A045991 -id:A045991 - cp's OEIS frontend

A212971 Number of triples (w,x,y) with all terms in {0,...,n} and w < floor((x+y)/3).

0, 0, 3, 11, 25, 48, 82, 128, 189, 267, 363, 480, 620, 784, 975, 1195, 1445, 1728, 2046, 2400, 2793, 3227, 3703, 4224, 4792, 5408, 6075, 6795, 7569, 8400, 9290, 10240, 11253, 12331, 13475, 14688, 15972, 17328, 18759, 20267, 21853, 23520

Offset: 0

Clark Kimberling, Jun 03 2012

For a guide to related sequences, see A212959.

Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).

Cf. A212959, A212972.

Cf. A045991, A212974.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w < Floor[(x + y)/3], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 60]]   (* A212971*)
LinearRecurrence[{3,-3,2,-3,3,-1},{0,0,3,11,25,48},50] (* Harvey P. Dale, Aug 24 2021 *)

a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 3*a(n-4) + 3*a(n-5) - a(n-6).
G.f.: (x^2)*(3 + 2*x + x^2)/((1 + x + x^2)*(1-x)^4).
a(n) = (n+1)^3 - A212972(n).
From Ayoub Saber Rguez, Dec 11 2023: (Start)
a(n) = A045991(n+1) - A212974(n).
a(n) = (n^3 + n^2 - n - 1 + (((n+1) mod 3) mod 2))/3. (End)

A304487 a(n) = (3 + 2n - 3n^2 + 4n^3 - 3((-1 + n) mod 2))/6.

Original entry on oeis.org

1, 4, 15, 36, 73, 128, 207, 312, 449, 620, 831, 1084, 1385, 1736, 2143, 2608, 3137, 3732, 4399, 5140, 5961, 6864, 7855, 8936, 10113, 11388, 12767, 14252, 15849, 17560, 19391, 21344, 23425, 25636, 27983, 30468, 33097, 35872, 38799, 41880, 45121, 48524, 52095

Offset: 1

Stefano Spezia, Aug 17 2018

a(n) is the trace of an n X n matrix A in which the entries are 1 through n^2, spiraling inward starting with 1 in the (1,1)-entry (proved).

The first three terms of a(n) coincide with those of A317614.

			For n = 1 the matrix A is
   1
with trace Tr(A) = a(1) = 1.
For n = 2 the matrix A is
   1, 2
   4, 3
with Tr(A) = a(2) = 4.
For n = 3 the matrix A is
   1, 2, 3
   8, 9, 4
   7, 6, 5
with Tr(A) = a(3) = 15.
For n = 4 the matrix A is
   1,  2,  3, 4
  12, 13, 14, 5
  11, 16, 15, 6
  10,  9,  8, 7
with Tr(A) = a(4) = 36.

Stefano Spezia, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A317614, A228958, A045991, A085046.

Cf. A126224 (determinant of the matrix A), A317298 (first differences).

a_n:=List([1..43], n->(3 + 2*n - 3*n^2 + 4*n^3 - 3*RemInt(-1 + n, 2))/6);

List([1..43],n->(3+2*n-3*n^2+4*n^3-3*((-1+n) mod 2))/6); # Muniru A Asiru, Sep 17 2018

I:=[1,4,15,36,73]; [n le 5 select I[n] else 3*Self(n-1)-2*Self(n-2)-2*Self(n-3)+3*Self(n-4)-Self(n-5): n in [1..43]]; // Vincenzo Librandi, Aug 26 2018

seq((3+2*n-3*n^2+4*n^3-3*modp((-1+n),2))/6,n=1..43); # Muniru A Asiru, Sep 17 2018

Table[1/6 (3 + 2 n - 3 n^2 + 4 n^3 - 3 Mod[-1 + n, 2]), {n, 1, 43}] (* or *)
CoefficientList[ Series[x*(1 + x + 5 x^2 + x^3)/((-1 + x)^4 (1 + x)), {x, 0, 43}], x] (* or *)
LinearRecurrence[{3, -2, -2, 3, -1}, {1, 4, 15, 36, 73}, 43]

a(n):=(3 + 2*n - 3*n^2 + 4*n^3 - 3*mod(-1 + n, 2))/6$ makelist(a(n), n, 1, 43);

Vec(x*(1 + x + 5*x^2 + x^3)/((-1 + x)^4*(1 + x)) + O(x^44))

a(n) = (3 + 2*n - 3*n^2 + 4*n^3 - 3*((-1 + n)%2))/6

a(n) = A045991(n) - Sum_{k=2..n-1} A085046(k) for n > 2 (proved).
G.f.: x*(1 + x + 5 x^2 + x^3)/((-1 + x)^4 (1 + x)).
a(n) + a(n + 1) = A228958(2*n + 1).
From Colin Barker, Aug 17 2018: (Start)
a(n) = (2*n - 3*n^2 + 4*n^3) / 6 for n even.
a(n) = (3 + 2*n - 3*n^2 + 4*n^3) / 6 for n odd.
a(n) = 3*a(n - 1) - 2*a(n - 2) - 2*a(n - 3) + 3*a(n - 4) - a(n - 5) for n > 5.
(End)
E.g.f.: (1/12)*exp(-x)*(-3 + exp(2*x)*(3 + 6*x + 18*x^2 + 8*x^3)). - Stefano Spezia, Feb 10 2019

A132998 a(n) = n^4 - n^3 - n^2.

