A062708 -id:A062708 - cp's OEIS frontend

A226488 a(n) = n(13n - 9)/2.

0, 2, 17, 45, 86, 140, 207, 287, 380, 486, 605, 737, 882, 1040, 1211, 1395, 1592, 1802, 2025, 2261, 2510, 2772, 3047, 3335, 3636, 3950, 4277, 4617, 4970, 5336, 5715, 6107, 6512, 6930, 7361, 7805, 8262, 8732, 9215, 9711, 10220, 10742, 11277, 11825, 12386, 12960

Offset: 0

Bruno Berselli, Jun 09 2013

Sum of n-th octagonal number and n-th 9-gonal (nonagonal) number.

Sum of reciprocals of a(n), for n>0: 0.629618994194109711163742089971688...

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

Cf. A000567, A001106, A153080 (first differences).

Cf. numbers of the form n*(n*k-k+4)/2 listed in A005843 (k=0), A000096 (k=1), A002378 (k=2), A005449 (k=3), A001105 (k=4), A005476 (k=5), A049450 (k=6), A218471 (k=7), A002939 (k=8), A062708 (k=9), A135706 (k=10), A180223 (k=11), A139267 (n=12), this sequence (k=13), A139268 (k=14), A226489 (k=15), A139271 (k=16), A180232 (k=17), A152995 (k=18), A226490 (k=19), A152965 (k=20), A226491 (k=21), A152997 (k=22).

List([0..50], n-> n*(13*n-9)/2); # G. C. Greubel, Aug 30 2019

Magma
```
[n*(13*n-9)/2: n in [0..50]];
```

I:=[0,2,17]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2) +Self(n-3): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

A226488:=n->n*(13*n - 9)/2; seq(A226488(n), n=0..50); # Wesley Ivan Hurt, Feb 25 2014

Table[n(13n-9)/2, {n, 0, 50}]
LinearRecurrence[{3, -3, 1}, {0, 2, 17}, 50] (* Harvey P. Dale, Jun 19 2013 *)
CoefficientList[Series[x(2+11x)/(1-x)^3, {x, 0, 45}], x] (* Vincenzo Librandi, Aug 18 2013 *)

a(n)=n*(13*n-9)/2 \\ Charles R Greathouse IV, Sep 24 2015

[n*(13*n-9)/2 for n in (0..50)] # G. C. Greubel, Aug 30 2019

G.f.: x*(2+11*x)/(1-x)^3.
a(n) + a(-n) = A152742(n).
a(0)=0, a(1)=2, a(2)=17; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jun 19 2013
E.g.f.: x*(4 + 13*x)*exp(x)/2. - G. C. Greubel, Aug 30 2019
a(n) = A000567(n) + A001106(n). - Michel Marcus, Aug 31 2019

A218470 Partial sums of floor(n/9).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 141, 147, 153, 159, 165, 171, 177, 183, 189, 196, 203, 210, 217, 224

Offset: 0

Philippe Deléham, Mar 26 2013

Apart from the initial zeros, the same as A008727.

			As square array:
..0....0....0....0....0....0....0....0....0....
..1....2....3....4....5....6....7....8....9....
.11...13...15...17...19...21...23...25...27....
.30...33...36...39...42...45...48...51...54....
.58...62...66...70...74...78...82...86...90....
.95..100..105..110..115..120..125..130..135....
141..147..153..159..165..171..177..183..189....
196..203..210..217..224..231..238..245..252....
...

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

Cf. similar sequences: A118729, A174109, A174738.

[&+[Floor(k/9): k in [0..n]]: n in [0..70]]; // Bruno Berselli, Mar 27 2013

Accumulate[Floor[Range[0, 100]/9]] (* Jean-François Alcover, Mar 27 2013 *)

for(n=0,50, print1(sum(k=0,n, floor(k/9)), ", ")) \\ G. C. Greubel, Dec 13 2016

a(n)=my(k=n\9); k*(9*k-7)/2 + k*(n-9*k) \\ Charles R Greathouse IV, Dec 13 2016

a(9n)   = A051682(n).
a(9n+1) = A062708(n).
a(9n+2) = A062741(n).
a(9n+3) = A022266(n).
a(9n+4) = A022267(n).
a(9n+5) = A081266(n).
a(9n+6) = A062725(n).
a(9n+7) = A062728(n).
a(9n+8) = A027468(n).
G.f.: x^9/((1-x)^2*(1-x^9)). - Bruno Berselli, Mar 27 2013

A226492 a(n) = n(11n-5)/2.

