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Search: a028569 -id:a028569
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A056000
a(n) = n*(n+9)/2.
+10
30
0, 5, 11, 18, 26, 35, 45, 56, 68, 81, 95, 110, 126, 143, 161, 180, 200, 221, 243, 266, 290, 315, 341, 368, 396, 425, 455, 486, 518, 551, 585, 620, 656, 693, 731, 770, 810, 851, 893, 936, 980, 1025, 1071, 1118, 1166, 1215, 1265, 1316, 1368, 1421, 1475
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Numbers m >= 0 such that 8m+81 is a square. - Bruce J. Nicholson, Jul 29 2017
REFERENCES
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 193.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..5000
Leo Tavares, Illustration: Triangular Pairs.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = A000217(n+4) - 10.
G.f.: x(5-4x)/(1-x)^3.
From Zerinvary Lajos, Oct 01 2006: (Start)
a(n) = A000096(n) + 3*n.
a(n) = A055999(n) + n.
a(n) = A056115(n) - n.
(End)
a(n) = binomial(n,2) - 4*n, n >= 9. - Zerinvary Lajos, Nov 25 2006
a(n) = A126890(n,4) for n > 3. - Reinhard Zumkeller, Dec 30 2006
a(n) = A028569(n)/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-1,5), for n >= 1. - Milan Janjic, Dec 20 2008
a(n) = n + a(n-1) + 4. - Vincenzo Librandi, Aug 07 2010
a(n) = Sum_{k=1..n} (k+4). - Gary Detlefs, Aug 10 2010
Sum_{n>=1} 1/a(n) = 7129/11340. - R. J. Mathar, Jul 14 2012
a(n) = 5n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013
E.g.f.: (1/2)*(x^2 + 10*x)*exp(x). - G. C. Greubel, Jul 17 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/9 - 1879/11340. - Amiram Eldar, Jul 03 2020
a(n) = A000217(n+1) + A008585(n) - 1. - Leo Tavares, Sep 22 2022
From Amiram Eldar, Feb 12 2024: (Start)
Product_{n>=1} (1 - 1/a(n)) = -567*cos(sqrt(89)*Pi/2)/(220*Pi).
Product_{n>=1} (1 + 1/a(n)) = 35*cos(sqrt(73)*Pi/2)/(4*Pi). (End)
MATHEMATICA
Table[n (n + 9)/2, {n, 0, 50}] (* or *)
FoldList[#1 + #2 + 4 &, Range[0, 50]] (* or *)
Table[PolygonalNumber[n + 4] - 10, {n, 0, 50}] (* or *)
CoefficientList[Series[x (5 - 4 x)/(1 - x)^3, {x, 0, 50}], x] (* Michael De Vlieger, Jul 30 2017 *)
PROG
(PARI) a(n)=n*(n+9)/2 \\ Charles R Greathouse IV, Sep 24 2015
CROSSREFS
Cf. A000096, A055998, A055999, A001477.
Column m=2 of (1, 5)-Pascal triangle A096940.
Cf. numbers of the form n*(d*n+10-d)/2 indexed in A140090.
Cf. A000217, A008585.
KEYWORD
easy,nonn
AUTHOR
Barry E. Williams, Jun 16 2000
EXTENSIONS
More terms from James A. Sellers, Jul 04 2000
STATUS
approved
A120071
a(n) = n*(n+20).
+10
18
0, 21, 44, 69, 96, 125, 156, 189, 224, 261, 300, 341, 384, 429, 476, 525, 576, 629, 684, 741, 800, 861, 924, 989, 1056, 1125, 1196, 1269, 1344, 1421, 1500, 1581, 1664, 1749, 1836, 1925, 2016, 2109, 2204, 2301, 2400, 2501, 2604, 2709, 2816, 2925, 3036, 3149, 3264
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..48.
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).
FORMULA
a(n) = (n+10)^2 - 10^2 = n*(n+20), n>=0.
