A228958 -id:A228958 - cp's OEIS frontend

A305189 a(n) = 12 + 3 + 45 + 6 + 78 + 9 + 1011 + 12 + ... + (up to n).

1, 2, 5, 9, 25, 31, 38, 87, 96, 106, 206, 218, 231, 400, 415, 431, 687, 705, 724, 1085, 1106, 1128, 1612, 1636, 1661, 2286, 2313, 2341, 3125, 3155, 3186, 4147, 4180, 4214, 5370, 5406, 5443, 6812, 6851, 6891, 8491, 8533, 8576, 10425, 10470, 10516, 12632

Offset: 1

Wesley Ivan Hurt, Sep 15 2018

			a(1) = 1;
a(2) = 1*2 = 2;
a(3) = 1*2 + 3 = 5;
a(4) = 1*2 + 3 + 4 = 9;
a(5) = 1*2 + 3 + 4*5 = 25;
a(6) = 1*2 + 3 + 4*5 + 6 = 31;
a(7) = 1*2 + 3 + 4*5 + 6 + 7 = 38;
a(8) = 1*2 + 3 + 4*5 + 6 + 7*8 = 87;
a(9) = 1*2 + 3 + 4*5 + 6 + 7*8 + 9 = 96;
a(10) = 1*2 + 3 + 4*5 + 6 + 7*8 + 9 + 10 = 106;
a(11) = 1*2 + 3 + 4*5 + 6 + 7*8 + 9 + 10*11 = 206;
a(12) = 1*2 + 3 + 4*5 + 6 + 7*8 + 9 + 10*11 + 12 = 218; etc.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,3,-3,0,-3,3,0,1,-1).

Cf. A093361, A228958, A319014.

seq(coeff(series((x*(1+x+3*x^2+x^3+13*x^4-3*x^5-2*x^6+4*x^7))/((1-x)^4*(1+x+x^2)^3),x,n+1), x, n), n = 1 .. 50); # Muniru A Asiru, Sep 16 2018

Table[3*Floor[n/3]*(Floor[n/3] + 1)/2 + Floor[(n + 1)/3]*(3*Floor[(n + 1)/3]^2 - 1) + n*(Floor[(n - 1)/3] - Floor[(n - 2)/3]), {n, 50}]
LinearRecurrence[{1,0,3,-3,0,-3,3,0,1,-1 }, {1, 2, 5, 9, 25, 31, 38, 87, 96, 106}, 50] (* Stefano Spezia, Sep 16 2018 *)

Vec(x*(1 + x + 3*x^2 + x^3 + 13*x^4 - 3*x^5 - 2*x^6 + 4*x^7) / ((1 - x)^4*(1 + x + x^2)^3) + O(x^40)) \\ Colin Barker, Sep 16 2018

a(n) = 3*floor(n/3)*(floor(n/3) + 1)/2 + floor((n+1)/3)*(3*floor((n+1)/3)^2 - 1) + n*(floor((n-1)/3) - floor((n-2)/3)).
From Colin Barker, Sep 16 2018: (Start)
G.f.: x*(1 + x + 3*x^2 + x^3 + 13*x^4 - 3*x^5 - 2*x^6 + 4*x^7) / ((1 - x)^4*(1 + x + x^2)^3).
a(n) = a(n-1) + 3*a(n-3) - 3*a(n-4) - 3*a(n-6) + 3*a(n-7) + a(n-9) - a(n-10) for n>10.
(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

A318868 a(n) = 1^2 + 3^4 + 5^6 + 7^8 + 9^10 + 11^12 + 13^14 + ... + (up to n).

Original entry on oeis.org

1, 1, 4, 82, 87, 15707, 15714, 5780508, 5780517, 3492564909, 3492564920, 3141920941630, 3141920941643, 3940518306640919, 3940518306640934, 6572348874019531544, 6572348874019531561, 14069656800941744522553, 14069656800941744522572, 37604043114346899937878154

Offset: 1

Wesley Ivan Hurt, Sep 16 2018

			a(1) = 1;
a(2) = 1^2 = 1;
a(3) = 1^2 + 3 = 4;
a(4) = 1^2 + 3^4 = 82;
a(5) = 1^2 + 3^4 + 5 = 87;
a(6) = 1^2 + 3^4 + 5^6 = 15707;
a(7) = 1^2 + 3^4 + 5^6 + 7 = 15714;
a(8) = 1^2 + 3^4 + 5^6 + 7^8 = 5780508;
a(9) = 1^2 + 3^4 + 5^6 + 7^8 + 9 = 5780517;
a(10) = 1^2 + 3^4 + 5^6 + 7^8 + 9^10 = 3492564909;
a(11) = 1^2 + 3^4 + 5^6 + 7^8 + 9^10 + 11 = 3492564920;
a(12) = 1^2 + 3^4 + 5^6 + 7^8 + 9^10 + 11^12 = 3141920941630, etc.

