A085788 -id:A085788 - cp's OEIS frontend

A081266 Staggered diagonal of triangular spiral in A051682.

0, 6, 21, 45, 78, 120, 171, 231, 300, 378, 465, 561, 666, 780, 903, 1035, 1176, 1326, 1485, 1653, 1830, 2016, 2211, 2415, 2628, 2850, 3081, 3321, 3570, 3828, 4095, 4371, 4656, 4950, 5253, 5565, 5886, 6216, 6555, 6903, 7260, 7626, 8001, 8385, 8778, 9180

Offset: 0

Paul Barry, Mar 15 2003

Staggered diagonal of triangular spiral in A051682, between (0,4,17) spoke and (0,7,23) spoke.
Binomial transform of (0, 6, 9, 0, 0, 0, ...).
If Y is a fixed 3-subset of a (3n+1)-set X then a(n) is the number of (3n-1)-subsets of X intersecting Y. - Milan Janjic, Oct 28 2007
Partial sums give A085788. - Leo Tavares, Nov 23 2023

			a(1)=9*1+0-3=6, a(2)=9*2+6-3=21, a(3)=9*3+21-3=45.
For n=3, a(3) = -0^2+1^2-2^2+3^2-4^2+5^2-6^2+7^2-8^2+9^2 = 45.

Muniru A Asiru, Table of n, a(n) for n = 0..10000
Tomislav Došlić and Luka Podrug, Sweet division problems: from chocolate bars to honeycomb strips and back, arXiv:2304.12121 [math.CO], 2023.
Milan Janjic, Two Enumerative Functions
Milan Janjic and B. Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
Amelia Carolina Sparavigna, The groupoid of the Triangular Numbers and the generation of related integer sequences, Politecnico di Torino, Italy (2019).
Leo Tavares, Star illustration
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000290, A005449, A014105, A022266, A033585, A062725, A144312, A144314, A218470.

Cf. A003154, A133694.

List([0..50],n->Binomial(3*n+1,2)); # Muniru A Asiru, Feb 28 2019

seq(binomial(3*n+1,2), n=0..45); # Zerinvary Lajos, Jan 21 2007

LinearRecurrence[{3,-3,1},{0,6,21},50] (* Harvey P. Dale, Aug 29 2015 *)

a(n)=3*n*(3*n+1)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 6*C(n,1) + 9*C(n,2).
a(n) = 3*n*(3*n+1)/2.
G.f.: (6*x+3*x^2)/(1-x)^3.
a(n) = A000217(3*n); a(2*n) = A144314(n). - Reinhard Zumkeller, Sep 17 2008
a(n) = 3*A005449(n). - R. J. Mathar, Mar 27 2009
a(n) = 9*n+a(n-1)-3 for n>0, a(0)=0. - Vincenzo Librandi, Aug 08 2010
a(n) = A218470(9n+5). - Philippe Deléham, Mar 27 2013
a(n) = Sum_{k=0..3n} (-1)^(n+k)*k^2. - Bruno Berselli, Aug 29 2013
E.g.f.: 3*exp(x)*x*(4 + 3*x)/2. - Stefano Spezia, Jun 06 2021
From Amiram Eldar, Aug 11 2022: (Start)
Sum_{n>=1} 1/a(n) = 2 - Pi/(3*sqrt(3)) - log(3).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/(3*sqrt(3)) + 4*log(2)/3 - 2. (End)
From Leo Tavares, Nov 23 2023: (Start)
a(n) = 3*A000217(n) + 3*A000290(n).
a(n) = A003154(n+1) - A133694(n+1). (End)

A254407 a(n) = n(n+1)(11*n +10)/6.

Original entry on oeis.org

0, 7, 32, 86, 180, 325, 532, 812, 1176, 1635, 2200, 2882, 3692, 4641, 5740, 7000, 8432, 10047, 11856, 13870, 16100, 18557, 21252, 24196, 27400, 30875, 34632, 38682, 43036, 47705, 52700, 58032, 63712, 69751, 76160, 82950, 90132, 97717, 105716, 114140, 123000

Offset: 0

Bruno Berselli, Jan 30 2015

Similar sequences of the type m*P(s,m) - Sum_{i=1..m} P(s-1,i), where P(s,m) is the m-th s-gonal number:
s=3: A027480(n) = (n+1)*A000217(n+1) - Sum_{i=1..n+1} i;
s=4: A162148(n) = (n+1)*A000290(n+1) - Sum_{i=1..n+1} A000217(i);
s=5: A245301(n) = (n+1)*A000326(n+1) - Sum_{i=1..n+1} A000290(i);
s=6: A085788(n) = (n+1)*A000384(n+1) - Sum_{i=1..n+1} A000326(i);
s=7:       a(n) = (n+1)*A000566(n+1) - Sum_{i=1..n+1} A000384(i).