G.f.: x*(21-19*x)/(1-x)^3.
a(n) = 2*n + a(n-1) + 19 (with a(0)=0). - Vincenzo Librandi, Nov 13 2010
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(20)/20 = A001008(20)/A102928(20) = 11167027/62078016, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 155685007/4655851200. (End)
MAPLE
seq(sum(4*k-n, k=7..n), n=6..51); # Zerinvary Lajos, Feb 17 2008
MATHEMATICA
s=0; lst={}; Do[s+=n; AppendTo[lst, s], {n, 21, 6!, 2}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 26 2009 *)
PROG
(PARI) a(n)=n*(n+20) \\ Charles R Greathouse IV, Jan 21 2015
CROSSREFS
a(n-10), n>=11, tenth column (used for the n=10 series of the hydrogen atom) of triangle A120070.
For n*(n+18) see A098850.
Cf. A001008, A001477, A098849, A102928, A120061, A132765, A132760, A132761, A132762, A005563, A028552, A028347, A028557, A028560, A028563, A028566, A028569, A098603, A098847, A132759, A098848.
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jul 20 2006
STATUS
approved
A132765
a(n) = n*(n + 23).
+10
17
0, 24, 50, 78, 108, 140, 174, 210, 248, 288, 330, 374, 420, 468, 518, 570, 624, 680, 738, 798, 860, 924, 990, 1058, 1128, 1200, 1274, 1350, 1428, 1508, 1590, 1674, 1760, 1848, 1938, 2030, 2124, 2220, 2318, 2418, 2520, 2624, 2730, 2838, 2948, 3060, 3174, 3290, 3408
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
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, ResearchGate, 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = n*(n + 23).
a(n) = 2*n + a(n-1) + 22 for n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
From Chai Wah Wu, Dec 17 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
G.f.: 2*x*(12 - 11*x)/(1-x)^3. (End)
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(23)/23 = A001008(23)/A102928(23) = 444316699/2736605872, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/23 - 3825136961/123147264240. (End)
E.g.f.: x*(24 + x)*exp(x). - G. C. Greubel, Mar 14 2022
MATHEMATICA
Table[n (n + 23), {n, 0, 50}] (* Bruno Berselli, Sep 03 2018 *)
PROG
(PARI) a(n)=n*(n+23) \\ Charles R Greathouse IV, Jun 17 2017
(Sage) [n*(n+23) for n in (0..50)] # G. C. Greubel, Mar 14 2022
CROSSREFS
Cf. A001008, A001477, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A056126, A098603, A098847, A098848, A098849, A098850, A102928, A120071, A132759, A132760, A132761, A132762, A132763, A132764, A132765, A132766.
KEYWORD
nonn,easy
AUTHOR
Omar E. Pol, Aug 28 2007
STATUS
approved
A132762
a(n) = n*(n + 19).
+10
13
0, 20, 42, 66, 92, 120, 150, 182, 216, 252, 290, 330, 372, 416, 462, 510, 560, 612, 666, 722, 780, 840, 902, 966, 1032, 1100, 1170, 1242, 1316, 1392, 1470, 1550, 1632, 1716, 1802, 1890, 1980, 2072, 2166, 2262, 2360, 2460, 2562, 2666, 2772, 2880, 2990, 3102, 3216
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
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).
