A147874 -id:A147874 - cp's OEIS frontend

A294828 Numerators of the partial sums of the reciprocals of the numbers (k + 1)(5k + 3) = A147874(k+2), for k >= 0.

1, 19, 263, 815, 95597, 678149, 7531399, 18016577, 259695727, 4173941423, 222039686299, 2153029760377, 19428099753313, 331021112488901, 24211723390477517, 12126560607807901, 1008024074147249303, 168229609596886043, 1740462375346732831, 1219642439745618215

Offset: 0

Wolfdieter Lang, Nov 16 2017

The corresponding denominators are given in A294829.
For the general case V(m,r;n) = Sum_{k=0..n} 1/((k + 1)*(m*k + r)) = (1/(m - r))*Sum_{k=0..n} (m/(m*k + r) - 1/(k+1)), for r = 1, ..., m-1 and m = 2, 3, ..., and their limits see a comment in A294512. Here [m,r] = [5,3].
The limit of the series is V(5,3) = lim_{n -> oo} V(5,3;n) = ((5/2)*log(5) - (2*phi-1)*(log(phi) + (Pi/5)*sqrt(7-4*phi)))/4, with the golden section phi:= (1 + sqrt(5))/2 = A001622. The value is 0.48170177449... given in A294830.
In the Koecher reference v_5(3) = (2/5)*V(5,3) = 0.19268070979833151082... given there by ((1/4)*log(5) - (1/(2*sqrt(5)))*log((1+sqrt(5))/2) - (Pi/10)*sqrt((5 - 2*sqrt(5))/5)).

			The rationals V(5,3;n), n >= 0, begin: 1/3, 19/48, 263/624, 815/1872, 95597/215280, 678149/1506960, 7531399/16576560, 18016577/39369330, 259695727/564293730, 4173941423/9028699680, 222039686299/478521083040, 2153029760377/4625703802720, 19428099753313/41631334224480, ...
V(5,3;10^6) = 0.4817015746 to be compared with 0.4817017745 from A294830 with 10 digits.

Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189 - 193.

Robert Israel, Table of n, a(n) for n = 0..891
Eric Weisstein's World of Mathematics, Digamma Function

Cf. A001620, A001622, A147874, A294512, A294826/A294827 (V(5,2;n)), A294829, A294830,

map(numer,ListTools:-PartialSums([seq(1/(k+1)/(5*k+3),k=0..50)])); # Robert Israel, Nov 17 2017

a(n) = numerator(sum(k=0, n, 1/((k + 1)*(5*k + 3)))); \\ Michel Marcus, Nov 17 2017

a(n) = numerator(V(5,3;n)) with V(5,3;n) = Sum_{k=0..n} 1/((k + 1)*(5*k + 3)) = Sum_{k=0..n} 1/A147874(k+2) = (1/2)*Sum_{k=0..n} (1/(k + 3/5) - 1/(k+1)) = (-Psi(3/5) + Psi(n+8/5) - (gamma + Psi(n+2)))/2 with the digamma function Psi and the Euler-Mascheroni constant gamma = -Psi(1) from A001620.

A294829 Denominators of the partial sums of the reciprocals of the numbers (k + 1)(5k + 3) = A147874(k+2), for k >= 0.

Original entry on oeis.org

3, 48, 624, 1872, 215280, 1506960, 16576560, 39369330, 564293730, 9028699680, 478521083040, 4625703802720, 41631334224480, 707732681816160, 51664485772579680, 25832242886289840, 2144076159562056720, 357346026593676120, 3692575608134653240, 2584802925694257268

Offset: 0

Wolfdieter Lang, Nov 16 2017

The corresponding numerators are given in A294828. Details are found there.

			For the rationals see A294828.

