A028895 -id:A028895 - cp's OEIS frontend

A195015 Main axis of the square spiral whose edges have length A195013 and whose vertices are the numbers A195014.

0, 2, 12, 24, 44, 66, 96, 128, 168, 210, 260, 312, 372, 434, 504, 576, 656, 738, 828, 920, 1020, 1122, 1232, 1344, 1464, 1586, 1716, 1848, 1988, 2130, 2280, 2432, 2592, 2754, 2924, 3096, 3276, 3458, 3648, 3840, 4040, 4242, 4452, 4664, 4884

Offset: 0

Omar E. Pol, Sep 26 2011

Sequence found by reading the line from 0, in the direction 0, 2, ..., and the same line from 0, in the direction 0, 12, ..., in the square spiral mentioned above. Axis perpendicular to A195016 in the same spiral.

Also four times A005475 and positives A152965 interleaved.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000566, A005475, A005476, A028895, A033571, A152965, A153126, A195013, A195014, A195016.

[(2*n*(5*n+2)+3*(-1)^n-3)/4: n in [0..50]]; // Vincenzo Librandi, Oct 28 2011

LinearRecurrence[{2, 0, -2, 1}, {0, 2, 12, 24}, 50] (* Paolo Xausa, Feb 09 2024 *)

From Bruno Berselli, Oct 14 2011: (Start)
G.f.: 2*x*(1+4*x)/((1+x)*(1-x)^3).
a(n) = (2*n*(5*n+2) + 3*(-1)^n-3)/4.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
a(n) + a(n-1) = A135706(n). (End)

A267370 Partial sums of A140091.

Original entry on oeis.org

0, 6, 21, 48, 90, 150, 231, 336, 468, 630, 825, 1056, 1326, 1638, 1995, 2400, 2856, 3366, 3933, 4560, 5250, 6006, 6831, 7728, 8700, 9750, 10881, 12096, 13398, 14790, 16275, 17856, 19536, 21318, 23205, 25200, 27306, 29526, 31863, 34320, 36900, 39606, 42441, 45408, 48510

Offset: 0

Bruno Berselli, Jan 13 2016

After 0, this sequence is the third column of the array in A185874.

Sequence is related to A051744 by A051744(n) = n*a(n)/3 - Sum_{i=0..n-1} a(i) for n>0.

			The sequence is also provided by the row sums of the following triangle (see the fourth formula above):
.  0;
.  1,  5;
.  4,  7, 10;
.  9, 11, 13, 15;
. 16, 17, 18, 19, 20;
. 25, 25, 25, 25, 25, 25;
. 36, 35, 34, 33, 32, 31, 30;
. 49, 47, 45, 43, 41, 39, 37, 35;
. 64, 61, 58, 55, 52, 49, 46, 43, 40;
. 81, 77, 73, 69, 65, 61, 57, 53, 49, 45, etc.
First column is A000290.
Second column is A027690.
Third column is included in A189834.
Main diagonal is A008587; other parallel diagonals: A016921, A017029, A017077, A017245, etc.
Diagonal 1, 11, 25, 43, 65, 91, 121, ... is A161532.

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

Cf. A005581, A051744, A105163, A140091, A185874.

Cf. similar sequences of the type n*(n+1)*(n+k)/2: A002411 (k=0), A006002 (k=1), A027480 (k=2), A077414 (k=3, with offset 1), A212343 (k=4, without the initial 0), this sequence (k=5).

Magma
```
[n*(n+1)*(n+5)/2: n in [0..50]];
```

Table[n (n + 1) (n + 5)/2, {n, 0, 50}]
LinearRecurrence[{4,-6,4,-1},{0,6,21,48},50] (* Harvey P. Dale, Jul 18 2019 *)

PARI
```
vector(50, n, n--; n*(n+1)*(n+5)/2)
```
Sage
```
[n*(n+1)*(n+5)/2 for n in (0..50)]
```

O.g.f.: 3*x*(2 - x)/(1 - x)^4.
E.g.f.: x*(12 + 9*x + x^2)*exp(x)/2.
a(n) = n*(n + 1)*(n + 5)/2.
a(n) = Sum_{i=0..n} n*(n - i) + 5*i, that is: a(n) = A002411(n) + A028895(n). More generally, Sum_{i=0..n} n*(n - i) + k*i = n*(n + 1)*(n + k)/2.
a(n) = 3*A005581(n+1).
a(n+1) - 3*a(n) + 3*a(n-1) = 3*A105163(n) for n>0.
From Amiram Eldar, Jan 06 2021: (Start)
Sum_{n>=1} 1/a(n) = 163/600.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/5 - 253/600. (End)

A195016 a(n) = (n(5n+7)-(-1)^n+1)/2.

