A144391 -id:A144391 - cp's OEIS frontend

A144390 a(n) = 3*n^2 - n - 1.

1, 9, 23, 43, 69, 101, 139, 183, 233, 289, 351, 419, 493, 573, 659, 751, 849, 953, 1063, 1179, 1301, 1429, 1563, 1703, 1849, 2001, 2159, 2323, 2493, 2669, 2851, 3039, 3233, 3433, 3639, 3851, 4069, 4293, 4523, 4759, 5001, 5249, 5503, 5763, 6029, 6301, 6579

Offset: 1

Paul Curtz, Oct 02 2008

Sequence's original Name was "First bisection of A135370."

The partial sums of this sequence give A081437. - Leo Tavares, Dec 26 2021

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
John Elias, Illustration: Belted Hexagrams
Leo Tavares, Illustration: Bounded Hexagons
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016933, A049450, A144391.
Cf. A003215, A005408, A016754, A002378.
Cf. A081437 (partial sums).

[3*n^2-n-1: n in [1..60]]; // Vincenzo Librandi, Jun 14 2011

A144390:=n->3*n^2-n-1; seq(A144390(n), n=1..50); # Wesley Ivan Hurt, Mar 26 2014

Table[3*n^2 -n -1 , {n,0,50}] (* G. C. Greubel, Jul 19 2017 *)

a(n)=3*n^2-n-1 \\ Charles R Greathouse IV, Oct 07 2015

a(n+1) = a(n) + 6*n + 2; see A016933.
G.f.: x*(1+6*x-x^2)/(1-x)^3. a(n) = A049450(n)-1. - R. J. Mathar, Oct 24 2008
a(-n) = A144391(n). - Michael Somos, Mar 27 2014
E.g.f.: (3*x^2 + 2*x -1)*exp(x) + 1. - G. C. Greubel, Jul 19 2017
From Leo Tavares, Dec 26 2021: (Start)
a(n) = A003215(n) - 2*A005408(n). See Bounded Hexagons illustration.
a(n) = A016754(n-1) - A002378(n-2). (End)
a(n) = A003154(n) - A049451(n-1). - John Elias, Dec 22 2022

Edited by R. J. Mathar, Oct 24 2008

More terms from Vladimir Joseph Stephan Orlovsky, Oct 25 2008

A185877 Array T given by T(n,k) = k^2 +(2n-3)k -2*n +3, by antidiagonals.

Original entry on oeis.org

1, 3, 1, 7, 5, 1, 13, 11, 7, 1, 21, 19, 15, 9, 1, 31, 29, 25, 19, 11, 1, 43, 41, 37, 31, 23, 13, 1, 57, 55, 51, 45, 37, 27, 15, 1, 73, 71, 67, 61, 53, 43, 31, 17, 1, 91, 89, 85, 79, 71, 61, 49, 35, 19, 1, 111, 109, 105, 99, 91, 81, 69, 55, 39, 21, 1, 133, 131, 127, 121, 113, 103, 91, 77, 61, 43, 23, 1, 157, 155, 151, 145, 137, 127, 115, 101, 85, 67, 47, 25, 1, 183, 181, 177, 171, 163, 153, 141, 127, 111, 93, 73, 51, 27, 1

Offset: 1

Clark Kimberling, Feb 05 2011

A member of the accumulation chain ... < A185879 < A185877 < A185878 < A185880 < ... (See A144112 for the definition of accumulation array).

			Northwest corner:
  1, 3,  7, 13, 21
  1, 5, 11, 19, 29
  1, 7, 15, 25, 45
  1, 9, 19, 31, 45

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A144112, A185878, A185879, A185880.
Row 1 to 3: A002061, A028387, A082111.
diag (1,5,...): A056108;
diag (3,11,...): A056106;
diag (7,19,...): A003215;
diag (13,29,...): A144391;
diag (1,7,...): A003215;
diag (1,9,...): A144390.

(* This program generates A185877, its accumulation array A185878, and its weight array A185879. *)
f[n_,0]:=0;f[0,k_]:=0;
f[n_,k_]:=k^2+(2n-3)k-2n+3;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A185877 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
FullSimplify[s[n,k]]  (* formula for A185878 *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]]
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
w[m_,n_]:=f[m,n]+f[m-1,n-1]-f[m,n-1]-f[m-1,n]/;Or[m>0,n>0];
TableForm[Table[w[n,k],{n,1,10},{k,1,15}]] (* A185879 *)
Table[w[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

T(n,k) = k^2 + (2*n-3)*k - 2*n + 3, k>=1, n>=1.

A185880 Second accumulation array of A185877, by antidiagonals.

Original entry on oeis.org

1, 5, 3, 16, 17, 6, 40, 56, 38, 10, 85, 140, 128, 70, 15, 161, 295, 320, 240, 115, 21, 280, 553, 670, 600, 400, 175, 28, 456, 952, 1246, 1250, 1000, 616, 252, 36, 705, 1536, 2128, 2310, 2075, 1540, 896, 348, 45, 1045, 2355, 3408, 3920, 3815, 3185, 2240, 1248, 465, 55, 1496, 3465, 5190, 6240, 6440, 5831, 4620, 3120, 1680, 605, 66, 2080, 4928, 7590, 9450, 10200, 9800, 8428, 6420, 4200, 2200

Offset: 1

Clark Kimberling, Feb 05 2011

A member of the accumulation chain ... < A185879 < A185877 < A185878 < A185880 < ... See A144112 for the definition of accumulation array.