Robert Israel, Table of n, a(n) for n = 0..891

Cf. A147874, A294828, A294830.

map(denom,ListTools:-PartialSums([seq(1/(k+1)/(5*k+3),k=0..50)])); # Robert Israel, Nov 17 2017

a(n) = denominator(sum(k=0, n, 1/((k + 1)*(5*k + 3)))); \\ Michel Marcus, Nov 17 2017

a(n) = denominator(V(5,3;n)) with V(5,3;n) = Sum_{k=0..n} 1/((k + 1)*(5*k + 3)) = Sum_{k=0..n} 1/A147874(k+2) = (1/2)*Sum_{k=0..n} (1/(k + 3/5) - 1/(k+1)). For this sum in terms of the digamma function see A294828.

A294830 Decimal expansion of the sum of the reciprocals of the numbers (k+1)(5k+3) = A147874(k+2) for k >= 0.

Original entry on oeis.org

4, 8, 1, 7, 0, 1, 7, 7, 4, 4, 9, 5, 8, 2, 8, 7, 7, 7, 0, 7, 7, 0, 7, 5, 9, 2, 9, 3, 6, 1, 9, 1, 4, 7, 5, 5, 2, 3, 4, 1, 8, 7, 4, 5, 9, 3, 7, 4, 8, 4, 1, 8, 0, 4, 7, 3, 0, 4, 5, 9, 0, 1, 4, 1, 8, 8, 1, 5, 0, 5, 5, 7, 2, 3, 1, 7, 1, 8, 8, 9, 7, 5, 6, 8, 1, 9, 7, 7, 0, 2, 2, 1, 4, 0, 1, 6, 0, 3, 5

Offset: 0

Wolfdieter Lang, Nov 16 2017

In the Koecher reference v_5(3) = (2/5)*(present value V(5,3)) = 0.192680709798338..., given there as (1/4)*log(5) - (1/(2*sqrt(5)))*log((1 + sqrt(5))/2) - (Pi/10)*sqrt((5 - 2*sqrt(5))/5).

			0.481701774495828777077075929361914755234187459374841804730459014188150...

Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189 - 193.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Digamma Function

Cf. A001620, A001622, A147874, A294828/A294829.

SetDefaultRealField(RealField(100)); R:= RealField(); phi:= (1 + Sqrt(5))/2; ((5/2)*Log(5) - (2*phi-1)*(Log(phi) + (Pi(R)/5)*Sqrt(7 - 4*phi)))/4; // G. C. Greubel, Aug 30 2018

RealDigits[((5/2)*Log[5] - (2*GoldenRatio - 1)*(Log[GoldenRatio] + (Pi/5)*Sqrt[7 - 4*GoldenRatio]))/4, 10, 100][[1]] (* G. C. Greubel, Aug 30 2018 *)

default(realprecision, 100); phi=(1+sqrt(5))/2; ((5/2)*log(5) - (2*phi-1)*(log(phi) + (Pi/5)*sqrt(7-4*phi)))/4 \\ G. C. Greubel, Aug 30 2018

Sum_{k>=0} 1/((5*n + 3)*(n + 1)) =: V(5,3) = ((5/2)*log(5) - (2*phi-1)*(log(phi) + (Pi/5)*sqrt(7-4*phi)))/4 = (1/2)*(-Psi(3/5) + Psi(1)) with the golden section phi =(1 + sqrt(5))/2 = A001622 with the digamma function Psi and Psi(1) = -gamma = A001620.

The partial sums of this series are given in A294828/A294829.

A131242 Partial sums of A059995: a(n) = sum_{k=0..n} floor(k/10).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 156, 162, 168, 174, 180, 186, 192, 198

Offset: 0

Hieronymus Fischer, Jun 21 2007

Complementary with A130488 regarding triangular numbers, in that A130488(n)+10*a(n)=n(n+1)/2=A000217(n).

			As square array :
    0,   0,   0,   0,   0,   0,   0,   0,   0,    0
    1,   2,   3,   4,   5,   6,   7,   8,   9,   10
   12,  14,  16,  18,  20,  22,  24,  26,  28,   30
   33,  36,  39,  42,  45,  48,  51,  54,  57,   60
   64,  68,  72,  76,  80,  84,  88,  92,  96,  100
  105, 110, 115, 120, 125, 130, 135, 140, 145,  150
  156, 162, 168, 174, 180, 186, 192, 198, 204,  210
... - _Philippe Deléham_, Mar 27 2013

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

Cf. A008728, A059995, A010879, A002266, A130488, A000217, A002620, A130518, A130519, A130520, A174709, A174738, A118729, A218470.