Original entry on oeis.org

0, 7, 17, 34, 54, 81, 111, 148, 188, 235, 285, 342, 402, 469, 539, 616, 696, 783, 873, 970, 1070, 1177, 1287, 1404, 1524, 1651, 1781, 1918, 2058, 2205, 2355, 2512, 2672, 2839, 3009, 3186, 3366, 3553, 3743, 3940, 4140, 4347, 4557, 4774, 4994

Offset: 0

Omar E. Pol, Sep 26 2011

Sequence found by reading the line from 0, in the direction 0, 7,..., and the same line from 0, in the direction 0, 17,..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. Axis perpendicular to the main axis A195015 in the same spiral.

Also sequence found by reading the line from 0, in the direction 0, 7,..., and the same line from 0, in the direction 0, 17,..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. This line is parallel to A153126 in the same spiral.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000566, A005476, A028895, A033571, A085787, A152965, A153126, A195013, A195014, A195015.

&cat[[n*t,(n+1)*t] where t is 10*n+7: n in [0..22]]; // Bruno Berselli, Oct 14 2011

LinearRecurrence[{2, 0, -2, 1}, {0, 7, 17, 34}, 50] (* Paolo Xausa, Feb 09 2024 *)

n*(10*n-3), if n >= 1, and (2*n+1)*(5*n+1)-1, if n >= 0, interleaved.

G.f.: x*(7+3*x)/((1+x)*(1-x)^3). - Bruno Berselli, Oct 14 2011

Concise definition by Bruno Berselli, Oct 14 2011

A124110 Primes of the form A124080 (10 times triangular numbers) +- 1.

Original entry on oeis.org

11, 29, 31, 59, 61, 101, 149, 151, 211, 281, 359, 449, 659, 661, 911, 1049, 1051, 1201, 1361, 1531, 1709, 1901, 2099, 2309, 2311, 2531, 2999, 3001, 3251, 3511, 3779, 4349, 4649, 4651, 5279, 5281, 6299, 6301, 6659, 6661, 7411, 8609, 9029, 9461, 9901, 11279

Offset: 1

Jonathan Vos Post, Nov 26 2006

Numbers j such that A124080(j)-1 is prime or A124080(j)+1 is prime, where repetition means a twin prime, are 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 8, 9, 11, 11, 13, 14, 14, 15, 16, 17, 18, 19, 20, 21, 21, 22, 24, 24, 25, ..., . - Robert G. Wilson v, Nov 29 2006

			a(1) = A124080(1)+1 = (10*T(1)) - 1 = 10*(1*(1+1)/2) + 1 = 10+1 = 11 is prime.
a(2) = A124080(2)-1 = (10*T(2))-1 = 10*(2*(2+1)/2) - 1 = 30-1 = 29 is prime.
a(3) = A124080(2)+1 = (10*T(2))+1 = 10*(2*(2+1)/2) + 1 = 30+1 = 31 is prime.

Cf. A000040, A000217, A028895, A046092, A045943, A002378, A028896, A024966, A033996, A027468.

s = {}; Do[t = 5n(n + 1); If[PrimeQ[t - 1], AppendTo[s, t - 1]]; If[PrimeQ[t + 1], AppendTo[s, t + 1]], {n, 47}]; s (* Robert G. Wilson v *)

{A124080(j)-1 when prime} U {A124080(j)+1 when prime} = {i = 10*T(j)-1 such that i is prime} U {i = 10*T(j)+1 such that i is prime} where T(j) = A000217(j) = j*(j+1)/2.

More terms from Robert G. Wilson v, Nov 29 2006

A238738 Expansion of (1 + 2x + 2x^2)/(1 - x - 2x^3 + 2x^4 + x^6 - x^7).

Original entry on oeis.org

1, 3, 5, 7, 11, 15, 18, 24, 30, 34, 42, 50, 55, 65, 75, 81, 93, 105, 112, 126, 140, 148, 164, 180, 189, 207, 225, 235, 255, 275, 286, 308, 330, 342, 366, 390, 403, 429, 455, 469, 497, 525, 540, 570, 600, 616, 648, 680, 697, 731, 765, 783, 819, 855, 874

Offset: 0

Bruno Berselli, Mar 04 2014

Subsequence of A008732: a(n) = A008732(A047212(n+1)).

See also Deléham's example in A008732: these numbers are in the first (A000566), third (A005475) and fifth (A028895) column.

