A033582 -id:A033582 - cp's OEIS frontend

A092277 a(n) = 7*n^2 + n.

0, 8, 30, 66, 116, 180, 258, 350, 456, 576, 710, 858, 1020, 1196, 1386, 1590, 1808, 2040, 2286, 2546, 2820, 3108, 3410, 3726, 4056, 4400, 4758, 5130, 5516, 5916, 6330, 6758, 7200, 7656, 8126, 8610, 9108, 9620, 10146, 10686, 11240, 11808, 12390, 12986, 13596

Offset: 0

Evgeniy A. Chukhlomin (dkea(AT)yandex.ru), Feb 18 2004

First bisection of A219191. - Bruno Berselli, Nov 15 2012

			From _Bruno Berselli_, Oct 27 2017: (Start)
After 0:
8   =       -(1) + (2+3+4).
30  =     -(1+2) + (3+4+5+6+7+8).
66  =   -(1+2+3) + (4+5+6+7+8+9+10+11+12).
116 = -(1+2+3+4) + (5+6+7+8+9+10+11+12+13+14+15+16). (End)

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

Cf. A000290, A033582. - Omar E. Pol, Dec 22 2008

Cf. A022265, A219191.

a(n)=7*n^2+n \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 7*A000290(n) + n = A033582(n) + n. - Omar E. Pol, Dec 22 2008
a(n) = a(n-1) + 14*n - 6 with n > 0, a(0)=0. - Vincenzo Librandi, Nov 17 2010
From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: 2*x*(4 + 3*x)/(1-x)^3.
E.g.f.: exp(x)*x*(8 + 7*x).
a(n) = 2*A022265(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A103218 Triangle read by rows: T(n, k) = (2k+1)(n+1-k)^2.

Original entry on oeis.org

1, 4, 3, 9, 12, 5, 16, 27, 20, 7, 25, 48, 45, 28, 9, 36, 75, 80, 63, 36, 11, 49, 108, 125, 112, 81, 44, 13, 64, 147, 180, 175, 144, 99, 52, 15, 81, 192, 245, 252, 225, 176, 117, 60, 17, 100, 243, 320, 343, 324, 275, 208, 135, 68, 19, 121, 300, 405, 448, 441, 396, 325, 240

Offset: 0

Lambert Klasen (lambert.klasen(AT)gmx.de) and Gary W. Adamson, Jan 25 2005

The triangle is generated from the product A * B of the infinite lower triangular matrix A =
0 0 0...
1 0 0...
3 1 0...
5 3 1...
... and B =
0 0 0...
3 0 0...
3 5 0...
3 5 7...
...

			Triangle begins:
1,
4,3,
9,12,5,
16,27,20,7,
25,48,45,28,9,

Row sums give A002412 (hexagonal pyramidal numbers).
T(n, 0)=A000290(n+1) (the squares);
T(n, 1)=3*n^2=A033428(n);
T(n, 2)=5*n^2=A033429(n+1);
T(n, 3)=7*n^2=A033582(n+2);
Cf. A103219 (product B*A), A002412, A000290.

T[n_, k_] := (2*k + 1)*(n + 1 - k)^2; Flatten[ Table[ T[n, k], {n, 0, 10}, {k, 0, n}]] (* Robert G. Wilson v, Feb 10 2005 *)

T(n, k) = (2*k+1)*(n+1-k)^2; for(i=0,10, for(j=0,i,print1(T(i,j),","));print())

A296636 Sequences n(n+1)(6n+1)/2 and n(n+1)(7n+1)/2 interleaved.

Original entry on oeis.org

0, 7, 8, 39, 45, 114, 132, 250, 290, 465, 540, 777, 903, 1204, 1400, 1764, 2052, 2475, 2880, 3355, 3905, 4422, 5148, 5694, 6630, 7189, 8372, 8925, 10395, 10920, 12720, 13192, 15368, 15759, 18360, 18639, 21717, 21850, 25460, 25410, 29610, 29337, 34188, 33649, 39215, 38364, 44712

Offset: 0

Luce ETIENNE, Dec 17 2017

Difference between these subsequences is A002411.

