A059100 -id:A059100 - cp's OEIS frontend

A114949 a(n) = n^2 + 6.

6, 7, 10, 15, 22, 31, 42, 55, 70, 87, 106, 127, 150, 175, 202, 231, 262, 295, 330, 367, 406, 447, 490, 535, 582, 631, 682, 735, 790, 847, 906, 967, 1030, 1095, 1162, 1231, 1302, 1375, 1450, 1527, 1606, 1687, 1770, 1855, 1942, 2031, 2122, 2215, 2310, 2407, 2506

Offset: 0

Cino Hilliard, Feb 21 2006

2/a(n) = R(n)/r, n >= 0, with R(n) the n-th radius of the counterclockwise Pappus chain of the arbelos with semicircle radii r, r1 = 2r/3, r2 = r - r1 = r/3. See the MathWorld link for such a Pappus chain. The clockwise chain companion has circle radii R'(n)/r = 2/A222465(n), n >= 0. - Wolfdieter Lang, Mar 01 2013

			The arbelos chain defined in a comment above has circle radii [1/3, 2/7, 1/5, 2/15, 1/11, 2/31, 1/21, 2/55, 1/35, 2/87, 1/53,...], for n >= 0. - _Wolfdieter Lang_, Mar 01 2013

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Pappus chain.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A002522, A016742, A059100, A082044, A087475, A114964 (see comment), A117619, A117950, A117951, A222465.

A114949:=n->n^2+6: seq(A114949(n), n=0..100); # Wesley Ivan Hurt, Apr 28 2017

Range[0, 49]^2 + 6 (* Alonso del Arte, Jan 30 2013 *)

From R. J. Mathar, May 17 2009: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: -(6 - 11*x + 7*x^2)/(x - 1)^3. (End)
a(n) = 2*n + a(n - 1) - 1, with n > 0, a(0)=6. - Vincenzo Librandi, Nov 13 2010
a(n) = A000290(n) + 6. - Omar E. Pol, Mar 02 2013
a(n) = ((n-2)^3 + (n-1)^3 + n^3 + (n+1)^3 + (n+2)^3)/(5*n) for n>=1. - Bruno Berselli, May 12 2014
For n >= 1, a(n) = (A016742(n) + A082044(n) - 1)/A000290(n). - Bruce J. Nicholson, Apr 19 2017
From Amiram Eldar, Nov 02 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(6)*Pi*coth(sqrt(6)*Pi))/12.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(6)*Pi*cosech(sqrt(6)*Pi))/12. (End)
From Amiram Eldar, Feb 05 2024: (Start)
Product_{n>=0} (1 - 1/a(n)) = sqrt(5/6)*sinh(sqrt(5)*Pi)/sinh(sqrt(6)*Pi).
Product_{n>=0} (1 + 1/a(n)) = sqrt(7/6)*sinh(sqrt(7)*Pi)/sinh(sqrt(6)*Pi). (End)
E.g.f.: exp(x)*(6 + x + x^2). - Elmo R. Oliveira, Jan 17 2025

A114962 a(n) = n^2 + 14.

Original entry on oeis.org

14, 15, 18, 23, 30, 39, 50, 63, 78, 95, 114, 135, 158, 183, 210, 239, 270, 303, 338, 375, 414, 455, 498, 543, 590, 639, 690, 743, 798, 855, 914, 975, 1038, 1103, 1170, 1239, 1310, 1383, 1458, 1535, 1614, 1695, 1778, 1863, 1950, 2039, 2130, 2223, 2318, 2415, 2514

Offset: 0

Cino Hilliard, Feb 21 2006

Old name was: "Numbers of the form x^2 + 14".

x^2 + 14 != y^n for all x,y and n > 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
J. H. E. Cohn, The diophantine equation x^2 + C = y^n, Acta Arithmetica LXV.4 (1993).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A155136, n^2 - 28; A000290, n^2; A114948, n^2 + 10.