			G.f.: 1 + 3*x + 5*x^2 + 7*x^3 + 11*x^4 + 15*x^5 + 18*x^6 + 24*x^7 + ...

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of the initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Cf. A000212 (see illustration above), A000217, A008732, A211538.

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+2*x+2*x^2)/(1-x-2*x^3+2*x^4+x^6-x^7)));

CoefficientList[Series[(1 + 2 x + 2 x^2)/(1 - x - 2 x^3 + 2 x^4 + x^6 - x^7), {x, 0, 60}], x]

makelist(coeff(taylor((1+2*x+2*x^2)/(1-x-2*x^3+2*x^4+x^6-x^7), x, 0, n), x, n), n, 0, 60);

Vec((1+2*x+2*x^2)/(1-x-2*x^3+2*x^4+x^6-x^7)+O(x^60))

m = 60; L. = PowerSeriesRing(ZZ, m); f = (1+2*x+2*x^2)/(1-x-2*x^3+2*x^4+x^6-x^7); print(f.coefficients())

G.f.: (1 + 2*x + 2*x^2) / ((1 - x)^3*(1 + x + x^2)^2).
a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7), with n>6.
a(3k)   = k*(5*k +  7)/2 + 1 (A000566);
a(3k+1) = k*(5*k + 11)/2 + 3 (A005475);
a(3k+2) = k*(5*k + 15)/2 + 5 (A028895).
a(n) = (floor(n/3)+1)*(4*n-7*floor(n/3)+2)/2. [Luce ETIENNE, Jun 14 2014]

A269457 a(n) = 5(n + 1)(n + 4)/2.

Original entry on oeis.org

10, 25, 45, 70, 100, 135, 175, 220, 270, 325, 385, 450, 520, 595, 675, 760, 850, 945, 1045, 1150, 1260, 1375, 1495, 1620, 1750, 1885, 2025, 2170, 2320, 2475, 2635, 2800, 2970, 3145, 3325, 3510, 3700, 3895, 4095, 4300, 4510, 4725, 4945, 5170, 5400, 5635

Offset: 0

Ilya Gutkovskiy, Feb 27 2016

More generally, the ordinary generating function for the sequences of the form k*(n + 1)*(n - 1 + k)/2 is (k*(k - 1)/2 + (k*(3 - k)/2)*x)/(1 - x)^3 (see links section).

			a(0) = 0 + 1 + 2 + 3 + 4 = 10;
a(1) = 0 + 1 + 2 + 3 + 4 + 1 + 2 + 3 + 4 + 5 = 25;
a(2) = 0 + 1 + 2 + 3 + 4 + 1 + 2 + 3 + 4 + 5 + 2 + 3 + 4 + 5 + 6 = 45, etc.

Ilya Gutkovskiy, Sequences of the form k*(n + 1)*(n - 1 + k)/2.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000096, A008587, A028895, A040002, A045943, A054000, A055998, A067728.

[5*(n+1)*(n+4)/2: n in [0..50]]; // Vincenzo Librandi, Feb 28 2016

Table[5 (n + 1) ((n + 4)/2), {n, 0, 45}]
Table[Sum[5 (k + 2), {k, 0, n}], {n, 0, 45}]
LinearRecurrence[{3, -3, 1}, {10, 25, 45}, 46]

a(n) = 5*(n + 1)*(n + 4)/2; \\ Michel Marcus, Feb 29 2016

Vec(5*(2-x)/(1-x)^3 + O(x^100)) \\ Altug Alkan, Mar 04 2016

G.f.: 5*(2 - x)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = Sum_{k=0..n} 5*(k + 2) = Sum_{k=0..n} A008587(k + 2).
Sum_{n>=0} 1/a(n) = 11/45 = 0.24444444444... = A040002.
a(n) = 5*A000096(n+1).
a(n) = A055998(2*n+2) + A055998(n+1). - Bruno Berselli, Sep 23 2016
E.g.f.: 5*exp(x)*(4 + 6*x + x^2)/2. - Elmo R. Oliveira, Dec 24 2024

A360176 Triangle read by rows. T(n, k) = Sum_{j=k..n} binomial(n, j) * (-j)^(n - j) * (-1)^(j - k)* A360177(j, k).

