 2^n given by A165409.

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A157482 Number of primes between n^3-n^2-n^1 and (n+1)^3-(n+1)^2-(n+1)^1.

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Mathematica

A257144 Numbers n not of the form x+y*x^2 for x>1 and y>0.

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A306242 Number of ways to write n as x(3x+1) + y(3y-1) + z(3z+2) + w(3w-2), where x,y,z,w are nonnegative integers.

A306250 Number of ways to write n as x(3x+1) + y(3y-1) + z(3z+2) + w(3w-2), where x,y,z,w are nonnegative integers with xyz = 0.

A027903 a(n) = n(n + 1)(3*n + 1).