Original entry on oeis.org

0, -1, 4, 45, 176, 475, 1044, 2009, 3520, 5751, 8900, 13189, 18864, 26195, 35476, 47025, 61184, 78319, 98820, 123101, 151600, 184779, 223124, 267145, 317376, 374375, 438724, 511029, 591920, 682051, 782100, 892769

Offset: 0

Omar E. Pol, Nov 01 2007

			a(7)=2009 because 7^4=2401, 7^3=343, 7^2=49 and we can write 2401-343-49=2009.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000290, A000578, A000583, A011379, A045991.

[n^4-n^3-n^2: n in [0..50]]; // Vincenzo Librandi, Dec 15 2010

A132998:=n->n^4-n^3-n^2; seq(A132998(n), n=0..50); # Wesley Ivan Hurt, May 21 2014

f[n_]:=n^4-n^3-n^2; Table[f[n],{n,5!}] (* Vladimir Joseph Stephan Orlovsky, Dec 04 2009 *)
CoefficientList[Series[- x (-1 + 9 x + 15 x^2 + x^3)/(-1 + x)^5, {x, 0, 50}], x] (* Vincenzo Librandi, May 21 2014 *)
LinearRecurrence[{5,-10,10,-5,1}, {0, -1, 4, 45, 176}, 50] (* G. C. Greubel, Sep 28 2017 *)

x='x+O('x^50); Vec(x*(-1+9*x+15*x^2+x^3)/(1-x)^5) \\ G. C. Greubel, Sep 28 2017

a(n) = n^4 - n^3 - n^2.

G.f.: x*(-1+9*x+15*x^2+x^3)/(1-x)^5. - R. J. Mathar, Nov 14 2007

A134631 a(n) = 5n^5 - 3n^3 + 2*n^2. Coefficients and exponents are the prime numbers in decreasing order.

Original entry on oeis.org

0, 4, 144, 1152, 4960, 15300, 38304, 83104, 162432, 293220, 497200, 801504, 1239264, 1850212, 2681280, 3787200, 5231104, 7085124, 9430992, 12360640, 15976800, 20393604, 25737184, 32146272, 39772800, 48782500, 59355504, 71686944, 85987552, 102484260, 121420800, 143058304, 167675904, 195571332, 227061520

Offset: 0

Omar E. Pol, Nov 04 2007

nonn

			a(4)=4960 because 4^5=1024, 5*1024=5120, 4^3=64, 3*64=192, 4^2=16, 2*16=32 and we can write 5120 - 192 + 32 = 4960.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6, -15, 20, -15, 6, -1).

Cf. A000290, A000578, A000584, A045991, A100019, A133071.

[5*n^5-3*n^3+2*n^2: n in [0..50]]; // Vincenzo Librandi, Dec 14 2010

Table[5n^5-3n^3+2n^2,{n,0,40}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{0,4,144,1152,4960,15300},40] (* Harvey P. Dale, Jan 20 2023 *)

a(n) = 5*n^5 - 3*n^3 + 2*n^2.

G.f.: 4x*(1+30x+87x^2+32x^3)/(1-x)^6. - R. J. Mathar, Nov 14 2007

More terms from Vincenzo Librandi, Dec 14 2010

A138667 Values of x in solutions (x,y,z) to the Diophantine equation x^3-x^2+y^3-y^2=z^3-z^2, with 1

Original entry on oeis.org

64, 94, 160, 256, 268, 301, 333, 390, 682, 864, 1015, 1151, 1188, 2068, 3094, 4165, 4177, 5452, 5959, 6201, 6490, 8181, 9334, 11440, 11561, 11628, 14116, 14416, 17220, 18684, 18940, 19360, 20856, 21825, 25880, 26865, 27501, 28630, 28850, 28858

Offset: 1

Manuel Valdivia, Apr 17 2008

nonn

A045991(x)=A045991(z)-A045991(y), y=A138668(n) and z=A138669(n).

			Solutions to the Diophantine equation in lexicographic order: {x, y, z}: {64, 100, 108}, {94, 301, 304}, {160, 226, 250}, {256, 305, 356}, {268, 341, 389}, {301, 770, 785}, {333, 370, 444}, {390, 876, 901},...

Cf. A138668, A138669, A045991.

