A168668 -id:A168668 - cp's OEIS frontend

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

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

A254963 a(n) = n(11n + 3)/2.

Original entry on oeis.org

0, 7, 25, 54, 94, 145, 207, 280, 364, 459, 565, 682, 810, 949, 1099, 1260, 1432, 1615, 1809, 2014, 2230, 2457, 2695, 2944, 3204, 3475, 3757, 4050, 4354, 4669, 4995, 5332, 5680, 6039, 6409, 6790, 7182, 7585, 7999, 8424, 8860, 9307, 9765, 10234, 10714, 11205, 11707

Offset: 0

Bruno Berselli, Feb 11 2015

This sequence provides the first differences of A254407 and the partial sums of A017473.
Also:
a(n) - n         = A022269(n);
a(n) + n         = n*(11*n+5)/2: 0, 8, 27, 57, 98, 150, 213, 287, ...;
a(n) - 2*n       = A022268(n);
a(n) + 2*n       = n*(11*n+7)/2: 0, 9, 29, 60, 102, 155, 219, 294, ...;
a(n) - 3*n       = n*(11*n-3)/2: 0, 4, 19, 45, 82, 130, 189, 259, ...;
a(n) + 3*n       = A211013(n);
a(n) - 4*n       = A226492(n);
a(n) + 4*n       = A152740(n);
a(n) - 5*n       = A180223(n);
a(n) + 5*n       = n*(11*n+13)/2: 0, 12, 35, 69, 114, 170, 237, 315, ...;
a(n) - 6*n       = A051865(n);
a(n) + 6*n       = n*(11*n+15)/2: 0, 13, 37, 72, 118, 175, 243, 322, ...;
a(n) - 7*n       = A152740(n-1) with A152740(-1) = 0;
a(n) + 7*n       = n*(11*n+17)/2: 0, 14, 39, 75, 122, 180, 249, 329, ...;
a(n) - n*(n-1)/2 = A168668(n);
a(n) + n*(n-1)/2 = A049453(n);
a(n) - n*(n+1)/2 = A202803(n);
a(n) + n*(n+1)/2 = A033580(n).

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

Cf. A008729 and A218530 (seventh column); A017473, A254407.
Cf. similar sequences of the type 4*n^2 + k*n*(n+1)/2: A055999 (k=-7, n>6), A028552 (k=-6, n>2), A095794 (k=-5, n>1), A046092 (k=-4, n>0), A000566 (k=-3), A049450 (k=-2), A022264 (k=-1), A016742 (k=0), A022267 (k=1), A202803 (k=2), this sequence (k=3), A033580 (k=4).
Cf. A069125: (2*n+1)^2 + 3*n*(n+1)/2; A147875: n^2 + 3*n*(n+1)/2.
Cf. A022268, A022269, A049453, A051865, A152740, A168668, A180223, A211013, A226492.

Magma
```
[n*(11*n+3)/2: n in [0..50]];
```

Table[n (11 n + 3)/2, {n, 0, 50}]
LinearRecurrence[{3,-3,1},{0,7,25},50] (* Harvey P. Dale, Mar 25 2018 *)

Maxima
```
makelist(n*(11*n+3)/2, n, 0, 50);
```
PARI
```
vector(50, n, n--; n*(11*n+3)/2)
```
Sage
```
[n*(11*n+3)/2 for n in (0..50)]
```

G.f.: x*(7 + 4*x)/(1 - x)^3.
From Elmo R. Oliveira, Dec 15 2024: (Start)
E.g.f.: exp(x)*x*(14 + 11*x)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

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

A002789 Number of integer points in a certain quadrilateral scaled by a factor of n.

Original entry on oeis.org

2, 4, 7, 11, 16, 21, 28, 35, 43, 52, 62, 72, 84, 96, 109, 123, 138, 153, 170, 187, 205, 224, 244, 264, 286, 308, 331, 355, 380, 405, 432, 459, 487, 516, 546, 576, 608, 640, 673, 707, 742, 777, 814, 851, 889, 928, 968, 1008, 1050, 1092, 1135, 1179, 1224, 1269

Offset: 1

N. J. A. Sloane

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 closed line segment from (1/2, 1/3) to (0, 1) is not included but the rest of the boundary is. The sequence is denoted by d(n). - Michael Somos, May 22 2014

			G.f. = 2*x + 4*x^2 + 7*x^3 + 11*x^4 + 16*x^5 + 21*x^6 + 28*x^7 + 35*x^8 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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. A168668, A242771, A242774.