Table[(1/2)*Floor[n/10]*(2*n - 8 - 10*Floor[n/10]), {n,0,50}] (* G. C. Greubel, Dec 13 2016 *)
Accumulate[Table[FromDigits[Most[IntegerDigits[n]]],{n,0,110}]] (* or *) LinearRecurrence[{2,-1,0,0,0,0,0,0,0,1,-2,1},{0,0,0,0,0,0,0,0,0,0,1,2},120] (* Harvey P. Dale, Apr 06 2017 *)

for(n=0,50, print1((1/2)*floor(n/10)*(2n-8-10*floor(n/10)), ", ")) \\ G. C. Greubel, Dec 13 2016

a(n)=my(k=n\10); k*(n-5*k-4) \\ Charles R Greathouse IV, Dec 13 2016

a(n) = (1/2)*floor(n/10)*(2n-8-10*floor(n/10)).
a(n) = A059995(n)*(2n-8-10*A059995(n))/2.
a(n) = (1/2)*A059995(n)*(n-8+A010879(n)).
a(n) = (n-A010879(n))*(n+A010879(n)-8)/20.
G.f.: x^10/((1-x^10)(1-x)^2).
From Philippe Deléham, Mar 27 2013: (Start)
a(10n)   = A051624(n).
a(10n+1) = A135706(n).
a(10n+2) = A147874(n+1).
a(10n+3) = 2*A005476(n).
a(10n+4) = A033429(n).
a(10n+5) = A202803(n).
a(10n+6) = A168668(n).
a(10n+7) = 2*A147875(n).
a(10n+8) = A135705(n).
a(10n+9) = A124080(n). (End)
a(n) = A008728(n-10) for n>= 10. - Georg Fischer, Nov 03 2018

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)

A115006 Row 2 of array in A114999.

Original entry on oeis.org

0, 3, 8, 16, 26, 39, 54, 72, 92, 115, 140, 168, 198, 231, 266, 304, 344, 387, 432, 480, 530, 583, 638, 696, 756, 819, 884, 952, 1022, 1095, 1170, 1248, 1328, 1411, 1496, 1584, 1674, 1767, 1862, 1960, 2060, 2163, 2268, 2376, 2486, 2599, 2714, 2832, 2952, 3075, 3200

Offset: 0

N. J. A. Sloane, Feb 23 2006

Number of lattice points (x,y) in the region of the coordinate plane bounded by y < 3x+1, y > x/2 and x <= n. - Wesley Ivan Hurt, Oct 27 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A114999, A000217 (triangular numbers), A002620 (quarter-squares), A001859 (triangular numbers plus quarter-squares), A017305 (10n+3), A147874 (zero followed by partial sums of A017305).

Partial Sums of A047218.

[ n*(n+1) + (n+1)^2 div 4: n in [0..50] ];

A115006:=n->(10*n^2 + 12*n + 1 - (-1)^n)/8: seq(A115006(n), n=0..50); # Wesley Ivan Hurt, Oct 27 2014

Table[(10*n^2 + 12*n + 1 - (-1)^n)/8, {n, 0, 50}] (* Wesley Ivan Hurt, Oct 27 2014 *)
LinearRecurrence[{2,0,-2,1},{0,3,8,16},60] (* Harvey P. Dale, Jan 13 2015 *)

{for(n=0, 50, print1(n*(n+1)+floor((n+1)^2/4), ","))}

a(n) = floor((n+1)^2/4)+n*(n+1).
G.f.: x*(2*x+3)/((1-x)^3*(1+x)).
From Wesley Ivan Hurt, Oct 27 2014: (Start)
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4).
a(n) = (10*n^2 + 12*n + 1 - (-1)^n)/8.
a(n) = Sum_{i=1..n+1} (10*i + (-1)^i - 9)/4. (End)
E.g.f.: (x*(11 + 5*x)*cosh(x) + (1 + 11*x + 5*x^2)*sinh(x))/4. - Stefano Spezia, Aug 22 2023

Edited by Klaus Brockhaus, Nov 18 2008

A242771 Number of integer points in a certain quadrilateral scaled by a factor of n (another version).

