A119412 -id:A119412 - cp's OEIS frontend

A056115 a(n) = n*(n+11)/2.

0, 6, 13, 21, 30, 40, 51, 63, 76, 90, 105, 121, 138, 156, 175, 195, 216, 238, 261, 285, 310, 336, 363, 391, 420, 450, 481, 513, 546, 580, 615, 651, 688, 726, 765, 805, 846, 888, 931, 975, 1020, 1066, 1113, 1161, 1210, 1260, 1311, 1363, 1416, 1470, 1525

Offset: 0

Barry E. Williams, Jul 04 2000

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

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

Cf. A000096, A001477, A055999, A056000, A056119.

Third column of Pascal (1, 6) triangle A096956.

List([0..50], n-> n*(n+11)/2 ); # G. C. Greubel, Jan 18 2020

[n*(n+11)/2: n in [0..50]]; // G. C. Greubel, Jan 18 2020

((2*Range[0,50]+11)^2 -11^2)/8 (* G. C. Greubel, Jan 18 2020 *)

a(n)=n*(n+11)/2; \\ Joerg Arndt, Oct 25 2014

[n*(n+11)/2 for n in (0..50)] # G. C. Greubel, Jan 18 2020

G.f.: x*(6-5*x)/(1-x)^3.
a(n) = A000096(n) + 4*A001477(n) = A056000(n) + A001477(n) = A056119(n) - A001477(n). - Zerinvary Lajos, Oct 01 2006
a(n) = A126890(n,5) for n>4. - Reinhard Zumkeller, Dec 30 2006
Equals A119412/2. - Zerinvary Lajos, Feb 12 2007
If we define f(n,i,a) = Sum_{k=0..n-i} ( binomial(n,k)*stirling1(n-k,i) *Product_{j=0..k-1} (-a-j) ), then a(n) = -f(n,n-1,6), for n>=1. - Milan Janjic, Dec 20 2008
a(n) = a(n-1) + n + 5 (with a(0)=0). - Vincenzo Librandi, Aug 07 2010
Sum_{n>=1} 1/a(n) = 83711/152460. - R. J. Mathar, Jul 14 2012
a(n) = 6*n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013
E.g.f.: x*(12 + x)*exp(x)/2. - G. C. Greubel, Jan 18 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/11 - 20417/152460. - Amiram Eldar, Jan 10 2021

A164012 Zero together with row 12 of the array in A163280.

Original entry on oeis.org

0, 31, 62, 87, 124, 115, 186, 147, 232, 207, 250, 242, 372, 312, 364, 405, 464, 476, 558, 570, 640, 693, 726, 782, 888, 925, 962, 1026, 1092, 1160, 1230, 1302, 1376, 1452, 1530, 1610, 1692, 1776, 1862, 1950, 2040, 2132, 2226, 2322, 2420, 2520, 2622, 2726

Offset: 0

Omar E. Pol, Aug 08 2009

nonn

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

Cf. A008578, A161344, A161345, A163280, A164000, A164011.

Conjecture: a(n) = A119412(n), n > 36. [R. J. Mathar, Jul 31 2010]

Terms beyond a(12) from R. J. Mathar, Jul 31 2010

A243138 a(n) = n^2 + 15*n + 13.

Original entry on oeis.org

13, 29, 47, 67, 89, 113, 139, 167, 197, 229, 263, 299, 337, 377, 419, 463, 509, 557, 607, 659, 713, 769, 827, 887, 949, 1013, 1079, 1147, 1217, 1289, 1363, 1439, 1517, 1597, 1679, 1763, 1849, 1937, 2027, 2119, 2213, 2309, 2407, 2507, 2609, 2713, 2819, 2927, 3037, 3149

Offset: 0

Vincenzo Librandi, Jun 02 2014

From Klaus Purath, Dec 13 2022: (Start)
Numbers m such that 4*m + 173 is a square.
The product of two consecutive terms belongs to the sequence, a(n)*a(n+1) = a(a(n)+n).
The prime terms in this sequence are listed in A153422. Each prime factor p divides exactly 2 out of any p consecutive terms. If a(i) and a(k) form such a pair that are divisible by p, then i + k == -15 (mod p). (End)

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

Cf. A098847, A119412, A126719, A132759, A153422.

Magma
```
[n^2+15*n+13: n in [0..50]];
```

Table[n^2 + 15 n + 13, {n, 0, 50}] (* or *) CoefficientList[Series[(13 - 10 x - x^2)/(1 - x)^3, {x, 0, 50}], x]
LinearRecurrence[{3,-3,1},{13,29,47},50] (* Harvey P. Dale, Sep 06 2020 *)

a(n)=n^2+15*n+13 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: (13 - 10*x - x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
From Klaus Purath, Dec 13 2022: (Start)
a(n) = A119412(n+2) - 13.
a(n) = A132759(n+1) - 1.
a(n) = A098847(n+1) + n. (End)
Sum_{n>=0} 1/a(n) = tan(sqrt(173)*Pi/2)*Pi/sqrt(173) + 742077303/604626139. - Amiram Eldar, Feb 14 2023
E.g.f.: (13 + 16*x + x^2)*exp(x). - Elmo R. Oliveira, Oct 18 2024

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A056115 a(n) = n*(n+11)/2.

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A164012 Zero together with row 12 of the array in A163280.

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A243138 a(n) = n^2 + 15*n + 13.

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