a[ n_] := Quotient[ 7 + 12 n + 5 n^2, 12]; (* Michael Somos, May 22 2014 *)
a[ n_] := Length @ With[{o = Boole[ 0 < n], c = Boole[ 0 >= n], m = Abs@n}, FindInstance[ 0 < o + x && 0 < o + y && (2 x < o + m && 4 x + 3 y < c + 3 m || m < c + 2 x && 2 x + 3 y < o + 2 m), {x, y}, Integers, 10^9]]; (* Michael Somos, May 22 2014 *)

{a(n) = (7 + 12*n + 5*n^2) \ 12}; /* Michael Somos, May 22 2014 */

{a(n) = if( n<0, polcoeff( x^3 * (1 + x + x^2 + 2*x^4) / ((1 - x)^2 * (1 - x^6)) + x * O(x^-n), -n), polcoeff( x * (2 + x^2 + x^3 + x^4) / ((1 - x)^2 * (1 - x^6)) + x * O(x^n), n))}; /* Michael Somos, May 22 2014 */

G.f.: x * (2 + 2*x + x^2) / (1 - x - x^2 + x^4 + x^5 - x^6) = (2*x + x^3 + x^4 + x^5) / ((1 - x)^2 * (1 - x^6)). - Michael Somos, May 22 2014

a(n) = floor( A168668(n+1) / 12), a(n) = A242771(-n), a(n) - a(n-1) = A242774(n) for all n in Z. - Michael Somos, May 22 2014

A249547 a(n) = (10n^2+8n-1+(-1)^n)/8.

Original entry on oeis.org

0, 2, 7, 14, 24, 36, 51, 68, 88, 110, 135, 162, 192, 224, 259, 296, 336, 378, 423, 470, 520, 572, 627, 684, 744, 806, 871, 938, 1008, 1080, 1155, 1232, 1312, 1394, 1479, 1566, 1656, 1748, 1843, 1940, 2040, 2142, 2247, 2354, 2464, 2576, 2691, 2808, 2928, 3050

Offset: 0

Wesley Ivan Hurt, Oct 31 2014

a(n) is the number of lattice points (x,y) in the coordinate plane bounded by y < 3x, y >= x/2 and x <= n.

a(n)+1 is the number of lattice points bounded by y <= 3x, y >= x/2 and x <= n.

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A001068, A004526, A135706, A168668, A226292.

[(10*n^2+8*n-1+(-1)^n)/8 : n in [0..50]];

A249547:=n->(10*n^2+8*n-1+(-1)^n)/8: seq(A249547(n), n=0..100);

Table[(10*n^2 + 8 n - 1 + (-1)^n)/8 , {n, 0, 50}]

a(n) = (10*n^2+8*n-1+(-1)^n)/8; \\ Michel Marcus, Nov 04 2014

concat(0, Vec(x*(2+3*x)/((1-x)^3*(1+x)) + O(x^100))) \\ Altug Alkan, Oct 28 2015

G.f.: x*(2+3*x)/((1-x)^3*(1+x)).
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n>3.
a(n) = A004526(n) + A226292(n), for n>0.
a(n) = Sum_{i=0..n} A001068(2*i). - Wesley Ivan Hurt, May 06 2016
E.g.f.: (x*(9 + 5*x)*exp(x) - sinh(x))/4. - Ilya Gutkovskiy, May 06 2016
a(2n) = A168668(n). a(2n-1) = A135706(n). - Wesley Ivan Hurt, May 09 2016

A272039 a(n) = 10n^2 + 4n + 1.

Original entry on oeis.org

1, 15, 49, 103, 177, 271, 385, 519, 673, 847, 1041, 1255, 1489, 1743, 2017, 2311, 2625, 2959, 3313, 3687, 4081, 4495, 4929, 5383, 5857, 6351, 6865, 7399, 7953, 8527, 9121, 9735, 10369, 11023, 11697, 12391, 13105, 13839, 14593, 15367, 16161, 16975, 17809, 18663, 19537

Offset: 0

Vincenzo Librandi, Apr 20 2016

Polynomials from the table "Coefficients and roots of Ehrhart polynomials" in Beck et al. paper (see Links section):
. Cube:                   A000578;
. Cube minus corner:      A004068;
. Prism:                  A002411;
. Octahedron:             A005900;
. Square pyramid:         A000330;
. Bypyramid:              A006003;
. Unimodular tetrahedron: A000292;
. Fat tetrahedron:        A167875;
. Cyclic(2,5), which has the same polynomial form of this sequence.
a(n) for n = 0, -1, 1, -2, 2, -3, 3, ... gives all x such that (5*x - 3)/2 is a square.
Squares in sequence: 1, 49, 1385329, 101263969, 2880599856289, ...
Is this 1 followed by A228219?