Original entry on oeis.org

0, 0, 1, 3, 6, 9, 14, 19, 25, 32, 40, 48, 58, 68, 79, 91, 104, 117, 132, 147, 163, 180, 198, 216, 236, 256, 277, 299, 322, 345, 370, 395, 421, 448, 476, 504, 534, 564, 595, 627, 660, 693, 728, 763, 799, 836, 874, 912, 952, 992, 1033, 1075, 1118, 1161, 1206

Offset: 1

Michael Somos, May 22 2014

nonn

The quadrilateral is given by four vertices [(1/2, 1/3), (0, 1), (0, 0), (1, 0)] as an example on page 22 of Ehrhart 1967. Here the open line segment from (1/2, 1/3) to (0, 1) is included but the rest of the boundary is not. The sequence is denoted by d'(n).
From Gus Wiseman, Oct 18 2020: (Start)
Also the number of ordered triples of positive integers summing to n that are not strictly increasing. For example, the a(3) = 1 through a(7) = 14 triples are:
  (1,1,1)  (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)
           (1,2,1)  (1,2,2)  (1,3,2)  (1,3,3)
           (2,1,1)  (1,3,1)  (1,4,1)  (1,4,2)
                    (2,1,2)  (2,1,3)  (1,5,1)
                    (2,2,1)  (2,2,2)  (2,1,4)
                    (3,1,1)  (2,3,1)  (2,2,3)
                             (3,1,2)  (2,3,2)
                             (3,2,1)  (2,4,1)
                             (4,1,1)  (3,1,3)
                                      (3,2,2)
                                      (3,3,1)
                                      (4,1,2)
                                      (4,2,1)
                                      (5,1,1)
A001399(n-6) counts the complement (unordered strict triples).
A014311 \ A333255 ranks these compositions.
A140106 is the unordered version.
A337484 is the case not strictly decreasing either.
A337698 counts these compositions of any length, with complement A000009.
A001399(n-6) counts unordered strict triples.
A001523 counts unimodal compositions, with complement A115981.
A007318 and A097805 count compositions by length.
A069905 counts unordered triples.
A218004 counts strictly increasing or weakly decreasing compositions.
A337483 counts triples either weakly increasing or weakly decreasing.
Cf. A332834, A337461, A337481, A337482, A337604.
(End)

			G.f. = x^3 + 3*x^4 + 6*x^5 + 9*x^6 + 14*x^7 + 19*x^8 + 25*x^9 + 32*x^10 + ...

E. Ehrhart, Sur un problème de géométrie diophantienne linéaire I, (Polyèdres et réseaux), J. Reine Angew. Math. 226 1967 1-29. MR0213320 (35 #4184).
E. Ehrhart, Sur un problème de géométrie diophantienne linéaire I, (Polyèdres et réseaux), J. Reine Angew. Math. 226 1967 1-29. MR0213320 (35 #4184). [Annotated scanned copy of pages 16 and 22 only]
E. Ehrhart, Sur un problème de géométrie diophantienne linéaire II. Systemes diophantiens lineaires, J. Reine Angew. Math. 227 1967 25-49. [Annotated scanned copy of pages 47-49 only]
Wikipedia, Ehrhart polynomial
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Cf. A002789, A010761, A147874.

Cf. A000212, A000217, A001840, A046691, A128422, A156040.