			532 is the 7th term because A000566(7)=112 and Sum_{i=1..7} A000384(i)=252, therefore 7*112-252 = 532.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wikipedia, Polygonal numbers: Table of values.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A011875, A027480, A057145, A085788, A132112, A162148, A245301, A254963.

Magma
```
[n*(n+1)*(11*n+10)/6: n in [0..40]];
```

A254407:= n-> n*(n+1)*(11*n+10)/6; seq(A254407(n), n=0..50); # G. C. Greubel, Mar 31 2021

Table[n (n + 1) (11 n + 10)/6, {n, 0, 40}]
Column[CoefficientList[Series[x (7 + 4 x) / (1 - x)^4, {x, 0, 60}], x]] (* Vincenzo Librandi, Jan 31 2015 *)

makelist(n*(n+1)*(11*n+10)/6, n, 0, 40);

vector(40, n, n--; n*(n+1)*(11*n+10)/6)

Sage
```
[n*(n+1)*(11*n+10)/6 for n in (0..40)]
```

G.f.: x*(7 + 4*x)/(1 - x)^4.
a(-n) = -A132112(n-1).
a(n) = Sum_{k=0..n} A011875(11*k+2).
Equivalently, partial sums of A254963.
E.g.f.: x*(42 + 54*x + 11*x^2)*exp(x)/6. - G. C. Greubel, Mar 31 2021

A001283 Triangle read by rows, in which row n consists of n(n+m) for m = 1 .. n-1.

Original entry on oeis.org

6, 12, 15, 20, 24, 28, 30, 35, 40, 45, 42, 48, 54, 60, 66, 56, 63, 70, 77, 84, 91, 72, 80, 88, 96, 104, 112, 120, 90, 99, 108, 117, 126, 135, 144, 153, 110, 120, 130, 140, 150, 160, 170, 180, 190, 132, 143, 154, 165, 176, 187, 198, 209, 220, 231, 156, 168, 180

Offset: 2

N. J. A. Sloane

With a different offset: triangle read by rows: t(n, m) = T(n+1, m) = (n+1)(n+m+1) = radius of C-excircle of Pythagorean triangle with sides a=(n+1)^2-m^2, b=2*(n+1)*m and c=(n+1)^2+m^2. - Floor van Lamoen, Aug 21 2001

			The triangle T(n, m) begins:
n\m   1   2   3   4   5   6   7   8   9  10  11  12  13  14 ...
2:    6
3:   12  15
4:   20  24  28
5:   30  35  40  45
6:   42  48  54  60  66
7:   56  63  70  77  84  91
8:   72  80  88  96 104 112 120
9:   90  99 108 117 126 135 144 153
10: 110 120 130 140 150 160 170 180 190
11: 132 143 154 165 176 187 198 209 220 231
12: 156 168 180 192 204 216 228 240 252 264 276
13: 182 195 208 221 234 247 260 273 286 299 312 325
14: 210 224 238 252 266 280 294 308 322 336 350 364 378
15: 240 255 270 285 300 315 330 345 360 375 390 405 420 435
...
[Reformatted and extended by _Wolfdieter Lang_, Dec 02 2014]
----------------------------------------------------------------

T. D. Noe, Rows n = 2..100, flattened

Cf. A003991, A063929, A063930.

Row sums are in A085788. Central column is A033581.

Flatten[Table[n*(n+m), {n, 2, 10}, {m, n-1}]] (* T. D. Noe, Jun 27 2012 *)

T(n, m) = n*(n+m), n-1 >= m >= 1.

Edited comment by Wolfdieter Lang, Dec 02 2014

cp's OEIS Frontend

A081266 Staggered diagonal of triangular spiral in A051682.

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A254407 a(n) = n*(n+1)*(11*n +10)/6.

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A001283 Triangle read by rows, in which row n consists of n(n+m) for m = 1 .. n-1.

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A254407 a(n) = n(n+1)(11*n +10)/6.