Cf. sequences of the type n^2 + k: A002522 (k=1), A059100 (k=2), A117950 (k=3), A087475 (k=4), A117951 (k=5), A114949 (k=6), A117619 (k=7), A189833 (k=8), A189834 (k=9), A114948 (k=10), A189836 (k=11), A241748 (k=12), A241749 (k=13), this sequence (k=14), A241750 (k=15), A241751 (k=16), A241847 (k=17), A241848 (k=18), A241849 (k=19), A241850 (k=20), A241851 (k=21), A114963 (k=22), A241889 (k=23), A241890 (k=24), A114964 (k=30).

[n^2+14: n in [0..60]]; // Vincenzo Librandi, Apr 30 2014

Table[n^2 + 14, {n, 0, 50}]  (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)
CoefficientList[Series[(14 - 27 x + 15 x^2)/(1 - x)^3, {x, 0, 80}], x] (* Vincenzo Librandi, Apr 30 2014 *)

G.f.: (14-27*x+15*x^2)/(1-x)^3. - Colin Barker, Jan 11 2012
From Amiram Eldar, Nov 02 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(14)*Pi*coth(sqrt(14)*Pi))/28.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(14)*Pi*cosech(sqrt(14)*Pi))/28. (End)
From Elmo R. Oliveira, Nov 29 2024: (Start)
E.g.f.: exp(x)*(14 + x + x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

Added 14 from Vincenzo Librandi, Apr 30 2014
Definition changed by Bruno Berselli, Mar 13 2015
Offset corrected by Amiram Eldar, Nov 02 2020

A255843 a(n) = 2*n^2 + 4.

Original entry on oeis.org

4, 6, 12, 22, 36, 54, 76, 102, 132, 166, 204, 246, 292, 342, 396, 454, 516, 582, 652, 726, 804, 886, 972, 1062, 1156, 1254, 1356, 1462, 1572, 1686, 1804, 1926, 2052, 2182, 2316, 2454, 2596, 2742, 2892, 3046, 3204, 3366, 3532, 3702, 3876, 4054, 4236, 4422

Offset: 0

Avi Friedlich, Mar 08 2015

This is the case k=2 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2.

Equivalently, numbers m such that 2*m - 8 is a square.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A059100.
Cf. unsigned A147973: numbers of the form 2*m^2-4.
Cf. sequences of the form 2*m^2+2*k: A005893 (k=1), this sequence (k=2), A255844 (k=3), A155966 (k=4), A255845 (k=5), A255842 (k=6), A255846 (k=7), A255847 (k=8), A255848 (k=9).

Magma
```
[2*n^2+4: n in [0..50]];
```
Mathematica
```
Table[2 n^2 + 4, {n, 0, 50}]
```
PARI
```
vector(50, n, n--; 2*n^2+4)
```
Sage
```
[2*n^2+4 for n in (0..50)]
```

G.f.: 2*(2 - 3*x + 3*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*A059100(n).
a(n) = a(n-1) + 4n - 2. - Bob Selcoe, Mar 25 2020
From Amiram Eldar, Mar 28 2023: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(2)*Pi*coth(sqrt(2)*Pi))/8.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(2)*Pi*cosech(sqrt(2)*Pi))/8. (End)
E.g.f.: 2*exp(x)*(2 + x + x^2). - Stefano Spezia, Aug 07 2024

Edited by Bruno Berselli, Mar 13 2015

A027602 a(n) = n^3 + (n+1)^3 + (n+2)^3.

Original entry on oeis.org

9, 36, 99, 216, 405, 684, 1071, 1584, 2241, 3060, 4059, 5256, 6669, 8316, 10215, 12384, 14841, 17604, 20691, 24120, 27909, 32076, 36639, 41616, 47025, 52884, 59211, 66024, 73341, 81180, 89559, 98496, 108009, 118116, 128835, 140184

Offset: 0

Patrick De Geest

a(3) = 216 = 6^3 (a cube). - Howard Berman (howard_berman(AT)hotmail.com), Nov 07 2008
Pairs [n,a(n)] for n<=10^7 such that a(n) is a perfect power are [0, 9], [1, 36], [3, 216], [23, 41616]. - Joerg Arndt, Jan 25 2011
Sums of three consecutive cubes. - Al Hakanson (hawkuu(AT)gmail.com), May 20 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..750
Patrick De Geest, Palindromic Sums of Cubes of Consecutive Integers
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000537, A003215, A006527, A008585, A059100.