Original entry on oeis.org

0, 3, 17, 42, 78, 125, 183, 252, 332, 423, 525, 638, 762, 897, 1043, 1200, 1368, 1547, 1737, 1938, 2150, 2373, 2607, 2852, 3108, 3375, 3653, 3942, 4242, 4553, 4875, 5208, 5552, 5907, 6273, 6650, 7038, 7437, 7847, 8268, 8700, 9143, 9597, 10062, 10538, 11025, 11523

Offset: 0

Bruno Berselli, Jun 11 2013

Sequences of numbers of the form n*(n*k - k + 6)/2:
. k from 0 to 10, respectively: A008585, A055998, A005563, A045943, A014105, A005475, A033428, A022264, A033991, A062741, A147874;
. k=11: a(n);
. k=12: A094159;
. k=13: 0, 3, 19, 48, 90, 145, 213, 294, 388, 495, 615, 748, 894, ...;
. k=14: 0, 3, 20, 51, 96, 155, 228, 315, 416, 531, 660, 803, 960, ...;
. k=15: A152773;
. k=16: A139272;
. k=17: 0, 3, 23, 60, 114, 185, 273, 378, 500, 639, 795, 968, ...;
. k=18: A152751;
. k=19: 0, 3, 25, 66, 126, 205, 303, 420, 556, 711, 885, 1078, ...;
. k=20: 0, 3, 26, 69, 132, 215, 318, 441, 584, 747, 930, 1133, ...;
. k=21: A152759;
. k=22: 0, 3, 28, 75, 144, 235, 348, 483, 640, 819, 1020, 1243, ...;
. k=23: 0, 3, 29, 78, 150, 245, 363, 504, 668, 855, 1065, 1298, ...;
. k=24: A152767;
. k=25: 0, 3, 31, 84, 162, 265, 393, 546, 724, 927, 1155, 1408, ...;
. k=26: 0, 3, 32, 87, 168, 275, 408, 567, 752, 963, 1200, 1463, ...;
. k=27: A153783;
. k=28: A195021;
. k=29: 0, 3, 35, 96, 186, 305, 453, 630, 836, 1071, 1335, 1628, ...;
. k=30: A153448;
. k=31: 0, 3, 37, 102, 198, 325, 483, 672, 892, 1143, 1425, 1738, ...;
. k=32: 0, 3, 38, 105, 204, 335, 498, 693, 920, 1179, 1470, 1793, ...;
. k=33: A153875.
Also:
a(n) - n         = A180223(n);
a(n) + n         = n*(11*n-3)/2 = 0, 4, 19, 45, 82, 130, 189, 259, ...;
a(n) - 2*n       = A051865(n);
a(n) + 2*n       = A022268(n);
a(n) - 3*n       = A152740(n-1);
a(n) + 3*n       = A022269(n);
a(n) - 4*n       = n*(11*n-13)/2 = 0, -1, 9, 30, 62, 105, 159, 224, ...;
a(n) + 4*n       = A254963(n);
a(n) - n*(n-1)/2 = A147874(n+1);
a(n) + n*(n-1)/2 = A094159(n) (case k=12);
a(n) - n*(n-1)   = A062741(n) (see above, this is the case k=9);
a(n) + n*(n-1)   = n*(13*n-7)/2 (case k=13);
a(n) - n*(n+1)/2 = A135706(n);
a(n) + n*(n+1)/2 = A033579(n);
a(n) - n*(n+1)   = A051682(n);
a(n) + n*(n+1)   = A186030(n);
a(n) - n^2       = A062708(n);
a(n) + n^2       = n*(13*n-5)/2 = 0, 4, 21, 51, 94, 150, 219, ..., etc.
Sum of reciprocals of a(n), for n > 0: 0.47118857003113149692081665034891...

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

Cf. sequences in Comments lines.
First differences are in A017425.
Cf. A033584, A180223.