FORMULA
a(n) = 2*n + a(n-1) + 18 for n > 0, a(0) = 0. - Vincenzo Librandi, Aug 03 2010
From Chai Wah Wu, Dec 17 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
G.f.: 2*x*(10 - 9*x)/(1-x)^3. (End)
a(n) = 2*A051942(n+9). - R. J. Mathar, Sep 05 2018
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(19)/19 = A001008(19)/A102928(19) = 275295799/1474352880, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/19 - 33464927/884611728. (End)
E.g.f.: x*(20 + x)*exp(x). - G. C. Greubel, Mar 14 2022
MATHEMATICA
Table[n (n + 19), {n, 0, 50}] (* Bruno Berselli, Sep 05 2018 *)
LinearRecurrence[{3, -3, 1}, {0, 20, 42}, 60] (* Harvey P. Dale, Jun 03 2021 *)
PROG
(PARI) a(n)=n*(n+19) \\ Charles R Greathouse IV, Jun 17 2017
(Sage) [n*(n+19) for n in (0..50)] # G. C. Greubel, Mar 14 2022
CROSSREFS
Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A051942, A098603, A098847, A098848, A098849, A098850, A102928, A120071, A132759, A132760, A132761, A132763, A132764, A132765, A132766, A132767.
KEYWORD
nonn,easy
AUTHOR
Omar E. Pol, Aug 28 2007
STATUS
approved
A132764
a(n) = n*(n+22).
+10
13
0, 23, 48, 75, 104, 135, 168, 203, 240, 279, 320, 363, 408, 455, 504, 555, 608, 663, 720, 779, 840, 903, 968, 1035, 1104, 1175, 1248, 1323, 1400, 1479, 1560, 1643, 1728, 1815, 1904, 1995, 2088, 2183, 2280, 2379, 2480, 2583, 2688, 2795, 2904, 3015, 3128, 3243, 3360
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
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).
FORMULA
a(n) = n*(n + 22).
a(n) = 2*n + a(n-1) + 21 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
a(0)=0, a(1)=23, a(2)=48, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, May 02 2012
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(22)/22 = A001008(22)/A102928(22) = 19093197/113809696, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 156188887/5121436320. (End)
From G. C. Greubel, Mar 14 2022: (Start)
G.f.: x*(23 - 21*x)/(1-x)^3.
E.g.f.: x*(23 + x)*exp(x). (End)
EXAMPLE
a(1)=2*1+0+21=23; a(2)=2*2+23+21=48; a(3)=2*3+48+21=75. - Vincenzo Librandi, Aug 03 2010
MATHEMATICA
Table[n(n+22), {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 23, 48}, 50] (* Harvey P. Dale, May 02 2012 *)
PROG
(PARI) a(n)=n*(n+22) \\ Charles R Greathouse IV, Oct 07 2015
(Sage) [n*(n+22) for n in (0..50)] # G. C. Greubel, Mar 14 2022
CROSSREFS
Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A098603, A098847, A098848, A098849, A098850, A102928, A120071, A132759, A132760, A132761, A132762, A132763, A132765, A132766, A132767.
KEYWORD
easy,nonn
AUTHOR
Omar E. Pol, Aug 28 2007
STATUS
approved
A132763
a(n) = n*(n+21).
+10
12
0, 22, 46, 72, 100, 130, 162, 196, 232, 270, 310, 352, 396, 442, 490, 540, 592, 646, 702, 760, 820, 882, 946, 1012, 1080, 1150, 1222, 1296, 1372, 1450, 1530, 1612, 1696, 1782, 1870, 1960, 2052, 2146, 2242, 2340, 2440, 2542, 2646, 2752, 2860, 2970, 3082, 3196, 3312
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
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).
FORMULA
a(n) = n*(n + 21).
a(n) = 2*n + a(n-1) + 20 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
a(0)=0, a(1)=22, a(2)=46, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, May 25 2014
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(21)/21 = A001008(21)/A102928(21) = 18858053/108636528, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/21 - 166770367/4888643760. (End)
From Stefano Spezia, Jan 30 2021: (Start)
O.g.f.: 2*x*(11 - 10*x)/(1 - x)^3.