[Floor((5*n-7)*(n-1)/12): n in [1..60]]; // Vincenzo Librandi, Jun 27 2015

a[ n_] := Quotient[ 7 - 12 n + 5 n^2, 12];
a[ n_] := With[ {o = Boole[ 0 < n], c = Boole[ 0 >= n], m = Abs@n}, Length @ FindInstance[ 0 < c + x && 0 < c + y && (2 x < c + m && 4 x + 3 y < o + 3 m || m < o + 2 x && 2 x + 3 y < c + 2 m), {x, y}, Integers, 10^9]];
LinearRecurrence[{1,1,0,-1,-1,1},{0,0,1,3,6,9},90] (* Harvey P. Dale, May 28 2015 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],!Less@@#&]],{n,0,15}] (* Gus Wiseman, Oct 18 2020 *)

PARI
```
{a(n) = (7 - 12*n + 5*n^2) \ 12};
```

{a(n) = if( n<0, polcoeff( x * (2 + x^2 + x^3 + x^4) / ((1 - x)^2 * (1 - x^6)) + x * O(x^-n), -n), polcoeff( x^3 * (1 + x + x^2 + 2*x^4) / ((1 - x)^2 * (1 - x^6)) + x * O(x^n), n))};

G.f.: x^3 * (1 + 2*x + 2*x^2) / (1 - x - x^2 + x^4 + x^5 - x^6) = (x^3 + x^4 + x^5 + 2*x^7) / ((1 - x)^2 * (1 - x^6)).
a(n) = floor( A147874(n) / 12).
a(-n) = A002789(n).
a(n+1) - a(n) = A010761(n).
For n >= 6, a(n) = A000217(n-2) - A001399(n-6). - Gus Wiseman, Oct 18 2020

A193872 Even dodecagonal numbers: a(n) = 4n(5*n - 2).

Original entry on oeis.org

0, 12, 64, 156, 288, 460, 672, 924, 1216, 1548, 1920, 2332, 2784, 3276, 3808, 4380, 4992, 5644, 6336, 7068, 7840, 8652, 9504, 10396, 11328, 12300, 13312, 14364, 15456, 16588, 17760, 18972, 20224, 21516, 22848, 24220, 25632, 27084, 28576, 30108, 31680, 33292, 34944

Offset: 0

Omar E. Pol, Aug 19 2011

Even 12-gonal numbers. Bisection of A051624.

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

Cf. A016885, A051624, A147874.

PolygonalNumber[12,2*Range[0,40]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 02 2017 *)

a(n)=4*n*(5*n-2) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 4*A147874(n+1).
a(n) = 4*n*A016885(n-1), n >= 1.
From Elmo R. Oliveira, Dec 15 2024: (Start)
G.f.: 4*x*(3 + 7*x)/(1 - x)^3.
E.g.f.: 4*x*(3 + 5*x)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)

A180578 The Wiener index of the Dutch windmill graph D(6,n) (n>=1).

Original entry on oeis.org

27, 144, 351, 648, 1035, 1512, 2079, 2736, 3483, 4320, 5247, 6264, 7371, 8568, 9855, 11232, 12699, 14256, 15903, 17640, 19467, 21384, 23391, 25488, 27675, 29952, 32319, 34776, 37323, 39960, 42687, 45504, 48411, 51408, 54495, 57672, 60939, 64296, 67743, 71280, 74907

Offset: 1

Emeric Deutsch, Sep 30 2010

The Dutch windmill graph D(m,n) (also called friendship graph) is the graph obtained by taking n copies of the cycle graph C_m with a vertex in common (i.e., a bouquet of n C_m graphs).

The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.

			a(1)=27 because in D(6,1)=C_6 we have 6 distances equal to 1, 6 distances equal to 2, and 3 di stances equal to 3.

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., Vol. 60, 1996, pp. 959-969.
Eric Weisstein's World of Mathematics, Dutch Windmill Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A014642, A033991, A147874, A180579, A180867.

Maple
```
seq(9*n*(5*n-2), n = 1 .. 40);
```

a(n)=9*n*(5*n-2) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = A180867(6,n).
a(n) = 9*n*(5*n-2).
The Wiener polynomial of the graph D(6,n) is (1/2)nt(t^2+2t+2)((n-1)t^3+2(n-1)t^2+2(n-1)t+6).
G.f.: -9*x*(7*x+3)/(x-1)^3. - Colin Barker, Oct 31 2012
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: 9*exp(x)*x*(3 + 5*x).
a(n) = 9*A147874(n+1).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

More terms from Elmo R. Oliveira, Apr 03 2025

Duplicated a(38) removed by Sean A. Irvine, Apr 14 2025

A330082 a(n) = 5*A064038(n+1).