Magma
```
[n*(11*n-5)/2: n in [0..50]];
```

I:=[0,3,17]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..46]]; // Vincenzo Librandi, Aug 18 2013

Table[n (11 n - 5)/2, {n, 0, 50}]
CoefficientList[Series[x (3 + 8 x) / (1 - x)^3, {x, 0, 45}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{3,-3,1},{0,3,17},50] (* Harvey P. Dale, Jan 14 2019 *)

a(n)=n*(11*n-5)/2 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: x*(3+8*x)/(1-x)^3.
a(n) + a(-n) = A033584(n).
From Elmo R. Oliveira, Dec 27 2024: (Start)
E.g.f.: exp(x)*x*(6 + 11*x)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
a(n) = n + A180223(n). (End)

A238731 Riordan array ((1-2x)/(1-3x+x^2), x/(1-3*x+x^2)).

Original entry on oeis.org

1, 1, 1, 2, 4, 1, 5, 13, 7, 1, 13, 40, 33, 10, 1, 34, 120, 132, 62, 13, 1, 89, 354, 483, 308, 100, 16, 1, 233, 1031, 1671, 1345, 595, 147, 19, 1, 610, 2972, 5561, 5398, 3030, 1020, 203, 22, 1, 1597, 8495, 17984, 20410, 13893, 5943, 1610, 268, 25, 1, 4181

Offset: 0

Philippe Deléham, Mar 03 2014

Unsigned version of A124037 and A126126.
Subtriangle of the triangle given by (0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Row sums are A001075(n).
Diagonal sums are A133494(n).
Sum_{k=0..n} T(n,k)*x^k = A001519(n), A001075(n), A002320(n), A038723(n), A033889(n) for x = 0, 1, 2, 3, 4 respectively. - Philippe Deléham, Mar 05 2014

			Triangle begins:
1;
1, 1;
2, 4, 1;
5, 13, 7, 1;
13, 40, 33, 10, 1;
34, 120, 132, 62, 13, 1;
89, 354, 483, 308, 100, 16, 1;
233, 1031, 1671, 1345, 595, 147, 19, 1;...
Triangle (0, 1, 1, 1, 0, 0, 0, ...) DELTA (1, 0, 2, -2, 0, 0, ...) begins:
1;
0, 1;
0, 1, 1;
0, 2, 4, 1;
0, 5, 13, 7, 1;
0, 13, 40, 33, 10, 1;
0, 34, 120, 132, 62, 13, 1;
0, 89, 354, 483, 308, 100, 16, 1;
0, 233, 1031, 1671, 1345, 595, 147, 19, 1;...

Cf. A001519, A000012, A016777, A062708.

Cf. A001906, A124037, A126126.

(* The function RiordanArray is defined in A256893. *)
RiordanArray[(1-2#)/(1-3#+#^2)&, x/(1-3#+#^2)&, 10] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n.

G.f.: (1-2*x)/(1-(y+3)*x+x^2). - Philippe Deléham, Mar 05 2014

A139610 a(n) = 45*n + 10.

Original entry on oeis.org

10, 55, 100, 145, 190, 235, 280, 325, 370, 415, 460, 505, 550, 595, 640, 685, 730, 775, 820, 865, 910, 955, 1000, 1045, 1090, 1135, 1180, 1225, 1270, 1315, 1360, 1405, 1450, 1495, 1540, 1585, 1630, 1675, 1720, 1765, 1810, 1855, 1900

Offset: 0

Omar E. Pol, Apr 27 2008

Numbers of the 10th column of positive numbers in the square array of nonnegative and polygonal numbers A139600.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A017185, A057145, A062708, A139600.

[5*(9*n + 2): n in [0..60]]; // Vincenzo Librandi, Jul 23 2011

45*Range[0,50]+10 (* or *) LinearRecurrence[{2,-1},{10,55},50] (* Harvey P. Dale, Dec 27 2021 *)

a(n)=45*n+10 \\ Charles R Greathouse IV, Oct 05 2011

a(n) = A057145(n+2,10).
G.f.: 5*(2+7*x)/(x-1)^2. - R. J. Mathar, Jul 28 2016
From Elmo R. Oliveira, Apr 16 2024: (Start)
E.g.f.: 5*exp(x)*(2 + 9*x).
a(n) = 5*A017185(n) = 5*(A062708(n+1) - A062708(n)).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

A198392 a(n) = (6n(3n+7)+(2n+13)*(-1)^n+3)/16 + 1.