E.g.f.: x*(22 + x)*exp(x). (End)
MATHEMATICA
Table[n(n+21), {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 22, 46}, 50] (* Harvey P. Dale, May 25 2014 *)
PROG
(PARI) a(n)=n*(n+21) \\ Charles R Greathouse IV, Oct 07 2015
(Sage) [n*(n+21) for n in (0..50)] # G. C. Greubel, Mar 14 2022
CROSSREFS
Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A098603, A098847, A098848, A098849, A098850, A102928, A120071, A132759, A132760, A132761, A132762, A132764, A132765, A132766, A132767.
KEYWORD
easy,nonn
AUTHOR
Omar E. Pol, Aug 28 2007
STATUS
approved
A132766
a(n) = n*(n+24).
+10
12
0, 25, 52, 81, 112, 145, 180, 217, 256, 297, 340, 385, 432, 481, 532, 585, 640, 697, 756, 817, 880, 945, 1012, 1081, 1152, 1225, 1300, 1377, 1456, 1537, 1620, 1705, 1792, 1881, 1972, 2065, 2160, 2257, 2356, 2457, 2560, 2665, 2772, 2881, 2992, 3105, 3220, 3337
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
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).
FORMULA
a(n) = n*(n + 24).
a(n) = 2*n + a(n-1) + 23 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
a(0)=0, a(1)=25, a(2)=52; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Feb 11 2016
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(24)/24 = A001008(24)/A102928(24) = 1347822955/8566766208, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3602044091/128501493120. (End)
From G. C. Greubel, Mar 14 2022: (Start)
G.f.: 2*x*(13 - 12*x)/(1-x)^3.
E.g.f.: x*(26 + x)*exp(x). (End)
MATHEMATICA
Table[n (n + 24), {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 25, 52}, 50] (* Harvey P. Dale, Feb 11 2016 *)
PROG
(PARI) a(n)=n*(n+24) \\ Charles R Greathouse IV, Jun 17 2017
(Sage) [n*(n+24) for n in (0..50)] # G. C. Greubel, Mar 14 2022
CROSSREFS
Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A098603, A098849, A098850, A098847, A098848, A102928, A120071, A132759, A132760, A132761, A132762, A132763, A132764, A132765.
KEYWORD
easy,nonn
AUTHOR
Omar E. Pol, Aug 28 2007
STATUS
approved
A132767
a(n) = n*(n + 25).
+10
11
0, 26, 54, 84, 116, 150, 186, 224, 264, 306, 350, 396, 444, 494, 546, 600, 656, 714, 774, 836, 900, 966, 1034, 1104, 1176, 1250, 1326, 1404, 1484, 1566, 1650, 1736, 1824, 1914, 2006, 2100, 2196, 2294, 2394, 2496, 2600, 2706, 2814, 2924, 3036, 3150, 3266, 3384
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
a(n) is the Zagreb 1 index of the Mycielskian of the cycle graph C[n]. See p. 205 of the D. B. West reference. - Emeric Deutsch, Nov 04 2016
REFERENCES
Douglas B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001.
LINKS
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.
Wikipedia, Mycielskian.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = 2*n + a(n-1) + 24 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
a(n) = n^2 + 25*n. - Omar E. Pol, Nov 04 2016
From Chai Wah Wu, Dec 17 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
G.f.: 2*x*(13 - 12*x)/(1-x)^3. (End)
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(25)/25 = A001008(25)/A102928(25) = 34052522467/223092870000, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/25 - 19081066231/669278610000. (End)
E.g.f.: x*(26 + x)*exp(x). - G. C. Greubel, Mar 13 2022
MATHEMATICA
Table[n (n + 25), {n, 0, 50}] (* Bruno Berselli, Aug 22 2018 *)
LinearRecurrence[{3, -3, 1}, {0, 26, 54}, 60] (* Harvey P. Dale, Feb 20 2023 *)
PROG
(PARI) a(n)=n*(n+25) \\ Charles R Greathouse IV, Jun 17 2017
(Sage) [n*(n+25) for n in (0..50)] # G. C. Greubel, Mar 13 2022
CROSSREFS
Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A098603, A098847, A098848, A098849, A098850, A102928, A120071, A132759, A132760, A132761, A132762, A132763, A132764, A132765, A132766.