M. Beck, J. A. De Loera, M. Develin, J. Pfeifle and R. P. Stanley, Coefficients and roots of Ehrhart Polynomials, page 19.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000292, A000330, A000578, A002411, A004068, A005900, A006003, A167875, A168668, A228219, A272124, A272130.

Magma
```
[10*n^2+4*n+1: n in [0..50]];
```

Table[10 n^2 + 4 n + 1, {n, 0, 50}]
LinearRecurrence[{3,-3,1},{1,15,49},50] (* Harvey P. Dale, Dec 26 2021 *)

a(n)=10*n^2+4*n+1 \\ Charles R Greathouse IV, Jun 17 2017

O.g.f.: (1 + 12*x + 7*x^2)/(1 - x)^3.
E.g.f.: (1 + 14*x + 10*x^2)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*A168668(n) + 1.

Edited by Bruno Berselli, Apr 22 2016

A085047 a(n) is the least number not already used such that the arithmetic mean of the first n terms is a square.

Original entry on oeis.org

1, 7, 4, 24, 9, 51, 16, 88, 25, 135, 36, 192, 49, 259, 64, 336, 81, 423, 100, 520, 121, 627, 144, 744, 169, 871, 196, 1008, 225, 1155, 256, 1312, 289, 1479, 324, 1656, 361, 1843, 400, 2040, 441, 2247, 484, 2464, 529, 2691, 576, 2928, 625, 3175, 676, 3432, 729

Offset: 1

Amarnath Murthy, Jun 20 2003

nonn

This is very nearly a linear recurrence, but the distinctness requirement occasionally foils it. - Charles R Greathouse IV, Nov 07 2014

			(a(1) + a(2) + a(3) + a(4) + a(5))/5 = (1+7+4+24+9)/5 = 9 = 3^2.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A168668.

b:= proc(n) is(n>1) end:
s:= proc(n) option remember;
      `if`(n=1, 1, s(n-1)+a(n))
    end:
a:= proc(n) option remember; local k;
      if n=1 then 1
    else for k from n-irem(s(n-1),n) by n
         do if b(k) and issqr((s(n-1)+k)/n)
               then b(k):=false; return k
            fi
         od
      fi
    end:
seq(a(n), n=1..150);  # Alois P. Heinz, Nov 07 2014

Clear[a, b, s]; b[n_] := n>1; s[n_] := s[n] = If[n == 1, 1, s[n-1] + a[n]]; a[n_] := a[n] = Module[{k}, If [n == 1, 1, For[k = n - Mod[s[n-1], n], True, k = k+n, If[b[k] && IntegerQ[Sqrt[(s[n-1]+k)/n]], b[k] = False; Return[k]]]]]; Table[a[n], {n, 1, 150}] (* Jean-François Alcover, Jun 10 2015, after Alois P. Heinz *)

v=[1]; n=1; while(#v<50, s=(n+vecsum(v))/(#v+1); if(type(s)=="t_INT", if(issquare(s)&&!vecsearch(vecsort(v), n), v=concat(v, n); n=0)); n++); v \\ Derek Orr, Nov 05 2014, edited Jun 26 2015

a(2*n-1) = n^2; a(2*n) = n*(2+5*n). Also, (1/(2*n))*(Sum_{i=1..n} i^2 + i*(2+5*i)) = (n+1)^2 and (1/(2*n-1))*(Sum_{i=1..n} i^2 + (i-1)*(5*i-3)) = k^2. Thus the arithmetic mean of the first 2*n terms is (n+1)^2 and the arithmetic mean of the first 2*n-1 terms is n^2. - Derek Orr, Jun 26 2015

More terms from David Wasserman, Jan 11 2005

Incorrect formulas and programs removed by Charles R Greathouse IV, Nov 07 2014

A273982 Number of little cubes visible around an n X n X n cube with a face on a table.

Original entry on oeis.org

1, 8, 25, 52, 89, 136, 193, 260, 337, 424, 521, 628, 745, 872, 1009, 1156, 1313, 1480, 1657, 1844, 2041, 2248, 2465, 2692, 2929, 3176, 3433, 3700, 3977, 4264, 4561, 4868, 5185, 5512, 5849, 6196, 6553, 6920, 7297, 7684, 8081, 8488, 8905, 9332, 9769, 10216

Offset: 1

Sébastien Dumortier, Jun 05 2016

There are fewer visible cubes on the bottom than on the top.

			a(3)=25 because around a 3 X 3 X 3 cube, when it's on a table, it's possible to see only 25 little cubes (8 on each of the 2 bottom layers and 9 on the top layer).

Leo Tavares, Illustration: Diamond Branches
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005897, A168668, A000566.

