A098847 -id:A098847 - cp's OEIS frontend

A132769 a(n) = n*(n + 27).

0, 28, 58, 90, 124, 160, 198, 238, 280, 324, 370, 418, 468, 520, 574, 630, 688, 748, 810, 874, 940, 1008, 1078, 1150, 1224, 1300, 1378, 1458, 1540, 1624, 1710, 1798, 1888, 1980, 2074, 2170, 2268, 2368, 2470, 2574, 2680, 2788, 2898, 3010, 3124, 3240, 3358, 3478

Offset: 0

Omar E. Pol, Aug 28 2007

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566.
Cf. A028569, A098849, A098850, A098603, A098847, A098848, A102928, A120071, A132759.
Cf. A132760, A132761, A132762, A132763, A132764, A132765, A132766, A132767, A132768.
Cf. A132756.

Table[n(n+27),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,28,58},50] (* Harvey P. Dale, Oct 14 2012 *)

a(n)=n*(n+27) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 2*n + a(n-1) + 26, with a(0)=0. - Vincenzo Librandi, Aug 03 2010
a(0)=0, a(1)=28, a(2)=58; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Oct 14 2012
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(27)/27 = A001008(27)/A102928(27) = 312536252003/2168462696400, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/27 - 57128792093/2168462696400. (End)
From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: 2*x*(14 - 13*x)/(1 - x)^3.
E.g.f.: exp(x)*x*(28 + x).
a(n) = 2*A132756(n). (End)

A105020 Array read by antidiagonals: row n (n >= 0) contains the numbers m^2 - n^2, m >= n+1.

Original entry on oeis.org

1, 3, 4, 5, 8, 9, 7, 12, 15, 16, 9, 16, 21, 24, 25, 11, 20, 27, 32, 35, 36, 13, 24, 33, 40, 45, 48, 49, 15, 28, 39, 48, 55, 60, 63, 64, 17, 32, 45, 56, 65, 72, 77, 80, 81, 19, 36, 51, 64, 75, 84, 91, 96, 99, 100, 21, 40, 57, 72, 85, 96, 105, 112, 117, 120, 121

Offset: 0

Andrew S. Plewe and Franklin T. Adams-Watters, Jul 11 2005

A "Goldbach Conjecture" for this sequence: when there are n terms between consecutive odd integers (2n+1) and (2n+3) for n > 0, at least one will be the product of 2 primes (not necessarily distinct). Example: n=3 for consecutive odd integers a(7)=7 and a(11)=9 and of the 3 sequence entries a(8)=12, a(9)=15 and a(10)=16 between them, one is the product of 2 primes a(9)=15=3*5. - Michael Hiebl, Jul 15 2007
A024352 gives distinct values in the array, minus the first row (1, 4, 9, 16, etc.). a(n) gives all solutions to the equation x^2 + xy = n, with y mod 2 = 0, x > 0, y >= 0. - Andrew S. Plewe, Oct 19 2007
Alternatively, triangular sequence of coefficients of Dynkin diagram weights for the Cartan groups C_n: t(n,m) = m*(2*n - m). Row sums are A002412. - Roger L. Bagula, Aug 05 2008

			Array begins:
  1  4  9 16 25 36  49  64  81 100 ...
  3  8 15 24 35 48  63  80  99 120 ...
  5 12 21 32 45 60  77  96 117 140 ...
  7 16 27 40 55 72  91 112 135 160 ...
  9 20 33 48 65 84 105 128 153 180 ...
  ...
Triangle begins:
   1;
   3,  4;
   5,  8,  9;
   7, 12, 15, 16;
   9, 16, 21, 24, 25;
  11, 20, 27, 32, 35, 36;
  13, 24, 33, 40, 45, 48, 49;
  15, 28, 39, 48, 55, 60, 63, 64;
  17, 32, 45, 56, 65, 72, 77, 80, 81;
  19, 36, 51, 64, 75, 84, 91, 96, 99, 100;

R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.

G. C. Greubel, Antidiagonals n = 0..50, flattened

Rows give A000290, A005563, A028347, A028560, A028566, A098603, A098847, A098848, A098849, A098850.
Columns give A005408, A008586, A016945, A008590, A017329, A008594, A008598, A008602, A008606, A000567.
Diagonals give A033428, A045944, A067725.
Cf. A000384, A002412, A024352, A105020, A128076.