KEYWORD
nonn,easy
AUTHOR
Omar E. Pol, Aug 28 2007
STATUS
approved
A100345
Triangle read by rows: T(n,k) = n*(n+k), 0<=k<=n.
+10
9
0, 1, 2, 4, 6, 8, 9, 12, 15, 18, 16, 20, 24, 28, 32, 25, 30, 35, 40, 45, 50, 36, 42, 48, 54, 60, 66, 72, 49, 56, 63, 70, 77, 84, 91, 98, 64, 72, 80, 88, 96, 104, 112, 120, 128, 81, 90, 99, 108, 117, 126, 135, 144, 153, 162, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
T(n,0) = A000290(n);
T(n,1) = A002378(n) for n>0;
T(n,2) = A005563(n) for n>1;
T(n,3) = A028552(n) for n>2;
T(n,4) = A028347(n+2) for n>3;
T(n,5) = A028557(n) for n>4;
T(n,6) = A028560(n) for n>5;
T(n,7) = A028563(n) for n>6;
T(n,8) = A028566(n) for n>7;
T(n,9) = A028569(n) for n>8;
T(n,10) = A098603(n) for n>9;
T(n,n-5) = A071355(n-4) for n>4;
T(n,n-4) = A054000(n-1) for n>3;
T(n,n-3) = A014107(n) for n>2;
T(n,n-2) = A046092(n-1) for n>1;
T(n,n-1) = A000384(n) for n>0;
T(n,n) = A001105(n);
row-sums give A085789 for n>0.
LINKS
Table of n, a(n) for n=0..64.
EXAMPLE
0
1 2
4 6 8
9 12 15 18
16 20 24 28 32
25 30 35 40 45 50
36 42 48 54 60 66 72
49 56 63 70 77 84 91 98
64 72 80 88 96 104 112 120 128
MATHEMATICA
Table[n(n+k), {n, 0, 10}, {k, 0, n}]//Flatten (* Harvey P. Dale, Aug 16 2018 *)
CROSSREFS
Distinct members (except 0) are in A071562. Numbers occurring at least twice are in A175040. [Franklin T. Adams-Watters, Apr 04 2010]
KEYWORD
nonn,tabl,easy
AUTHOR
Reinhard Zumkeller, Nov 18 2004
STATUS
approved
A132768
a(n) = n*(n + 26).
+10
7
0, 27, 56, 87, 120, 155, 192, 231, 272, 315, 360, 407, 456, 507, 560, 615, 672, 731, 792, 855, 920, 987, 1056, 1127, 1200, 1275, 1352, 1431, 1512, 1595, 1680, 1767, 1856, 1947, 2040, 2135, 2232, 2331, 2432, 2535, 2640, 2747, 2856, 2967, 3080, 3195, 3312, 3431
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
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).
FORMULA
a(n) = n*(n + 26).
a(n) = 2*n + a(n-1) + 25, with a(0)=0. - Vincenzo Librandi, Aug 03 2010
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(26)/26 = A001008(26)/A102928(26) = 34395742267/232016584800, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 18051406831/696049754400. (End)
From G. C. Greubel, Mar 13 2022: (Start)
G.f.: x*(27 - 25*x)/(1-x)^3.
E.g.f.: x*(27 + x)*exp(x). (End)
MATHEMATICA
Table[n(n+26), {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 27, 56}, 50] (* Harvey P. Dale, Dec 15 2018 *)
PROG
(PARI) a(n)=n*(n+26) \\ Charles R Greathouse IV, Jun 17 2017
(Sage) [n*(n+26) for n in (0..50)] # G. C. Greubel, Mar 13 2022
CROSSREFS
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.
KEYWORD
nonn,easy
AUTHOR
Omar E. Pol, Aug 28 2007
STATUS
approved
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