Colin Barker, Table of n, a(n) for n = 1..350

Cf. A093361, A228958, A305189.

Table[(2*Floor[(n - 1)/2] + 1)*Mod[n, 2] + Sum[(2*i - 1)^(2*i), {i, Floor[n/2]}], {n, 25}]

a(n) = (2*((n-1)\2) + 1)*(n % 2) + sum(i=1, n\2, (2*i - 1)^(2*i)); \\ Michel Marcus, Sep 18 2018

a(n) = (2*floor((n-1)/2) + 1)*(n mod 2) + Sum_{i=1..floor(n/2)} (2*i - 1)^(2*i).

A319258 a(n) = 1 + 23 + 4 + 56 + 7 + 89 + 10 + 1112 + ... + (up to n).

Original entry on oeis.org

1, 3, 7, 11, 16, 41, 48, 56, 120, 130, 141, 262, 275, 289, 485, 501, 518, 807, 826, 846, 1246, 1268, 1291, 1820, 1845, 1871, 2547, 2575, 2604, 3445, 3476, 3508, 4532, 4566, 4601, 5826, 5863, 5901, 7345, 7385, 7426, 9107, 9150, 9194, 11130, 11176, 11223

Offset: 1

Wesley Ivan Hurt, Sep 16 2018

			a(1) = 1;
a(2) = 1 + 2 = 3;
a(3) = 1 + 2*3 = 7;
a(4) = 1 + 2*3 + 4 = 11;
a(5) = 1 + 2*3 + 4 + 5 = 16;
a(6) = 1 + 2*3 + 4 + 5*6 = 41;
a(7) = 1 + 2*3 + 4 + 5*6 + 7 = 48;
a(8) = 1 + 2*3 + 4 + 5*6 + 7 + 8 = 56;
a(9) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 = 120;
a(10) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 = 130;
a(11) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 + 11 = 141;
a(12) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 + 11*12 = 262; etc.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,3,-3,0,-3,3,0,1,-1).

Cf. A093361, A228958, A305189, A319014.

Table[n (1 + Floor[(n - 2)/3] - Floor[n/3]) + 3 Floor[n/3]^2 (1 + Floor[n/3]) + Floor[(n + 2)/3] (3 Floor[(n + 2)/3] - 1)/2, {n, 50}]

Vec(x*(1 + 2*x + 4*x^2 + x^3 - x^4 + 13*x^5 - 2*x^6 - x^7 + x^8) / ((1 - x)^4*(1 + x + x^2)^3) + O(x^40)) \\ Colin Barker, Sep 16 2018

a(n) = n*(1 + floor((n-2)/3) - floor(n/3)) + 3*floor(n/3)^2*(1 + floor(n/3)) + floor((n+2)/3)*(3*floor((n+2)/3) - 1)/2.
From Colin Barker, Sep 16 2018: (Start)
G.f.: x*(1 + 2*x + 4*x^2 + x^3 - x^4 + 13*x^5 - 2*x^6 - x^7 + x^8) / ((1 - x)^4*(1 + x + x^2)^3).
a(n) = a(n-1) + 3*a(n-3) - 3*a(n-4) - 3*a(n-6) + 3*a(n-7) + a(n-9) - a(n-10) for n>10.
(End)

A300254 a(n) = 25(n + 1)(4n + 3)(5*n + 4)/3.

Original entry on oeis.org

100, 1050, 3850, 9500, 19000, 33350, 53550, 80600, 115500, 159250, 212850, 277300, 353600, 442750, 545750, 663600, 797300, 947850, 1116250, 1303500, 1510600, 1738550, 1988350, 2261000, 2557500, 2878850, 3226050, 3600100, 4002000, 4432750, 4893350, 5384800, 5908100, 6464250

Offset: 0

Bruno Berselli, Mar 12 2018

Hirschhorn has discovered that p(20*n+11,4) + p(20*n+12,4) + p(20*n+13,4) = 25*(n + 1)*(4*n + 3)*(5*n + 4)/3, where p(m,k) denote the number of partitions of m into at most k parts. Therefore, p(20*n+11,4) + p(20*n+12,4) + p(20*n+13,4) == 0 (mod 50) [see Hirschhorn's paper in References section].
a(n) == 0 (mod 3) if n is of the form 2*h + 3*floor(h/3 + 2/3) + 1.
a(n) == 0 (mod 7) if n is a member of A047278.