[5*n^2-8*n+4: n in [1..60]]; // Vincenzo Librandi, Jun 06 2016

A273982:=n->5*n^2-8*n+4: seq(A273982(n), n=1..60); # Wesley Ivan Hurt, Oct 06 2017

Table[5 n^2 - 8 n + 4, {n, 46}] (* or *)
LinearRecurrence[{3, -3, 1}, {1, 8, 25}, 46] (* or *)
CoefficientList[Series[(-1 - 5 x - 4 x^2)/(-1 + x)^3, {x, 0, 45}], x] (* Michael De Vlieger, Oct 06 2017 *)

a(n) = 5*n^2 - 8*n + 4; \\ Altug Alkan, Oct 06 2017

a(n) = 5*n^2 - 8*n + 4.
a(n) = n^3 - (n-2)^3 - (n-2)^2. - Joerg Arndt, Jun 06 2016
a(n) = A168668(n-1) + 1. - Altug Alkan, Oct 06 2017
G.f.: (-1 - 5*x - 4*x^2)/(-1 + x)^3. - Michael De Vlieger, Oct 06 2017
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. - Wesley Ivan Hurt, Oct 06 2017
a(n) = A000566(n-1) + A000566(n), the sum of consecutive heptagonal numbers. - Charlie Marion, Jul 01 2021
a(n) = n^2 + 4*(n-1)^2. - Leo Tavares, Mar 24 2022

a(2) corrected and entry edited by Andrey Zabolotskiy, Oct 06 2017

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

A332495 a(n-2) = a(n-6) + 5(1+2n) with a(0)=0, a(1)=2, a(2)=7, a(3)=15 for n>=4.

Original entry on oeis.org

0, 2, 7, 15, 25, 37, 52, 70, 90, 112, 137, 165, 195, 227, 262, 300, 340, 382, 427, 475, 525, 577, 632, 690, 750, 812, 877, 945, 1015, 1087, 1162, 1240, 1320, 1402, 1487, 1575, 1665, 1757, 1852, 1950, 2050, 2152, 2257

Offset: 0

Paul Curtz, Feb 14 2020

a(-2)=2, a(-1)=0. 4 evens followed by 4 odds.
Last digit is only 0, 2, 5, 7.
The vertical spoke S-N of the pentagonal spiral for A004526.
                          37
                      37  25  25
                  36  24  15  15  26
              36  24  14   7   8  16  26
          35  23  14   7   2   3   8  16  27
      35  23  13   6   2   0   0   3   9  17  27
        34  22  13   6   1   1   4   9  17  28
          34  22  12   5   5   4  10  18  28
            33  21  12  11  11  10  18  29
              33  21  20  20  19  19  29
                32  32  31  31  30  30
Rank of multiples of 10: 0, 7, 8, 15, 16, ... = A047521. Compare to A154260 in the formula.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Cf. A004526, A033429, A062786, A168668, A135706, A147874, 2*A147875 (all in the spiral).

Cf. A005408, A047521, A154260

CoefficientList[Series[x (2 + x + 2 x^2)/((1 - x)^3*(1 + x^2)), {x, 0, 42}], x] (* Michael De Vlieger, Feb 14 2020 *)

concat(0, Vec(x*(2 + x + 2*x^2) / ((1 - x)^3*(1 + x^2)) + O(x^40))) \\ Colin Barker, Feb 14 2020

a(-1-n) = a(n).
a(2*n) + a(1+2*n) = 2, 22, 62, ... = A273366(n).
Second differences give the sequence of period 4: repeat [3, 3, 2, 2].
From Colin Barker, Feb 14 2020: (Start)
G.f.: x*(2 + x + 2*x^2) / ((1 - x)^3*(1 + x^2)).
a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5) for n>4.
(End)
Multiples of 10: 10*(0, 7, 9, 30, 34, ... = A154260).
4*a(n) = A087960(n) +5*n -1 +5*n^2. - R. J. Mathar, Feb 28 2020

cp's OEIS Frontend

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

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A254963 a(n) = n*(11*n + 3)/2.

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

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A002789 Number of integer points in a certain quadrilateral scaled by a factor of n.

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A249547 a(n) = (10*n^2+8*n-1+(-1)^n)/8.

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A272039 a(n) = 10*n^2 + 4*n + 1.

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A254963 a(n) = n(11n + 3)/2.

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

A249547 a(n) = (10n^2+8n-1+(-1)^n)/8.

A272039 a(n) = 10n^2 + 4n + 1.

A332495 a(n-2) = a(n-6) + 5(1+2n) with a(0)=0, a(1)=2, a(2)=7, a(3)=15 for n>=4.