Original entry on oeis.org

2, 4, 12, 18, 31, 41, 59, 73, 96, 114, 142, 164, 197, 223, 261, 291, 334, 368, 416, 454, 507, 549, 607, 653, 716, 766, 834, 888, 961, 1019, 1097, 1159, 1242, 1308, 1396, 1466, 1559, 1633, 1731, 1809, 1912, 1994, 2102, 2188, 2301, 2391, 2509, 2603, 2726, 2824, 2952

Offset: 0

Bruno Berselli, Oct 25 2011

For an origin of this sequence, see the triangular spiral illustrated in the Links section.

First bisection gives A117625 (without the initial term).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A152832 (by Superseeker).

Cf. sequences related to the triangular spiral: A022266, A022267, A027468, A038764, A045946, A051682, A062708, A062725, A062728, A062741, A064225, A064226, A081266-A081268, A081270-A081272, A081275 [incomplete list].

[(6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16+1: n in [0..50]];

LinearRecurrence[{1,2,-2,-1,1},{2,4,12,18,31},60] (* Harvey P. Dale, Jun 15 2022 *)

for(n=0, 50, print1((6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16+1", "));

G.f.: (2+2*x+4*x^2+2*x^3-x^4)/((1+x)^2*(1-x)^3).
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5).
a(n)-a(-n-1) = A168329(n+1).
a(n)+a(n-1) = A102214(n).
a(2n)-a(2n-1) = A016885(n).
a(2n+1)-a(2n) = A016825(n).

A304503 a(n) = 3(n+1)(9*n+4).

Original entry on oeis.org

12, 78, 198, 372, 600, 882, 1218, 1608, 2052, 2550, 3102, 3708, 4368, 5082, 5850, 6672, 7548, 8478, 9462, 10500, 11592, 12738, 13938, 15192, 16500, 17862, 19278, 20748, 22272, 23850, 25482, 27168, 28908, 30702, 32550, 34452, 36408, 38418, 40482, 42600, 44772

Offset: 0

Emeric Deutsch, May 13 2018

The first Zagreb index of the single-defect 3-gonal nanocone CNC(3,n) (see definition in the Doslic et al. reference, p. 27).
The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternatively, it is the sum of the degree sums d(i) + d(j) over all edges ij of the graph.
The M-polynomial of CNC(3,n) is M(CNC(3,n);x,y) = 3*x^2*y^2 + 6*n*x^2*y^3 + 3*n*(3*n+1)*x^3*y^3/2.
More generally, the M-polynomial of CNC(k,n) is M(CNC(k,n);x,y) = k*x^2*y^2 + 2*k*n*x^2*y^3 + k*n*(3*n + 1)*x^3*y^3/2.
12*a(n) + 25 is a square. - Bruno Berselli, May 14 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Emeric Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, Vol. 6, No. 2, 2015, pp. 93-102.
T. Doslic and M. Saheli, Augmented eccentric connectivity index of single-defect nanocones, J. of Mathematical Nanoscience, Vol. 1, No. 1, 2011, pp. 25-31.
A. Khaksar, M. Ghorbani, and H. R. Maimani, On atom bond connectivity and GA indices of nanocones, Optoelectronics and Advanced Materials - Rapid Communications, Vol. 4, No. 11, 2010, pp. 1868-1870.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A008585, A017209, A062708, A304504.

Maple
```
seq((3*(n+1))*(9*n+4), n = 0 .. 40);
```

Vec(6*(2 + 7*x) / (1 - x)^3 + O(x^40)) \\ Colin Barker, May 14 2018

From Colin Barker, May 14 2018: (Start)
G.f.: 6*(2 + 7*x)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
From Elmo R. Oliveira, Nov 15 2024: (Start)
E.g.f.: 3*exp(x)*(4 + 22*x + 9*x^2).
a(n) = 6*A062708(n+1) = A017209(n)*A008585(n+1). (End)

A304505 a(n) = 4(n+1)(9*n+4).

