A254963 -id:A254963 - cp's OEIS frontend

A168668 a(n) = n(2 + 5n).

0, 7, 24, 51, 88, 135, 192, 259, 336, 423, 520, 627, 744, 871, 1008, 1155, 1312, 1479, 1656, 1843, 2040, 2247, 2464, 2691, 2928, 3175, 3432, 3699, 3976, 4263, 4560, 4867, 5184, 5511, 5848, 6195, 6552, 6919, 7296, 7683, 8080, 8487, 8904, 9331, 9768, 10215, 10672

Offset: 0

Paul Curtz, Dec 02 2009

Appears on the main diagonal of the following table of terms of the Hydrogen series, A169603:
0,  3,  8, 15, 24,                        ... A005563
0,  7, 16,  1, 40,  55,                   ... A061039
0, 11, 24, 39, 56,   3,  96,              ... A061043
0, 15, 32, 51, 72,  95, 120,              ... A061047
0, 19, 40, 63, 88, 115, 144, 175, 208, 1, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature(3,-3,1).

Cf. A000217, A000384, A001622, A010692, A131242, A254963 (comment).

Cf. A005563, A061039, A061043, A061047, A169603.

[n*(2+5*n): n in [0..50] ]; // Vincenzo Librandi, Aug 06 2011

A168668:=n->n*(2+5*n): seq(A168668(n), n=0..50); # Wesley Ivan Hurt, Mar 28 2015

f[n_]:=n*(2+5*n);f[Range[0,60]] (* Vladimir Joseph Stephan Orlovsky, Feb 05 2011*)
LinearRecurrence[{3,-3,1},{0,7,24},50] (* Harvey P. Dale, Sep 09 2021 *)

vector(50,n,n--;n*(2+5*n)) \\ Derek Orr, Jun 26 2015

G.f.: x*(7 + 3*x)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
First differences: a(n) - a(n-1) = 10*n-3.
Second differences: a(n) - 2*a(n-1) + a(n-2) = 10 = A010692(n).
a(n) = A131242(10n+6). - Philippe Deléham, Mar 27 2013
a(n) = A000384(n) + 6*A000217(n). - Luciano Ancora, Mar 28 2015
a(n) = A000217(n) + A000217(3*n). - Bruno Berselli, Jul 01 2016
E.g.f.: x*(7 + 5*x)*exp(x). - G. C. Greubel, Jul 29 2016
Sum_{n>=1} 1/a(n) = 5/4 - sqrt(1-2/sqrt(5))*Pi/4 + sqrt(5)*log(phi)/4 - 5*log(5)/8, where phi is the golden ratio (A001622). - Amiram Eldar, Sep 17 2023

Edited and extended by R. J. Mathar, Dec 05 2009

A218530 Partial sums of floor(n/11).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 171

Offset: 0

Philippe Deléham, Mar 27 2013

Apart from the initial zeros, the same as A008729.

			As square array:
..0....0....0....0....0....0....0....0....0....0....0
..1....2....3....4....5....6....7....8....9...10...11
.13...15...17...19...21...23...25...27...29...31...33
.36...39...42...45...48...51...54...57...60...63...66
.70...74...78...82...86...90...94...98..102..106..110
115..120..125..130..135..140..145..150..155..160..165
171..177..183..189..195..201..207..213..219..225..231
238..245..252..259..266..273..280..287..294..301..308
316..324..332..340..348..356..364..372..380..388..396
405..414..423..432..441..450..459..468..477..486..495
505..515..525..535..545..555..565..575..585..595..605
...

Cf. similar sequences: A000217, A002820, A130518, A130519, A130520, A174709, A174738, A118729, A218470, A131242.

a(11n)    = A051865(n).
a(11n+1)  = A180223(n).
a(11n+4)  = A022268(n).
a(11n+5)  = A022269(n).
a(11n+6)  = A254963(n)
a(11n+9)  = A211013(n).
a(11n+10) = A152740(n).
G.f.: x^11/((1-x)^2*(1-x^11)).

A254407 a(n) = n(n+1)(11*n +10)/6.

Original entry on oeis.org

0, 7, 32, 86, 180, 325, 532, 812, 1176, 1635, 2200, 2882, 3692, 4641, 5740, 7000, 8432, 10047, 11856, 13870, 16100, 18557, 21252, 24196, 27400, 30875, 34632, 38682, 43036, 47705, 52700, 58032, 63712, 69751, 76160, 82950, 90132, 97717, 105716, 114140, 123000

Offset: 0

Bruno Berselli, Jan 30 2015

Similar sequences of the type m*P(s,m) - Sum_{i=1..m} P(s-1,i), where P(s,m) is the m-th s-gonal number:
s=3: A027480(n) = (n+1)*A000217(n+1) - Sum_{i=1..n+1} i;
s=4: A162148(n) = (n+1)*A000290(n+1) - Sum_{i=1..n+1} A000217(i);
s=5: A245301(n) = (n+1)*A000326(n+1) - Sum_{i=1..n+1} A000290(i);
s=6: A085788(n) = (n+1)*A000384(n+1) - Sum_{i=1..n+1} A000326(i);
s=7:       a(n) = (n+1)*A000566(n+1) - Sum_{i=1..n+1} A000384(i).