Cf. A000578, A005898, A027603, A027604.

[3*n^3 + 9*n^2 + 15*n + 9: n in [0..60]]; // Vincenzo Librandi, Apr 26 2011

f[n_]:=n^3; Table[f[n]+f[n+1]+f[n+2], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jan 03 2009 *)
Table[3n^3+9n^2+15n+9,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{9,36,99,216},40] (* Harvey P. Dale, Nov 27 2024 *)

a(n)=3*(n^3 + 3*n^2 + 5*n + 3) \\ Charles R Greathouse IV, Jun 11 2015

A027602_list, m = [], [18, 0, 9, 9]
for _ in range(10**2):
    A027602_list.append(m[-1])
    for i in range(3):
        m[i+1] += m[i] # Chai Wah Wu, Dec 15 2015

[i^3+(i+1)^3+(i+2)^3 for i in range(0,48)] # Zerinvary Lajos, Jul 03 2008

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - 1*a(n-4) for n>=4.
a(n) = 9*A006527(n+1). - Lekraj Beedassy, Feb 01 2007
a(n) = 3*n^3 + 9*n^2 + 15*n + 9.
G.f.: 9*(1+x^2)/(1-x)^4. - Bruno Berselli, Jan 21 2011
a(n) = A008585(n+1)*A059100(n+1). - Bruno Berselli, Jan 21 2011
E.g.f.: 3*(3 + 9*x + 6*x^2 + x^3)*exp(x). - G. C. Greubel, Aug 24 2022
Sum_{n>=0} 1/a(n) = (2*gamma + polygamma(0, 1-i*sqrt(2)) + polygamma(0, 1+i*sqrt(2)))/12 = 0.161383557127191633050394086192620963436504... where i denotes the imaginary unit. - Stefano Spezia, Aug 31 2023

A001846 Centered 4-dimensional orthoplex numbers (crystal ball sequence for 4-dimensional cubic lattice).

Original entry on oeis.org

1, 9, 41, 129, 321, 681, 1289, 2241, 3649, 5641, 8361, 11969, 16641, 22569, 29961, 39041, 50049, 63241, 78889, 97281, 118721, 143529, 172041, 204609, 241601, 283401, 330409, 383041, 441729, 506921, 579081, 658689, 746241, 842249, 947241, 1061761, 1186369

Offset: 0

N. J. A. Sloane

a(n) is the number of points in the Z^4 lattice that are at distance at most n from the origin in the adjacency graph. - N. J. A. Sloane, Feb 19 2013
Number of nodes of degree 8 in virtual, optimal, chordal graphs of diameter d(G)=n. - S. Bujnowski & B. Dubalski (slawb(AT)atr.bydgoszcz.pl), Mar 07 2002
If Y_i (i=1,2,3,4) are 2-blocks of an (n+4)-set X then a(n-4) is the number of 8-subsets of X intersecting each Y_i (i=1,2,3,4). - Milan Janjic, Oct 28 2007
Equals binomial transform of [1, 8, 24, 32, 16, 0, 0, 0, ...] where (1, 8, 24, 32, 16) = row 4 of the Chebyshev triangle A013609. - Gary W. Adamson, Jul 19 2008
Comment from Ben Thurston, Feb 18 2013: In the plane, if you make a picture by taking one unit step in each of the basic 8 directions from a central dot, then from each of those going one unit step in each of the eight directions, ... (see illustration), it appears that the number of dots in the picture after n steps is equal to a(n). Response from N. J. A. Sloane, Feb 19 2013: This is correct, and follows from the fact that the Z-module Z[1,i,(+-1+i)/sqrt(2)] is essentially a copy of the Z^4 lattice.
a(n) = D(4,n) where D are the Delannoy numbers (A008288). As such, a(n) gives the number of grid paths from (0,0) to (4,n) using steps that move one unit north, east, or northeast. - Jack W Grahl, Feb 15 2021
The first comment above can be re-expressed and generalized as follows: a(n) is the number of points in Z^4 that are L1 (Manhattan) distance <= n from any given point. Equivalently, due to a symmetry that is easier to see in the Delannoy numbers array (A008288), as a special case of Dmitry Zaitsev's Dec 10 2015 comment on A008288, a(n) is the number of points in Z^n that are L1 (Manhattan) distance <= 4 from any given point. - Shel Kaphan, Jan 02 2023

			a(6)=1289: (2*6^4 + 4*6^3 + 10*6^2 + 8*6 + 3) / 3 = (2592 + 864 + 360 + 48 + 3) / 3 = 3867 / 3 = 1289.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
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).