Original entry on oeis.org

1, 0, 1, 0, -5, 1, 0, 37, -15, 1, 0, -393, 223, -30, 1, 0, 5481, -3815, 745, -50, 1, 0, -95053, 76051, -18870, 1865, -75, 1, 0, 1975821, -1749811, 514381, -65730, 3920, -105, 1, 0, -47939601, 45876335, -15316854, 2358181, -183610, 7322, -140, 1

Offset: 0

Peter Luschny, Jan 28 2023

			Triangle T(n, k) starts:
[0] 1;
[1] 0,         1;
[2] 0,        -5,        1;
[3] 0,        37,      -15,         1;
[4] 0,      -393,      223,       -30,       1;
[5] 0,      5481,    -3815,       745,     -50,       1;
[6] 0,    -95053,    76051,    -18870,    1865,     -75,    1;
[7] 0,   1975821, -1749811,    514381,  -65730,    3920, -105,    1;
[8] 0, -47939601, 45876335, -15316854, 2358181, -183610, 7322, -140, 1;

Cf. A360177, A273954 (column 1), A028895 (subdiagonal).

T := (n, k) -> add(binomial(n, j) * (-j)^(n - j) * (-1)^(j - k) * A360177(j, k), j = k..n): for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
# Alternative:
egf := k -> (1 - exp(-LambertW(x*exp(-x))))^k / k!:
ser := k -> series(egf(k), x, 22): T := (n, k) -> n!*coeff(ser(k), x, n):
for n from 0 to 8 do seq(T(n, k), k = 0..n) od;

E.g.f. of column k: (1 - exp(-LambertW(x*exp(-x))))^k / k!.

A086922 Number of idempotent n X n (0,1) matrices over the reals.

Original entry on oeis.org

1, 2, 8, 50, 452, 5682, 96608, 2185738, 65108492

Offset: 0

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 19 2003

From Torlach Rush, Jun 18 2020: (Start)
Let m(n,k) be the number of idempotent n X n (0,1) matrices with k entries equal to 1. Then:
k    | m(n,k)
-----|------------------------------------------------------
0    | 1
1    | n
2    | A028895(n - 1)
3    | 19 * A000292(n - 2)
4    | ((n - 3) (n - 2) (n - 1) (35 n - 124))/8
5    | ((n - 4) (n - 3) (n - 2) (n - 1) (631 n - 2675))/120
...
Conjecture: There is no closed form expression for this sequence.
(End)

Cf. A000292, A028895, A132186, A222821.

a(5)-a(6) from Torlach Rush, Jun 17 2020

a(7)-a(8) from A222821 added by Giovanni Resta, Jun 23 2020

A330082 a(n) = 5*A064038(n+1).

Original entry on oeis.org

0, 5, 15, 15, 25, 75, 105, 70, 90, 225, 275, 165, 195, 455, 525, 300, 340, 765, 855, 475, 525, 1155, 1265, 690, 750, 1625, 1755, 945, 1015, 2175, 2325, 1240, 1320, 2805, 2975, 1575, 1665, 3515, 3705, 1950, 2050, 4305, 4515, 2365, 2475, 5175, 5405, 2820, 2940

Offset: 0

Paul Curtz, Dec 01 2019

Main column of a pentagonal spiral for A026741:
                       (25)
                    49 (15) 31
                24  29 (15)  8  16
            47  14   7 ( 5)  3  17  33
        23  27  13   2 ( 0)  1   7   9  17
          45  13   6   3   1   4  19  35
            22  25  11   5   9  10  18
              43  12  23  11  21  37
                21  41  20  39  19
a(n) = 5 * A064038(n+1) from a pentagonal spiral.
Compare to A319127 =  6 * A002620 in the hexagonal spiral:
                22  23  23  22 (24)
              20  12  13  13 (12) 25
            21  10   5   4 ( 6) 14  25
          21  11   5   1 ( 0)  7  15  24
        20  11   4   1 ( 0)  2   7  15  26
          18  10   2   3   3   6  14  27
            19   8   9   9   8  16  27
              19  18  16  17  17  26
                30  28  29  29  28

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-6,10,-12,12,-10,6,-3,1).

Cf. A026741, A028895, A064038. A033429, A062786, A087348, A147874, A158447, A168668 are in the spiral.