			Northwest corner:
   1,    5,   16,   40,   85
   3,   17,   56,  140,  295
   6,   38,  128,  320,  670
  10,   70,  240,  600, 1250

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A144112, A185877, A185878.
Antidiagonal sums: A037235.
diag (1,5,...): A056108 (4th spoke on hexagonal wheel);
diag (3,11,...): A056106 (2nd spoke on hexagonal wheel);
diag (7,19,...): A003215 (hex numbers);
diag (13,29,...): A144391.

(* This program generates A185878 first and then generates A185880 as the accumulation array of A185878. *)
f[n_,k_]:=(k*n/6)(7-3k+2k^2-3n+3kn);
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A185878 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}];
FullSimplify[s[n,k]]
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* A185880 *)
f[n_, k_] := (1/72)*k*(1 + k)*n*(1 + n)*(16 - k + 3 *k^2 + 4 *(-1 + k) *n); Table[f[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 21 2017 *)

T(n,k) = C(k,2)*C(n,2)*(3*k^2+4*k*n-k-4*n+16)/18, k>=1, n>=1.

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!}]

A274602 Triangle read by rows: T(n,k) = k*(n-k+1)^2 + n - k, 0 <= k <= n.

Original entry on oeis.org

0, 1, 1, 2, 5, 2, 3, 11, 9, 3, 4, 19, 20, 13, 4, 5, 29, 35, 29, 17, 5, 6, 41, 54, 51, 38, 21, 6, 7, 55, 77, 79, 67, 47, 25, 7, 8, 71, 104, 113, 104, 83, 56, 29, 8, 9, 89, 135, 153, 149, 129, 99, 65, 33, 9, 10, 109, 170, 199, 202, 185, 154, 115, 74, 37, 10

Offset: 1

Juri-Stepan Gerasimov, Dec 01 2016

Mirrored version of a(n) is T(n,k) = (n-k)*(k+1)^2+k, 0 <= k <= n, read by rows:
0
 1
 5   2
 9  11   3
13  20  19   4
17  29  35  29   5
As an infinite square array (matrix) with comments:
 1   2   3   4   5                      A001477
 5  11  19  29  41                      A028387
 9  20  35  54  77                      A014107
13  29  51  79 113                      A144391
17  38  67 104 149                      A182868
21  47  83 129 185

			0; 1,1; 2,5,2; 3,11,9,3; 4,19,20,13,4; 5,29,35,29,17,5; ...
As an infinite triangular array:
0
1   1
2   5   2
3  11   9    3
4  19  20   13    4
5  29  35   29   17    5
As an infinite square array (matrix) with comments:
0   1   2    3    4    5                   A001477
1   5   9   13   17   21                   A016813
2  11  20   29   38   47                   A017185
3  19  35   51   67   83
4  29  54   79  104  129
5  41  77  113  149  185

Cf. A002064, A001477, A016813, A017185, A062158 (central column). A028387, A014107, A144391, A182868.

Cf. Triangle read by rows: T(n,k) = k*(n-k+1)^m+n-k, 0 <= k <= n: A003056 (m = 0), A059036 (m = 1), A278910 (m = k).

/* As triangle */ [[k*(n-k+1)^2+n-k: k in [0..n]]: n in [0..10]];

Table[k (n - k + 1)^(k + #) + n - k &[2 - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 02 2016 *)

A319128 Interleave n(3n - 2), 3*n^2 + n - 1, n=0,0,1,1, ... .

Original entry on oeis.org

0, -1, 1, 3, 8, 13, 21, 29, 40, 51, 65, 79, 96, 113, 133, 153, 176, 199, 225, 251, 280, 309, 341, 373, 408, 443, 481, 519, 560, 601, 645, 689, 736, 783, 833, 883, 936, 989, 1045, 1101, 1160, 1219, 1281, 1343, 1408, 1473, 1541, 1609, 1680, 1751

Offset: 0

Paul Curtz, Sep 11 2018

A144391(n) = -1, 3, 13, 29, 51, ... is in the hexagonal spiral begining with -1 (like from 0 in A000567):
.
                55--54--53--52--51
                /                 \
              56  32--31--30--29  50
              /   /             \   \
            57  33  15--14--13  28  49
            /   /   /         \   \   \
          58  34  16   4---3  12  27  48
          /   /   /   /     \   \   \   \
        59  35  17   5  -1   2  11  26  47
            /   /   /   /   /   /   /   /
          36  18   6   0---1  10  25  46
            \   \   \         /   /   /
            37  19   7---8---9  24  45
              \   \             /   /
              38  20--21--22--23  44
                \                 /
                39--40--41--42--43
.
A000567(n) = 0, 1, 8, 21, 40, ... is in the first hexagonal spiral.
The bisections 0, 1, 8, 21, ... and -1, 3, 13, 29, ...  are on the respective main antidiagonals.
a(-n) = 0, 1, 5, 9, 16, 23, ... . The bisections n*(3*n + 2) and 3*n^2 - n - 1 are in both spirals on main diagonals.
The bisections of a(n) are in the second spiral: ... 29, 13, 3, -1, 0, 1, 8, 21, ... .
The bisections of a(-n) are in the first and in the second spiral: ...  33, 16, 5, 0, 1, 9, 23, ... .