[(k+1)*(2*n-k+1): k in [0..n], n in [0..15]]; // G. C. Greubel, Mar 15 2023

t[n_, m_]:= (n^2 - m^2); Flatten[Table[t[i, j], {i,12}, {j,i-1,0,-1}]]
(* to view table *) Table[t[i, j], {j,0,6}, {i,j+1,10}]//TableForm (* Robert G. Wilson v, Jul 11 2005 *)
Table[(k+1)*(2*n-k+1), {n,0,15}, {k,0,n}]//Flatten (* Roger L. Bagula, Aug 05 2008 *)

def A105020(n,k): return (k+1)*(2*n-k+1)
flatten([[A105020(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Mar 15 2023

a(n) = r^2 - (r^2 + r - m)^2/4, where r = round(sqrt(m)) and m = 2*n+2. - Wesley Ivan Hurt, Sep 04 2021
a(n) = A128076(n+1) * A105020(n+1). - Wesley Ivan Hurt, Jan 07 2022
From G. C. Greubel, Mar 15 2023: (Start)
Sum_{k=0..n} T(n, k) = A002412(n+1).
Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*((1+(-1)^n)*A000384((n+2)/2) - (1- (-1)^n)*A000384((n+1)/2)). (End)

More terms from Robert G. Wilson v, Jul 11 2005

A132770 a(n) = n*(n + 28).

Original entry on oeis.org

0, 29, 60, 93, 128, 165, 204, 245, 288, 333, 380, 429, 480, 533, 588, 645, 704, 765, 828, 893, 960, 1029, 1100, 1173, 1248, 1325, 1404, 1485, 1568, 1653, 1740, 1829, 1920, 2013, 2108, 2205, 2304, 2405, 2508, 2613, 2720, 2829, 2940, 3053, 3168, 3285, 3404, 3525

Offset: 0

Omar E. Pol, Aug 28 2007

G. C. Greubel, Table of n, a(n) for n = 0..5000
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569.
Cf. A098603, A098847, A098848, A098849, A098850, A102928, A120071, A132759, A132760, A132761.
Cf. A132762, A132763, A132764, A132765, A132766, A132767, A132768, A132769.

#(#+28)&/@Range[0,50] (* or *) LinearRecurrence[{3,-3,1},{0,29,60},50] (* Harvey P. Dale, Apr 30 2018 *)

a(n)=n*(n+28) \\ Charles R Greathouse IV, Jun 17 2017

[n*(n+28) for n in (0..50)] # G. C. Greubel, Mar 13 2022

a(n) = 2*n + a(n-1) + 27, with a(0)=0. - Vincenzo Librandi, Aug 03 2010
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(28)/28 = A001008(28)/A102928(28) = 315404588903/2248776129600, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 7751493599/321253732800. (End)
G.f.: x*(29 - 27*x)/(1-x)^3. - Harvey P. Dale, Aug 03 2021
E.g.f.: x*(29 + x)*exp(x). - G. C. Greubel, Mar 13 2022

A132771 a(n) = n*(n + 29).

Original entry on oeis.org

0, 30, 62, 96, 132, 170, 210, 252, 296, 342, 390, 440, 492, 546, 602, 660, 720, 782, 846, 912, 980, 1050, 1122, 1196, 1272, 1350, 1430, 1512, 1596, 1682, 1770, 1860, 1952, 2046, 2142, 2240, 2340, 2442, 2546, 2652, 2760, 2870, 2982, 3096, 3212, 3330, 3450, 3572

Offset: 0

Omar E. Pol, Aug 28 2007

G. C. Greubel, Table of n, a(n) for n = 0..5000
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569.
Cf. A098849, A098850, A098603, A098847, A098848, A102928, A120071, A132759, A132760, A132761.
Cf. A132762, A132763, A132764, A132765, A132766, A132767, A132768, A132769, A132770.

Table[n(n+29), {n,0,50}] (* Bruno Berselli, Apr 03 2015 *)
LinearRecurrence[{3,-3,1},{0,30,62},50] (* Harvey P. Dale, Oct 18 2024 *)

a(n)=n*(n+29) \\ Charles R Greathouse IV, Jun 17 2017

[n*(n+29) for n in (0..50)] # G. C. Greubel, Mar 13 2022

a(n) = 2*n + a(n-1) + 28 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(29)/29 = A001008(29)/A102928(29) = 9227046511387/67543597321200, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/29 - 236266661971/9649085331600. (End)
From G. C. Greubel, Mar 13 2022: (Start)
G.f.: 2*(15*x - 14*x^2)/(1-x)^3.
E.g.f.: x*(30 + x)*exp(x). (End)

A132772 a(n) = n*(n + 30).

