A202803 -id:A202803 - cp's OEIS frontend

A005475 a(n) = n(5n+1)/2.

0, 3, 11, 24, 42, 65, 93, 126, 164, 207, 255, 308, 366, 429, 497, 570, 648, 731, 819, 912, 1010, 1113, 1221, 1334, 1452, 1575, 1703, 1836, 1974, 2117, 2265, 2418, 2576, 2739, 2907, 3080, 3258, 3441, 3629

Offset: 0

N. J. A. Sloane

Sequence found by reading the line from 0, in the direction 0, 11, ..., and the line from 3, in the direction 3, 24, ..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. - Omar E. Pol, Sep 26 2011

For n >= 3, a(n) is the sum of the numbers appearing in the 3rd row of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows. - Wesley Ivan Hurt, May 17 2021

G. C. Greubel, Table of n, a(n) for n = 0..5000
A. Horzela, P. Blasiak, G. E. H. Duchamp, K. A. Penson and A. I. Solomon, A product formula and combinatorial field theory, arXiv:quant-ph/0409152, 2004.
Leo Tavares, Illustration: Hexagonal Trapeziums.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001622, A110449, A130520, A162147, A202803.

Cf. similar sequences listed in A022289.

seq(binomial(5*n+1,2)/5, n=0..34); # Zerinvary Lajos, Jan 21 2007
a:=n->sum(2*n+j, j=1..n): seq(a(n), n=0..38); # Zerinvary Lajos, Apr 29 2007

Table[n (5 n + 1)/2, {n, 0, 40}] (* Bruno Berselli, Oct 13 2016 *)

a(n)=n*(5*n+1)/2; \\ Joerg Arndt, Mar 27 2013

a(n) = A110449(n, 2) for n>1.
a(n) = a(n-1) + 5*n - 2 for n>0, a(0)=0. - Vincenzo Librandi, Nov 18 2010
a(n) = A130520(5*n+2). - Philippe Deléham, Mar 26 2013
a(n) = A202803(n)/2. - Philippe Deléham, Mar 27 2013
a(n) = A162147(n) - A162147(n-1). - J. M. Bergot, Jun 21 2013
a(n) = A000217(3*n) - A000217(2*n). - Bruno Berselli, Oct 13 2016
From G. C. Greubel, Aug 23 2017: (Start)
G.f.: x*(2*x + 3)/(1-x)^3.
E.g.f.: (x/2)*(5*x+6)*exp(x). (End)
Sum_{n>=1} 1/a(n) = 10+2*gamma+2*Psi(1/5) = 0.57635... see A001620 and A200135. - R. J. Mathar, May 30 2022
Sum_{n>=1} 1/a(n) = 10 - sqrt(1+2/sqrt(5))*Pi - sqrt(5)*log(phi) - 5*log(5)/2, where phi is the golden ratio (A001622). - Amiram Eldar, Sep 10 2022

Incorrect comment deleted and minor errors corrected by Johannes W. Meijer, Feb 04 2010

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

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)

A226292 (10n^2+4n+(1-(-1)^n))/8.

Original entry on oeis.org

2, 6, 13, 22, 34, 48, 65, 84, 106, 130, 157, 186, 218, 252, 289, 328, 370, 414, 461, 510, 562, 616, 673, 732, 794, 858, 925, 994, 1066, 1140, 1217, 1296, 1378, 1462, 1549, 1638, 1730, 1824, 1921, 2020, 2122, 2226, 2333, 2442, 2554, 2668, 2785, 2904, 3026, 3150

Offset: 1

Yosu Yurramendi, Jun 02 2013

The number of binary pattern classes in the (3,n)-rectangular grid with 2 '1's and (n-2) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other, n<10.
Column k=2 of A226290.
For n even, a(n) is A202803; for n odd, a(n) is A190816.
Number of lattice points (x,y) in the region bounded by y < 3x, y > x/2 and x <= n. - Wesley Ivan Hurt, Oct 31 2014

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

Cf. A190816, A202803, A226290.

[(10*n^2+4*n+(1-(-1)^n))/8: n in [1..50]]; // Vincenzo Librandi, Sep 04 2013

A226292:=n->(10*n^2+4*n+(1-(-1)^n))/8: seq(A226292(n), n=1..50); # Wesley Ivan Hurt, Oct 31 2014

CoefficientList[Series[(2 + 2 x + x^2) / ((1 + x) (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Sep 04 2013 *)
LinearRecurrence[{2,0,-2,1},{2,6,13,22},60] (* Harvey P. Dale, Feb 01 2019 *)

a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n>4, a(1)=2, a(2)=6, a(3)=13, a(4)=22.
a(n) = 2*a(n-2)-a(n-4)+10 for n>4, a(1)=2, a(2)=6, a(3)=13, a(4)=22.
a(n) = a(n-1)+a(n-2)-a(n-3)+5 for n>3, a(1)=2, a(2)=6, a(3)=13.
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3)+(-1)^n for n>3, a(1)=2, a(2)=6, a(3)=13.
a(n) = 2*a(n-1)-a(n-2)+2+(1-(-1)^n)/2 for n>2, a(1)=2, a(2)=6.
G.f.: x*(2+2*x+x^2)/((1+x)*(1-x)^3). - Bruno Berselli, Jun 03 2013

More terms from Vincenzo Librandi, Sep 04 2013

A257144 Numbers n not of the form x+y*x^2 for x>1 and y>0.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 37, 40, 41, 43, 44, 45, 47, 49, 51, 53, 59, 60, 61, 63, 64, 65, 67, 69, 71, 73, 76, 77, 79, 81, 83, 85, 87, 88, 89, 91, 92, 95, 96, 97, 99, 101, 103, 104, 107, 108, 109, 112, 113, 115, 117, 119, 121

Offset: 1

Gionata Neri, Apr 16 2015

nonn

Number n such that (d*k+1) /= (n/d), for k>0 and each value of d, where d is a divisor >1 of n.

For numbers of the form x+y*x^2 with 0A002378 (y=1), A014105 (y=2), A049451 (y=3), A007742 (y=4), A202803 (y=5), A049453 (y=6), A092277 (y=7), A139275 (y=8), A154517 (y=9), A055437 (y=10). - Danny Rorabaugh, Apr 20 2015

n = 71; Take[Complement[Range[n^2], DeleteDuplicates@ Sort@ Flatten@ Table[x + y x^2, {x, 2, n}, {y, 1, n}]], n] (* Michael De Vlieger, Apr 17 2015 *)

cp's OEIS Frontend

A005475 a(n) = n*(5*n+1)/2.

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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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A226292 (10*n^2+4*n+(1-(-1)^n))/8.

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Magma

Maple

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A257144 Numbers n not of the form x+y*x^2 for x>1 and y>0.

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Mathematica

A005475 a(n) = n(5n+1)/2.

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

A226292 (10n^2+4n+(1-(-1)^n))/8.