This sequence gives numbers of triangles all sizes in every n-th stage [of what? - N. J. A. Sloane, Feb 09 2018].

Colin Barker, Table of n, a(n) for n = 0..1000
Luce ETIENNE, Illustration
Eric Weisstein's World of Mathematics, Polyiamond
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-6,0,4,0,-1).

Cf. A002411, A002717, A006331, A033581, A033582, A255211.

List([0..50], n -> (2*n+1-(-1)^n)*(2*n+5-(-1)^n)*(13*n+10+(n-6)*(-1)^n)/128); # Bruno Berselli, Feb 12 2018

[(2*n+1-(-1)^n)*(2*n+5-(-1)^n)*(13*n+10+(n-6)*(-1)^n)/128: n in [0..50]]; // Bruno Berselli, Feb 12 2018

CoefficientList[Series[x (7 + 8 x + 11 x^2 + 13 x^3)/((1 - x)^4*(1 + x)^4), {x, 0, 46}], x] (* Michael De Vlieger, Dec 18 2017 *)
LinearRecurrence[{0,4,0,-6,0,4,0,-1},{0,7,8,39,45,114,132,250},50] (* Harvey P. Dale, May 01 2018 *)
Rest[Flatten[Table[With[{c=(n(n+1))/2},{c*(6n+1),c*(7n+1)}],{n,0,30}]]] (* Harvey P. Dale, Oct 11 2020 *)

concat(0, Vec(x*(7 + 8*x + 11*x^2 + 13*x^3) / ((1 - x)^4*(1 + x)^4) + O(x^80))) \\ Colin Barker, Dec 18 2017

a(n) = a(n-1)+4*a(n-2)-4*a(n-3)-6*a(n-4)+6*a(n-5)+4*a(n-6)-4*a(n-7)-a(n-8)+a(n-9).
a(n) = (2*n+1-(-1)^n)*(2*n+5-(-1)^n)*(13*n+10+(n-6)*(-1)^n)/128.
From Colin Barker, Dec 18 2017: (Start)
G.f.: x*(7 + 8*x + 11*x^2 + 13*x^3) / ((1 - x)^4*(1 + x)^4).
a(n) = 4*a(n-2) - 6*a(n-4) + 4*a(n-6) - a(n-8) for n>7.
(End)

A303302 a(n) = 34*n^2.

Original entry on oeis.org

0, 34, 136, 306, 544, 850, 1224, 1666, 2176, 2754, 3400, 4114, 4896, 5746, 6664, 7650, 8704, 9826, 11016, 12274, 13600, 14994, 16456, 17986, 19584, 21250, 22984, 24786, 26656, 28594, 30600, 32674, 34816, 37026, 39304, 41650, 44064, 46546, 49096, 51714, 54400, 57154, 59976, 62866, 65824, 68850, 71944

Offset: 0

Omar E. Pol, May 13 2018

Sequence found by reading the line from 0, in the direction 0, 34, ..., in the square spiral whose vertices are the generalized 19-gonal numbers A303813.

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

Cf. A005843, A008599, A303813.

Cf. similar sequences of the type k*n^2: A000290 (k=1), A001105 (k=2), A033428 (k=3), A016742 (k=4), A033429 (k=5), A033581 (k=6), A033582 (k=7), A139098 (k=8), A016766 (k=9), A033583 (k=10), A033584 (k=11), A135453 (k=12), A152742 (k=13), A144555 (k=14), A064761 (k=15), A016802 (k=16), A244630 (k=17), A195321 (k=18), A244631 (k=19), A195322 (k=20), A064762 (k=21), A195323 (k=22), A244632 (k=23), A195824 (k=24), A016850 (k=25), A244633 (k=26), A244634 (k=27), A064763 (k=28), A244635 (k=29), A244636 (k=30), A244082 (k=32), this sequence (k=34), A016910 (k=36), A016982 (k=49), A017066 (k=64), A017162 (k=81), A017270 (k=100), A017390 (k=121), A017522 (k=144).