Michael D. Hirschhorn, Congruences modulo 5 for partitions into at most four parts, The Fibonacci Quarterly, Vol. 56, Number 1, 2018, pages 32-37 [the equation 1.7 contains a typo].

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

Subsequence of A014112, A212964, A228958, A268684.

Cf. A300522, A300523.

List([0..40], n -> 25*(n+1)*(4*n+3)*(5*n+4)/3);

[div(25*(n+1)*(4*n+3)*(5*n+4), 3) for n in 0:40] |> println

[25*(n+1)*(4*n+3)*(5*n+4)/3: n in [0..40]];

Table[25 (n + 1) (4 n + 3) (5 n + 4)/3, {n, 0, 40}]

makelist(25*(n+1)*(4*n+3)*(5*n+4)/3, n, 0, 40);

vector(40, n, n--; 25*(n+1)*(4*n+3)*(5*n+4)/3)

Vec(50*(2 + 13*x + 5*x^2) / (1 - x)^4 + O(x^60)) \\ Colin Barker, Mar 13 2018

[25*(n+1)*(4*n+3)*(5*n+4)/3 for n in range(40)]

[25*(n+1)*(4*n+3)*(5*n+4)/3 for n in (0..40)]

O.g.f.: 50*(2 + 13*x + 5*x^2)/(1 - x)^4 [formula 4.3 in Hirschhorn's paper].
E.g.f.: 25*(12 + 114*x + 111*x^2 + 20*x^3)*exp(x)/3.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4)
a(n) = A014112(10*n+8) = A212964(10*n+9) = A228958(10*n+8) = A268684(5*n+4).

A319438 a(n) = 1^2 - 3^4 + 5^6 - 7^8 + 9^10 - 11^12 + 13^14 - ... + (up to n).

Original entry on oeis.org

1, 1, -2, -80, -75, 15545, 15538, -5749256, -5749247, 3481035145, 3481035134, -3134947341576, -3134947341563, 3934241438357713, 3934241438357698, -6564474114274532912, -6564474114274532895, 14056519977953450458097, 14056519977953450458078

Offset: 1

Wesley Ivan Hurt, Sep 18 2018

An alternating version of A318868.

			   a(1) = 1;
   a(2) = 1^2 = 1;
   a(3) = 1^2 - 3 = -2;
   a(4) = 1^2 - 3^4 = -80;
   a(5) = 1^2 - 3^4 + 5 = -75;
   a(6) = 1^2 - 3^4 + 5^6 = 15545;
   a(7) = 1^2 - 3^4 + 5^6 - 7 = 15538;
   a(8) = 1^2 - 3^4 + 5^6 - 7^8 = -5749256;
   a(9) = 1^2 - 3^4 + 5^6 - 7^8 + 9 = -5749247;
  a(10) = 1^2 - 3^4 + 5^6 - 7^8 + 9^10 = 3481035145;
  a(11) = 1^2 - 3^4 + 5^6 - 7^8 + 9^10 - 11 = 3481035134;
  a(12) = 1^2 - 3^4 + 5^6 - 7^8 + 9^10 - 11^12 = -3134947341576; etc .

Colin Barker, Table of n, a(n) for n = 1..350

Cf. A093361, A228958, A305189, A318868.

Table[n*Mod[n, 2]*(-1)^(Floor[n/2]) + Sum[(2*i - 1)^(2*i)*(-1)^(i - 1), {i, Floor[n/2]}], {n, 30}]

a(n) = n*(n mod 2)*(-1)^floor(n/2) + Sum_{i=1..floor(n/2)} (2*i - 1)^(2*i)*(-1)^(i - 1).

cp's OEIS Frontend

A305189 a(n) = 1*2 + 3 + 4*5 + 6 + 7*8 + 9 + 10*11 + 12 + ... + (up to n).

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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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A318868 a(n) = 1^2 + 3^4 + 5^6 + 7^8 + 9^10 + 11^12 + 13^14 + ... + (up to n).

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A319258 a(n) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 + 11*12 + ... + (up to n).

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A300254 a(n) = 25*(n + 1)*(4*n + 3)*(5*n + 4)/3.

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A319438 a(n) = 1^2 - 3^4 + 5^6 - 7^8 + 9^10 - 11^12 + 13^14 - ... + (up to n).

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A305189 a(n) = 12 + 3 + 45 + 6 + 78 + 9 + 1011 + 12 + ... + (up to n).

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

A319258 a(n) = 1 + 23 + 4 + 56 + 7 + 89 + 10 + 1112 + ... + (up to n).

A300254 a(n) = 25(n + 1)(4n + 3)(5*n + 4)/3.