Original entry on oeis.org

16, 104, 264, 496, 800, 1176, 1624, 2144, 2736, 3400, 4136, 4944, 5824, 6776, 7800, 8896, 10064, 11304, 12616, 14000, 15456, 16984, 18584, 20256, 22000, 23816, 25704, 27664, 29696, 31800, 33976, 36224, 38544, 40936, 43400, 45936, 48544, 51224, 53976, 56800, 59696

Offset: 0

Emeric Deutsch, May 14 2018

a(n) is the first Zagreb index of the single-defect 4-gonal nanocone CNC(4,n) (see definition in the Doslic et al. reference, p. 27).
The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternatively, it is the sum of the degree sums d(i) + d(j) over all edges ij of the graph.
The M-polynomial of CNC(4,n) is M(CNC(4,n); x,y) = 4*x^2*y^2 + 8*n*x^2*y^3 + 2*n*(3*n+1)*x^3*y^3.
More generally, the M-polynomial of CNC(k,n) is M(CNC(k,n); x,y) = k*x^2*y^2 + 2*k*n*x^2*y^3 + k*n*(3*n + 1)*x^3*y^3/2.
9*a(n) + 25 is a square. - Bruno Berselli, May 14 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Emeric Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, Vol. 6, No. 2, 2015, pp. 93-102.
T. Doslic and M. Saheli, Augmented eccentric connectivity index of single-defect nanocones, J. of Mathematical Nanoscience, Vol. 1, No. 1, 2011, pp. 25-31.
A. Khaksar, M. Ghorbani, and H. R. Maimani, On atom bond connectivity and GA indices of nanocones, Optoelectronics and Advanced Materials - Rapid Communications, Vol. 4, No. 11, 2010, pp. 1868-1870.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A008586, A017209, A062708, A304506.

List([0..50],n->4*(n+1)*(9*n+4)); # Muniru A Asiru, May 14 2018

Maple
```
seq((4*(n+1))*(9*n+4), n = 0 .. 40);
```

a(n) = 4*(n+1)*(9*n+4); \\ Altug Alkan, May 14 2018

Vec(8*(2 + 7*x) / (1 - x)^3 + O(x^40)) \\ Colin Barker, May 14 2018

From Colin Barker, May 14 2018: (Start)
G.f.: 8*(2 + 7*x)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
From Elmo R. Oliveira, Nov 15 2024: (Start)
E.g.f.: 4*exp(x)*(4 + 22*x + 9*x^2).
a(n) = 8*A062708(n+1) = A017209(n)*A008586(n+1). (End)

A304507 a(n) = 5(n+1)(9*n+4).

Original entry on oeis.org

20, 130, 330, 620, 1000, 1470, 2030, 2680, 3420, 4250, 5170, 6180, 7280, 8470, 9750, 11120, 12580, 14130, 15770, 17500, 19320, 21230, 23230, 25320, 27500, 29770, 32130, 34580, 37120, 39750, 42470, 45280, 48180, 51170, 54250, 57420, 60680, 64030, 67470, 71000, 74620

Offset: 0

Emeric Deutsch, May 14 2018

The first Zagreb index of the single-defect 5-gonal nanocone CNC(5,n) (see definition in the Doslic et al. reference, p. 27).
The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternatively, it is the sum of the degree sums d(i) + d(j) over all edges ij of the graph.
The M-polynomial of CNC(5,n) is M(CNC(5,n); x,y) = 5*x^2*y^2 + 10*n*x^2*y^3 + 5*n*(3*n+1)*x^3*y^3/2.
More generally, the M-polynomial of CNC(k,n) is M(CNC(k,n); x,y) = k*x^2*y^2 + 2*k*n*x^2*y^3 + k*n*(3*n + 1)*x^3*y^3/2.

Colin Barker, Table of n, a(n) for n = 0..1000
Emeric Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, Vol. 6, No. 2, 2015, pp. 93-102.
T. Doslic and M. Saheli, Augmented eccentric connectivity index of single-defect nanocones, Journal of Mathematical Nanoscience, Vol. 1, No. 1, 2011, pp. 25-31.
A. Khaksar, M. Ghorbani, and H. R. Maimani, On atom bond connectivity and GA indices of nanocones, Optoelectronics and Advanced Materials - Rapid Communications, Vol. 4, No. 11, 2010, pp. 1868-1870.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A008587, A011862, A017209, A062708, A108579, A304508.