Original entry on oeis.org

0, 31, 64, 99, 136, 175, 216, 259, 304, 351, 400, 451, 504, 559, 616, 675, 736, 799, 864, 931, 1000, 1071, 1144, 1219, 1296, 1375, 1456, 1539, 1624, 1711, 1800, 1891, 1984, 2079, 2176, 2275, 2376, 2479, 2584, 2691, 2800, 2911, 3024, 3139, 3256, 3375, 3496, 3619

Offset: 0

Omar E. Pol, Aug 28 2007

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566.
Cf. A028569, A067079, A098849, A098850, A098603, A098847, A098848, A102928, A120071.
Cf. A132759, A132760, A132761, A132762, A132763, A132764, A132765, A132766, A132767.
Cf. A132768, A132769, A132770, A132771.

Table[n(n+30),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,31,64},50] (* Harvey P. Dale, Mar 06 2015 *)

a(n)=n*(n+30) \\ Charles R Greathouse IV, Jun 17 2017

[n*(n+30) for n in (0..50)] # G. C. Greubel, Mar 13 2022

G.f.: x*(31-29*x)/(1-x)^3. - R. J. Mathar, Nov 14 2007
a(n) = 2*n + a(n-1) + 29 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
a(0)=0, a(1)=31, a(2)=64, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Mar 06 2015
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(30)/30 = A001008(30)/A102928(30) = 9304682830147/69872686884000, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 225175759291/9981812412000. (End)
E.g.f.: x*(31 + x)*exp(x). - G. C. Greubel, Mar 13 2022

A185739 Accumulation array of A185738, by antidiagonals.

Original entry on oeis.org

1, 3, 4, 6, 10, 11, 10, 18, 25, 26, 15, 28, 42, 56, 57, 21, 40, 62, 90, 119, 120, 28, 54, 85, 128, 186, 246, 247, 36, 70, 111, 170, 258, 378, 501, 502, 45, 88, 140, 216, 335, 516, 762, 1012, 1013, 55, 108, 172, 266, 417, 660, 1030, 1530, 2035, 2036, 66, 130, 207, 320, 504, 810, 1305, 2056, 3066, 4082, 4083, 78, 154, 245, 378, 596, 966, 1587, 2590, 4106, 6138, 8177, 8178, 91

Offset: 1

Clark Kimberling, Feb 02 2011

This arrays is a member of a chain; see A185738.

			Northwest corner:
1....3....6....10....15
4....10...18...28....40
11...25...42...62....85
26...56...90...128...170

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

Rows 1 to 4: A000217, A028562, A140675, 2*A098847
Columns 1 to 3: A000295, A000247, A068293.
Cf. A185738, A185740.

(* See A185738 *)
f[n_, k_] := (k/2)*(4*(2^n - 1) + (k - 3)*n);
TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 10}]]  (* Array A185739 *)
Table[f[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 11 2017 *)

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

A132773 a(n) = n*(n + 31).

Original entry on oeis.org

0, 32, 66, 102, 140, 180, 222, 266, 312, 360, 410, 462, 516, 572, 630, 690, 752, 816, 882, 950, 1020, 1092, 1166, 1242, 1320, 1400, 1482, 1566, 1652, 1740, 1830, 1922, 2016, 2112, 2210, 2310, 2412, 2516, 2622, 2730, 2840, 2952, 3066, 3182, 3300, 3420, 3542, 3666

Offset: 0

Omar E. Pol, Aug 28 2007

Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569.
Cf. A098603, A098847, A098848, A098849, A098850, A102928, A120071, A132758, A132759, A132760.
Cf. A132761, A132762, A132763, A132764, A132765, A132766, A132767, A132768, A132769, A132770.
Cf. A132771, A132772.

Table[n*(n + 31), {n, 0, 100}] (* Wesley Ivan Hurt, Apr 16 2024 *)

a(n)=n*(n+31) \\ Charles R Greathouse IV, Jun 17 2017

G.f.: 2*x*(-16+15*x)/(-1+x)^3. - R. J. Mathar, Nov 14 2007
a(n) = 2*A132758(n). - R. J. Mathar, Jul 22 2009
a(n) = 2*n + a(n-1) + 30, with n > 0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(31)/31 = A001008(31)/A102928(31) = 290774257297357/2238255069850800, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/31 - 7313175618421/319750724264400. (End)
From Elmo R. Oliveira, Dec 13 2024: (Start)
E.g.f.: exp(x)*x*(32 + x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

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

cp's OEIS Frontend

A132769 a(n) = n*(n + 27).

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A105020 Array read by antidiagonals: row n (n >= 0) contains the numbers m^2 - n^2, m >= n+1.

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A132770 a(n) = n*(n + 28).

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A132771 a(n) = n*(n + 29).

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A132772 a(n) = n*(n + 30).

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A185739 Accumulation array of A185738, by antidiagonals.

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A132773 a(n) = n*(n + 31).

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