[34*n^2: n in [0..50]]; // Vincenzo Librandi Jun 07 2018

Table[34 n^2, {n, 0, 40}]
LinearRecurrence[{3,-3,1},{0,34,136},50] (* Harvey P. Dale, Jul 23 2018 *)

PARI
```
a(n) = 34*n^2;
```

concat(0, Vec(34*x*(1 + x) / (1 - x)^3 + O(x^40))) \\ Colin Barker, Jun 12 2018

a(n) = 34*A000290(n) = 17*A001105(n) = 2*A244630(n).
G.f.: 34*x*(1 + x)/(1 - x)^3. - Vincenzo Librandi, Jun 07 2018
From Elmo R. Oliveira, Dec 02 2024: (Start)
E.g.f.: 34*x*(1 + x)*exp(x).
a(n) = A005843(n)*A008599(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A363436 Array read by ascending antidiagonals: A(n, k) = k*n^2, with k >= 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 4, 2, 0, 0, 9, 8, 3, 0, 0, 16, 18, 12, 4, 0, 0, 25, 32, 27, 16, 5, 0, 0, 36, 50, 48, 36, 20, 6, 0, 0, 49, 72, 75, 64, 45, 24, 7, 0, 0, 64, 98, 108, 100, 80, 54, 28, 8, 0, 0, 81, 128, 147, 144, 125, 96, 63, 32, 9, 0, 0, 100, 162, 192, 196, 180, 150, 112, 72, 36, 10, 0

Offset: 0

Stefano Spezia, Jul 08 2023

			The array begins:
  0,  0,  0,   0,   0,   0,   0, ...
  0,  1,  2,   3,   4,   5,   6, ...
  0,  4,  8,  12,  16,  20,  24, ...
  0,  9, 18,  27,  36,  45,  54, ...
  0, 16, 32,  48,  64,  80,  96, ...
  0, 25, 50,  75, 100, 125, 150, ...
  0, 36, 72, 108, 144, 180, 216, ...
  ...

Cf. A000290 (k = 1), A001105 (k = 2), A033428 (k = 3), A016742 (k = 4), A033429 (k = 5), A033581 (k = 6), A033582 (k = 7), A139098 (k = 8), A016766 (k = 9), A033583 (k = 10), A033584 (k = 11), A135453 (k = 12),  A152742 (k = 13), A144555 (k = 14), A064761 (k = 15), A016802 (k = 16), A244630 (k = 17), A195321 (k = 18), A244631 (k = 19), A195322 (k = 20), A064762 (k = 21), A195323 (k = 22), A244632 (k = 23), A195824 (k = 24), A016850 (k = 25), A244633 (k = 26), A244634 (k = 27), A064763 (k = 28), A244635 (k = 29), A244636 (k = 30).
Cf. A001477 (n = 1), A008586 (n = 2), A008591 (n = 3), A008598 (n = 4), A008607 (n = 5), A044102 (n = 6), A152691 (n = 8).
Cf. A000007 (n = 0 or k = 0), A000578 (main diagonal), A002415 (antidiagonal sums), A004247.

A[n_,k_]:=k n^2; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten

O.g.f.: x*y*(1 + x)/((1 - x)^3*(1 - y)^2).
E.g.f.: x*y*(1 + x)*exp(x + y).
A(n, k) = n*A004247(n, k).

cp's OEIS Frontend

A092277 a(n) = 7*n^2 + n.

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A103218 Triangle read by rows: T(n, k) = (2*k+1)*(n+1-k)^2.

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A296636 Sequences n*(n+1)*(6*n+1)/2 and n*(n+1)*(7*n+1)/2 interleaved.

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A303302 a(n) = 34*n^2.

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Magma

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PARI

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A363436 Array read by ascending antidiagonals: A(n, k) = k*n^2, with k >= 0.

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Mathematica

Formula

A103218 Triangle read by rows: T(n, k) = (2k+1)(n+1-k)^2.

A296636 Sequences n(n+1)(6n+1)/2 and n(n+1)(7n+1)/2 interleaved.