List([0..50], n -> 5*(n+1)*(9*n+4)); # Muniru A Asiru, May 15 2018

Maple
```
seq((5*(n+1))*(9*n+4), n = 0 .. 40);
```

Array[5 (# + 1) (9 # + 4) &, 41, 0] (* or *)
LinearRecurrence[{3, -3, 1}, {20, 130, 330}, 41] (* or *)
CoefficientList[Series[10 (2 + 7 x)/(1 - x)^3, {x, 0, 40}], x] (* Michael De Vlieger, May 14 2018 *)

a(n) = 5*(n+1)*(9*n+4); \\ Altug Alkan, May 14 2018

Vec(10*(2 + 7*x) / (1 - x)^3 + O(x^40)) \\ Colin Barker, May 14 2018

From Colin Barker, May 14 2018: (Start)
G.f.: 10*(2 + 7*x)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
a(n) = 10*A062708(n+1) for n >= 0. - Robert G. Wilson v, May 14 2018
a(n) = 5*A011862(9*n+7) = 5*A108579(6*n+7). - Bruno Berselli, May 15 2018
From Elmo R. Oliveira, Nov 15 2024: (Start)
E.g.f.: 5*exp(x)*(4 + 22*x + 9*x^2).
a(n) = 5*A017209(n)*A008587(n+1). (End)

A194582 Triangle T(n,k), read by rows, given by (0, 3, -7/3, -2/21, 3/7, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 2, 6, 1, 0, 2, 13, 9, 1, 0, 2, 16, 33, 12, 1, 0, 2, 20, 69, 62, 15, 1, 0, 2, 24, 108, 188, 100, 18, 1, 0, 2, 28, 156, 401, 400, 147, 21, 1, 0, 2, 32, 212, 704, 1115, 732, 203, 24, 1, 0, 2, 36, 276, 1120, 2433, 2547, 1211, 268, 27, 1

Offset: 0

Philippe Deléham, Jan 23 2012

Riordan array (1, x*(1+2x-x^2)/(1-x)).
Row sums are (Fibonacci(n+1))^2 = A007598(n+1).
T(n, k) is the number of ordered pairs of Fibonacci bit strings of length n with the number of matching 1 bits in the same position is k. A Fibonacci bit string begins a 1 bit and no two consecutive bits are 0 bits. - Michael Somos, Feb 28 2020

			Triangle begins:
  1;
  0,   1;
  0,   3,   1;
  0,   2,   6,   1;
  0,   2,  13,   9,   1;
  0,   2,  16,  33,  12,   1;
  0,   2,  20,  69,  62,  15,   1;
  0,   2,  24, 108, 188, 100,  18,   1;
  0,   2,  28, 156, 401, 400, 147,  21,   1;
T(3, 2) = 6 enumerates the pairs of Fibonacci bit string of length 3 with 2 matching 1 bits: (101, 101), (101, 111), (110, 110), (110, 111), (111, 101), (111, 110). - _Michael Somos_, Feb 28 2020

Indranil Ghosh, Rows 0..100, flattened

Cf. A000045, A007598. Diagonals: A000012, A008585, A062708.

nmax=10; Flatten[CoefficientList[Series[CoefficientList[Series[(1 - x)/(1 - x - x*y - 2*x^2*y + x^3*y)  , {x,  0, nmax}], x], {y, 0, nmax}], y]] (* Indranil Ghosh, Mar 10 2017, after  R. J. Mathar *)

T(n,k) = if(n==k, 1, if(k==0, 0, if(n>1 && k==n - 1, 3*k, T(n - 1, k) + T(n - 1,k - 1) + 2*T(n - 2,k - 1) - T(n-3,k-1))));
{for(n=0, 10, for(k=0, n, print1(T(n,k),", ");); print();); } \\ Indranil Ghosh, Mar 10 2017

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

G.f.: (1-x)/(1-x-x*y-2*x^2*y+x^3*y). - R. J. Mathar, Aug 11 2015

cp's OEIS Frontend

A226488 a(n) = n*(13*n - 9)/2.

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A218470 Partial sums of floor(n/9).

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A226492 a(n) = n*(11*n-5)/2.

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A238731 Riordan array ((1-2*x)/(1-3*x+x^2), x/(1-3*x+x^2)).

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A139610 a(n) = 45*n + 10.

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A198392 a(n) = (6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16 + 1.

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A304503 a(n) = 3*(n+1)*(9*n+4).

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A226488 a(n) = n(13n - 9)/2.

A226492 a(n) = n(11n-5)/2.

A238731 Riordan array ((1-2x)/(1-3x+x^2), x/(1-3*x+x^2)).

A198392 a(n) = (6n(3n+7)+(2n+13)*(-1)^n+3)/16 + 1.

A304503 a(n) = 3(n+1)(9*n+4).

A304505 a(n) = 4(n+1)(9*n+4).

A304507 a(n) = 5(n+1)(9*n+4).