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Main diagonal of A318958.

Cf. A000567, A144391, A168236, A045944, A144390.

Flat(List([0..30],n->[3*n^2-2*n,3*n^2+n-1])); # Muniru A Asiru, Sep 19 2018

seq(op([3*n^2-2*n,3*n^2+n-1]),n=0..30); # Muniru A Asiru, Sep 19 2018

LinearRecurrence[{2, 0, -2, 1}, {0, -1, 1, 3 }, 40] (* Stefano Spezia, Sep 16 2018 *)

concat(0, Vec(-x*(1 - 3*x - x^2) / ((1 - x)^3*(1 + x)) + O(x^40))) \\ Colin Barker, Sep 14 2018

a(n+1) = a(n) + (6*n^2 - 3*(-1)^n - 1)/4, n=0,1,2, ... , a(0) = 0.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), n>3.
From Colin Barker, Sep 14 2018: (Start)
G.f.: -x*(1 - 3*x - x^2) / ((1 - x)^3*(1 + x)).
a(n) = (-4*n + 3*n^2) / 4 for n even.
a(n) = (-3 - 4*n + 3*n^2) / 4 for n odd.
(End)
a(n) = (-3 + 3*(-1)^n - 8*n + 6*n^2)/8. - Colin Barker, Sep 14 2018
E.g.f.: (x*(3*x - 1)*cosh(x) + (3*x^2 - x - 3)*sinh(x))/4. - Stefano Spezia, Mar 15 2020

A370380 Array read by downward antidiagonals: A(n,k) = (k+2)*A(n-1,k+1) + Sum_{j=0..k} A(n-1,j) with A(0,k) = 1, n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 3, 1, 5, 13, 1, 7, 29, 71, 1, 9, 51, 195, 461, 1, 11, 79, 409, 1493, 3447, 1, 13, 113, 737, 3623, 12823, 29093, 1, 15, 153, 1203, 7427, 35285, 122125, 273343, 1, 17, 199, 1831, 13601, 81009, 375591, 1277991, 2829325, 1, 19, 251, 2645, 22961, 164371, 954419, 4344485, 14584789, 31998903

Offset: 0

Mikhail Kurkov, Feb 17 2024

			Array begins:
===========================================================
n\k|     0      1      2      3       4       5       6 ...
---+-------------------------------------------------------
0  |     1      1      1      1       1       1       1 ...
1  |     3      5      7      9      11      13      15 ...
2  |    13     29     51     79     113     153     199 ...
3  |    71    195    409    737    1203    1831    2645 ...
4  |   461   1493   3623   7427   13601   22961   36443 ...
5  |  3447  12823  35285  81009  164371  304667  526833 ...
6  | 29093 122125 375591 954419 2124937 4289433 8025755 ...
  ...

Row 2 appears to be essentially A144391. - Joerg Arndt, Feb 17 2024

Cf. A003319.

A(m, n=m)={my(r=vectorv(m+1), v=vector(n+m+1, k, 1)); r[1] = v[1..n+1];
for(i=1, m, v=vector(#v-1, k, (k+1)*v[k+1] + sum(j=1, k, v[j])); r[1+i] = v[1..n+1]); Mat(r)}
{ A(6) }

Conjecture: A(n,0) = A003319(n+2). - Mikhail Kurkov, Oct 27 2024

A(n,k) = A(n,k-1) - k*A(n-1,k) + (k+2)*A(n-1,k+1) with A(n,0) = A(n-1,0) + 2*A(n-1,1), A(0,k) = 1. - Mikhail Kurkov, Nov 23 2024

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!}]

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

A144390 a(n) = 3*n^2 - n - 1.

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A185877 Array T given by T(n,k) = k^2 +(2*n-3)*k -2*n +3, by antidiagonals.

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A185880 Second accumulation array of A185877, by antidiagonals.

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

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Mathematica

A274602 Triangle read by rows: T(n,k) = k*(n-k+1)^2 + n - k, 0 <= k <= n.

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Magma

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A319128 Interleave n*(3*n - 2), 3*n^2 + n - 1, n=0,0,1,1, ... .

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A370380 Array read by downward antidiagonals: A(n,k) = (k+2)*A(n-1,k+1) + Sum_{j=0..k} A(n-1,j) with A(0,k) = 1, n >= 0, k >= 0.

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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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A185877 Array T given by T(n,k) = k^2 +(2n-3)k -2*n +3, by antidiagonals.

A319128 Interleave n(3n - 2), 3*n^2 + n - 1, n=0,0,1,1, ... .