			532 is the 7th term because A000566(7)=112 and Sum_{i=1..7} A000384(i)=252, therefore 7*112-252 = 532.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wikipedia, Polygonal numbers: Table of values.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A011875, A027480, A057145, A085788, A132112, A162148, A245301, A254963.

Magma
```
[n*(n+1)*(11*n+10)/6: n in [0..40]];
```

A254407:= n-> n*(n+1)*(11*n+10)/6; seq(A254407(n), n=0..50); # G. C. Greubel, Mar 31 2021

Table[n (n + 1) (11 n + 10)/6, {n, 0, 40}]
Column[CoefficientList[Series[x (7 + 4 x) / (1 - x)^4, {x, 0, 60}], x]] (* Vincenzo Librandi, Jan 31 2015 *)

makelist(n*(n+1)*(11*n+10)/6, n, 0, 40);

vector(40, n, n--; n*(n+1)*(11*n+10)/6)

Sage
```
[n*(n+1)*(11*n+10)/6 for n in (0..40)]
```

G.f.: x*(7 + 4*x)/(1 - x)^4.
a(-n) = -A132112(n-1).
a(n) = Sum_{k=0..n} A011875(11*k+2).
Equivalently, partial sums of A254963.
E.g.f.: x*(42 + 54*x + 11*x^2)*exp(x)/6. - G. C. Greubel, Mar 31 2021

A202803 a(n) = n(5n+1).

Original entry on oeis.org

0, 6, 22, 48, 84, 130, 186, 252, 328, 414, 510, 616, 732, 858, 994, 1140, 1296, 1462, 1638, 1824, 2020, 2226, 2442, 2668, 2904, 3150, 3406, 3672, 3948, 4234, 4530, 4836, 5152, 5478, 5814, 6160, 6516, 6882, 7258, 7644, 8040, 8446, 8862, 9288, 9724, 10170

Offset: 0

Jeremy Gardiner, Dec 24 2011

First bisection of A219190. - Bruno Berselli, Nov 15 2012

a(n)*Pi is the total length of 5 points circle center spiral after n rotations. The spiral length at each rotation (L(n)) is A017341. The spiral length ratio rounded down [floor(L(n)/L(1))] is A032793. See illustration in links. - Kival Ngaokrajang, Dec 27 2013

			G.f. = 6*x + 22*x^2 + 48*x^3 + 84*x^4 + 130*x^5 +186*x^6 + 252*x^7 + 328*x^8 + ...

Jeremy Gardiner, Table of n, a(n) for n = 0..3000
Kival Ngaokrajang, Illustration of 5 points circle center spiral.
Leo Tavares, Illustration: Twin Trapezoids.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A033429, A168668, A126264, A135705, A124080.
Cf. sequences listed in A254963.
Cf. A001622, A005475.

List([0..50], n-> n*(5*n+1)); # G. C. Greubel, Jul 04 2019

[n*(5*n+1):n in [0..50]]; // Vincenzo Librandi, Aug 11 2014

CoefficientList[Series[2x(3+2x)/(1-x)^3, {x, 0, 50}] ,x] (* Vincenzo Librandi, Aug 11 2014 *)
Table[5*n^2+n, {n,0,50}] (* G. C. Greubel, Jul 04 2019 *)

a(n)=n*(5*n+1) \\ Charles R Greathouse IV, Jun 17 2017

[n*(5*n+1) for n in (0..50)] # G. C. Greubel, Jul 04 2019

a(n) = 5*n^2 + n.
a(n) = A033429(n) + n. - Omar E. Pol, Dec 24 2011
G.f.: 2*x*(3+2*x)/(1-x)^3. - Philippe Deléham, Mar 27 2013
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0) = 0, a(1) = 6, a(2) = 22. - Philippe Deléham, Mar 27 2013
a(n) = A131242(10n+5). - Philippe Deléham, Mar 27 2013
a(n) = 2*A005475(n). - Philippe Deléham, Mar 27 2013
a(n) = A168668(n) - n. - Philippe Deléham, Mar 27 2013
a(n) = (n+1)^3 - (1 + n + n*(n-1) + n*(n-1)*(n-2)). - Michael Somos, Aug 10 2014
E.g.f.: x*(6+5*x)*exp(x). - G. C. Greubel, Aug 22 2017
Sum_{n>=1} 1/a(n) = 5*(1-log(5)/4) - sqrt(1+2/sqrt(5))*Pi/2 -sqrt(5)*log(phi)/2, where phi is the golden ratio (A001622). - Amiram Eldar, Jul 19 2022

cp's OEIS Frontend

A168668 a(n) = n*(2 + 5*n).

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A218530 Partial sums of floor(n/11).

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A254407 a(n) = n*(n+1)*(11*n +10)/6.

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Magma

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A202803 a(n) = n*(5*n+1).

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GAP

Magma

Mathematica

PARI

Sage

Formula

A168668 a(n) = n(2 + 5n).

A254407 a(n) = n(n+1)(11*n +10)/6.

A202803 a(n) = n(5n+1).