A138668 Values of y in solutions (x,y,z) to the Diophantine equation x^3-x^2+y^3-y^2=z^3-z^2, with 1

Original entry on oeis.org

100, 301, 226, 305, 341, 770, 370, 876, 1232, 1800, 3451, 2044, 6816, 2666, 5876, 4459, 44982, 67082, 8350, 12840, 20724, 23571, 15840, 17100, 23001, 18190, 31093, 27756, 30176, 54245, 97019, 24708, 21582, 22584, 33429, 61201, 87814, 610305

Offset: 1

Manuel Valdivia, Apr 17 2008

nonn

A045991(y)=A045991(z)-A045991(x), x=A138667(n) and z=A138669(n).

Cf. A138667, A138669, A045991.

A138669 Values of z in solutions (x,y,z) to the Diophantine equation x^3-x^2+y^3-y^2=z^3-z^2, with 1

Original entry on oeis.org

108, 304, 250, 356, 389, 785, 444, 901, 1298, 1864, 3480, 2159, 6828, 3029, 6149, 5439, 44994, 67094, 9259, 13305, 20934, 23895, 16854, 18660, 23936, 19653, 32034, 28996, 31940, 54974, 97259, 28164, 26742, 27984, 37958, 62880, 88704, 610326

Offset: 1

Manuel Valdivia, Apr 17 2008

nonn

A045991(z)=A045991(x)+A045991(y), y=A138668(n) and x=A138667(n).

Cf. A138667, A138668, A045991.

A153258 n^3 - (n+2)^2.

Original entry on oeis.org

-4, -8, -8, 2, 28, 76, 152, 262, 412, 608, 856, 1162, 1532, 1972, 2488, 3086, 3772, 4552, 5432, 6418, 7516, 8732, 10072, 11542, 13148, 14896, 16792, 18842, 21052, 23428, 25976, 28702, 31612, 34712, 38008, 41506, 45212, 49132, 53272, 57638

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 22 2008

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

Cf. A045991, A081437, A069778, A153257

a[n_] := n^3-(n+2)^2; lst={}; Do[AppendTo[lst, a[n]], {n, 0, 5!}]; lst

G.f.: 2*x*(x^3+4*x-2)/(x-1)^4. [Colin Barker, Oct 08 2012]

A153259 a(n)=n^3-(3*(n+3))^2.

Original entry on oeis.org

-81, -143, -217, -297, -377, -451, -513, -557, -577, -567, -521, -433, -297, -107, 143, 459, 847, 1313, 1863, 2503, 3239, 4077, 5023, 6083, 7263, 8569, 10007, 11583, 13303, 15173, 17199, 19387, 21743, 24273, 26983, 29879, 32967, 36253, 39743

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 22 2008

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Cf. A045991, A081437, A069778, A153257, A153258.

a[n_]:=n^3-(3*(n+3))^2;lst={};Do[AppendTo[lst,a[n]],{n,0,5!}];lst
Table[n^3-(3(n+3))^2,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{-81,-143,-217,-297},40] (* Harvey P. Dale, Jul 10 2013 *)

a(n)=n^3-(3*n+9)^2 \\ Charles R Greathouse IV, Oct 18 2022

a(1)=-81, a(2)=-143, a(3)=-217, a(4)=-297, a(n)=4*a(n-1)-6*a(n-2)+ 4*a(n-3)- a(n-4). - Harvey P. Dale, Jul 10 2013

A154731 Integers of the form k = m^3-m^2 such that k-+1 are primes.

Original entry on oeis.org

4, 18, 180, 2028, 8820, 34848, 108288, 191748, 720900, 875520, 960498, 990000, 1355310, 1629108, 1713600, 1756920, 2334948, 2609028, 7376850, 8448048, 21639798, 37148148, 42023088, 48893940, 60544008, 63840000, 100328400

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 14 2009

nonn

Cf. A111503

lst={};Do[k=n^3-n^2;If[PrimeQ[k-1]&&PrimeQ[k+1],AppendTo[lst,p]],{n,7!}];lst

A045991 INTERSECT A014574. - R. J. Mathar, Jul 16 2022

cp's OEIS Frontend

A212971 Number of triples (w,x,y) with all terms in {0,...,n} and w < floor((x+y)/3).

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A304487 a(n) = (3 + 2*n - 3*n^2 + 4*n^3 - 3*((-1 + n) mod 2))/6.

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A132998 a(n) = n^4 - n^3 - n^2.

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A134631 a(n) = 5*n^5 - 3*n^3 + 2*n^2. Coefficients and exponents are the prime numbers in decreasing order.

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A138667 Values of x in solutions (x,y,z) to the Diophantine equation x^3-x^2+y^3-y^2=z^3-z^2, with 1

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A138668 Values of y in solutions (x,y,z) to the Diophantine equation x^3-x^2+y^3-y^2=z^3-z^2, with 1

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A138669 Values of z in solutions (x,y,z) to the Diophantine equation x^3-x^2+y^3-y^2=z^3-z^2, with 1

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A153258 n^3 - (n+2)^2.

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A304487 a(n) = (3 + 2n - 3n^2 + 4n^3 - 3((-1 + n) mod 2))/6.

A134631 a(n) = 5n^5 - 3n^3 + 2*n^2. Coefficients and exponents are the prime numbers in decreasing order.