Original entry on oeis.org

0, 5, 15, 15, 25, 75, 105, 70, 90, 225, 275, 165, 195, 455, 525, 300, 340, 765, 855, 475, 525, 1155, 1265, 690, 750, 1625, 1755, 945, 1015, 2175, 2325, 1240, 1320, 2805, 2975, 1575, 1665, 3515, 3705, 1950, 2050, 4305, 4515, 2365, 2475, 5175, 5405, 2820, 2940

Offset: 0

Paul Curtz, Dec 01 2019

Main column of a pentagonal spiral for A026741:
                       (25)
                    49 (15) 31
                24  29 (15)  8  16
            47  14   7 ( 5)  3  17  33
        23  27  13   2 ( 0)  1   7   9  17
          45  13   6   3   1   4  19  35
            22  25  11   5   9  10  18
              43  12  23  11  21  37
                21  41  20  39  19
a(n) = 5 * A064038(n+1) from a pentagonal spiral.
Compare to A319127 =  6 * A002620 in the hexagonal spiral:
                22  23  23  22 (24)
              20  12  13  13 (12) 25
            21  10   5   4 ( 6) 14  25
          21  11   5   1 ( 0)  7  15  24
        20  11   4   1 ( 0)  2   7  15  26
          18  10   2   3   3   6  14  27
            19   8   9   9   8  16  27
              19  18  16  17  17  26
                30  28  29  29  28

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-6,10,-12,12,-10,6,-3,1).

Cf. A026741, A028895, A064038. A033429, A062786, A087348, A147874, A158447, A168668 are in the spiral.

A330082[n_]:=5Numerator[n(n+1)/4];Array[A330082,100,0] (* Paolo Xausa, Dec 04 2023 *)

concat(0, Vec(5*x*(1 + 4*x^3 + x^6) / ((1 - x)^3*(1 + x^2)^3) + O(x^50))) \\ Colin Barker, Dec 08 2019

a(n) = A026741(A028895(n)).
From Colin Barker, Dec 08 2019: (Start)
G.f.: 5*x*(1 + 4*x^3 + x^6) / ((1 - x)^3*(1 + x^2)^3).
a(n) = 3*a(n-1) - 6*a(n-2) + 10*a(n-3) - 12*a(n-4) + 12*a(n-5) - 10*a(n-6) + 6*a(n-7) - 3*a(n-8) + a(n-9) for n>8.
a(n) = (-5/16 + (5*i)/16)*(((-3-3*i) + (-i)^n + i^(1+n))*n*(1+n)) where i=sqrt(-1).
(End)

More terms from Colin Barker, Dec 22 2019

Name corrected by Paolo Xausa, Dec 04 2023

cp's OEIS Frontend

A294828 Numerators of the partial sums of the reciprocals of the numbers (k + 1)*(5*k + 3) = A147874(k+2), for k >= 0.

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A294829 Denominators of the partial sums of the reciprocals of the numbers (k + 1)*(5*k + 3) = A147874(k+2), for k >= 0.

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A294830 Decimal expansion of the sum of the reciprocals of the numbers (k+1)*(5*k+3) = A147874(k+2) for k >= 0.

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A131242 Partial sums of A059995: a(n) = sum_{k=0..n} floor(k/10).

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

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A115006 Row 2 of array in A114999.

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A294828 Numerators of the partial sums of the reciprocals of the numbers (k + 1)(5k + 3) = A147874(k+2), for k >= 0.

A294829 Denominators of the partial sums of the reciprocals of the numbers (k + 1)(5k + 3) = A147874(k+2), for k >= 0.

A294830 Decimal expansion of the sum of the reciprocals of the numbers (k+1)(5k+3) = A147874(k+2) for k >= 0.

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

A193872 Even dodecagonal numbers: a(n) = 4n(5*n - 2).