A330082[n_]:=5Numerator[n(n+1)/4];Array[A330082,100,0] (* Paolo Xausa, Dec 04 2023 *)

concat(0, Vec(5*x*(1 + 4*x^3 + x^6) / ((1 - x)^3*(1 + x^2)^3) + O(x^50))) \\ Colin Barker, Dec 08 2019

a(n) = A026741(A028895(n)).
From Colin Barker, Dec 08 2019: (Start)
G.f.: 5*x*(1 + 4*x^3 + x^6) / ((1 - x)^3*(1 + x^2)^3).
a(n) = 3*a(n-1) - 6*a(n-2) + 10*a(n-3) - 12*a(n-4) + 12*a(n-5) - 10*a(n-6) + 6*a(n-7) - 3*a(n-8) + a(n-9) for n>8.
a(n) = (-5/16 + (5*i)/16)*(((-3-3*i) + (-i)^n + i^(1+n))*n*(1+n)) where i=sqrt(-1).
(End)

More terms from Colin Barker, Dec 22 2019

Name corrected by Paolo Xausa, Dec 04 2023

A342381 Triangle read by rows: T(n,k) is the number of symmetries of the n-dimensional hypercube that fix exactly 2*k facets; n,k >= 0.

Original entry on oeis.org

1, 1, 1, 5, 2, 1, 29, 15, 3, 1, 233, 116, 30, 4, 1, 2329, 1165, 290, 50, 5, 1, 27949, 13974, 3495, 580, 75, 6, 1, 391285, 195643, 48909, 8155, 1015, 105, 7, 1, 6260561, 3130280, 782572, 130424, 16310, 1624, 140, 8, 1, 112690097, 56345049, 14086260, 2347716, 293454, 29358, 2436, 180, 9, 1

Offset: 0

Peter Kagey, Mar 09 2021

Equivalently the number of symmetries of the n-dimensional cross-polytope that fix exactly 2*k vertices.

If a facet of the hypercube is fixed, then the opposite facet must also be fixed.

			Table begins:
n\k |         0        1        2       3      4     5    6   7 8 9
----+--------------------------------------------------------------
  0 |         1
  1 |         1        1
  2 |         5        2        1
  3 |        29       15        3       1
  4 |       233      116       30       4      1
  5 |      2329     1165      290      50      5     1
  6 |     27949    13974     3495     580     75     6    1
  7 |    391285   195643    48909    8155   1015   105    7   1
  8 |   6260561  3130280   782572  130424  16310  1624  140   8 1
  9 | 112690097 56345049 14086260 2347716 293454 29358 2436 180 9 1
For the cube in n=2 dimensions (the square) there is
T(2,2) = 1 symmetry that fixes all 2*2 = 4 sides, namely the identity:
     2
   +---+
  3|   |1;
   +---+
     4
T(2,1) = 2 symmetries that fix 2*1 = 2 sides, namely horizonal/vertical flips:
     4           2
   +---+       +---+
  3|   |1 and 1|   |3;
   +---+       +---+
     2           4
and T(2,0) = 5 symmetries that fix 2*0 = 0 sides, namely rotations or diagonal flips:
     1         4         3         3            1
   +---+     +---+     +---+     +---+        +---+
  2|   |4,  1|   |3,  4|   |2,  2|   |4, and 4|   |2.
   +---+     +---+     +---+     +---+        +---+
     3         2         1         1            3

Peter Kagey, Rows n = 0..100, flattened
Wikipedia, Cross-polytope
Wikipedia, Hypercube
Wikipedia, Hyperoctahedral group

Columns and diagonals: A000354 (k=0), A161937 (k=1), A028895 (n=k+2).
Row sums are A000165.
Cf. A001147, A114320.

f(n) = sum(k=0, n, (-1)^(n+k)*binomial(n, k)*k!*2^k); \\ A000354
T(n, k) = f(n-k)*binomial(n, k); \\ Michel Marcus, Mar 10 2021

T(n,k) = A114320(2n,k)/A001147(n).

T(n,k) = A000354(n-k)*binomial(n,k).

cp's OEIS Frontend

A195015 Main axis of the square spiral whose edges have length A195013 and whose vertices are the numbers A195014.

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A267370 Partial sums of A140091.

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A195016 a(n) = (n*(5*n+7)-(-1)^n+1)/2.

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A124110 Primes of the form A124080 (10 times triangular numbers) +- 1.

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A238738 Expansion of (1 + 2*x + 2*x^2)/(1 - x - 2*x^3 + 2*x^4 + x^6 - x^7).

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A269457 a(n) = 5*(n + 1)*(n + 4)/2.

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A360176 Triangle read by rows. T(n, k) = Sum_{j=k..n} binomial(n, j) * (-j)^(n - j) * (-1)^(j - k)* A360177(j, k).

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A195016 a(n) = (n(5n+7)-(-1)^n+1)/2.

A238738 Expansion of (1 + 2x + 2x^2)/(1 - x - 2x^3 + 2x^4 + x^6 - x^7).

A269457 a(n) = 5(n + 1)(n + 4)/2.