T. D. Noe, Table of n, a(n) for n = 0..1000
D. Bump, K. Choi, P. Kurlberg, and J. Vaaler, A local Riemann hypothesis, I pages 16 and 17
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Milan Janjic, Two Enumerative Functions. [Broken link; WayBackMachine archive]
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973).
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
R. G. Stanton and D. D. Cowan, Note on a "square" functional equation, SIAM Rev., 12 (1970), 277-279.
Ben Thurston, Illustration of the first four clusters of points in two dimensions
Index entries for crystal ball sequences
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A005408, A001844, A001845, A001847, A059100, A013609.
First differences are A008412.
Cf. A240876.
Row/column 4 of A008288.

for n from 1 to k do eval((2*n^4+4*n^3+10*n^2+8*n+3)/3) od;
A001846:=-(z+1)**4/(z-1)**5; # conjectured (correctly) by Simon Plouffe in his 1992 dissertation

CoefficientList[Series[(-z^4-4 z^3-6 z^2-4 z-1)/(z-1)^5, {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)
Table[(((2 n + 4) n + 10) n + 8) n/3 + 1, {n, 0, 30}] (* Robert A. Russell, Jul 02 2025 *)

G.f.: (1+x)^4 /(1-x)^5.
a(n) = (2*n^4 + 4*n^3 + 10*n^2 + 8*n + 3)/3. - S. Bujnowski & B. Dubalski (slawb(AT)atr.bydgoszcz.pl), Mar 07 2002
From Jonathan Vos Post, Mar 15 2006: (Start)
a(n) = Sum_{i=0..n} A008412(i);
a(n) = Sum_{i=0..n} 8*i*(i^2 + 2)/3;
a(n) = Sum_{i=0..n} 8*i*(A059100(i))/3. (End)
a(n) = Sum_{k=0..min(4,n)} 2^k * binomial(4,k)* binomial(n,k). See Bump et al. - Tom Copeland, Sep 05 2014
E.g.f.: exp(x)*(3 + 24*x + 36*x^2 + 16*x^3 + 2*x^4)/3. - Stefano Spezia, Mar 14 2024
Sum_{n >= 1} (-1)^(n+1)/(n*a(n-1)*a(n)) = log(2) - 7/12 = log(2) - (1 - 1/2 + 1/3 - 1/4). - Peter Bala, Mar 23 2024

A292627 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(kx)BesselI(0,2*x).

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 1, 2, 3, 0, 1, 3, 6, 7, 6, 1, 4, 11, 20, 19, 0, 1, 5, 18, 45, 70, 51, 20, 1, 6, 27, 88, 195, 252, 141, 0, 1, 7, 38, 155, 454, 873, 924, 393, 70, 1, 8, 51, 252, 931, 2424, 3989, 3432, 1107, 0, 1, 9, 66, 385, 1734, 5775, 13236, 18483, 12870, 3139, 252, 1, 10, 83, 560, 2995, 12276, 36645, 73392, 86515, 48620, 8953, 0

Offset: 0

Ilya Gutkovskiy, Sep 20 2017

A(n,k) is the k-th binomial transform of A126869 evaluated at n.

			E.g.f. of column k: A_k(x) =  1 + k*x/1! + (k^2 + 2)*x^2/2! + (k^3 + 6*k)*x^3/3! + (k^4 + 12*k^2 + 6)*x^4/4! + (k^5 + 20*k^3 + 30*k)*x^5/5! + ...
Square array begins:
  1,   1,    1,    1,     1,     1,  ...
  0,   1,    2,    3,     4,     5,  ...
  2,   3,    6,   11,    18,    27,  ...
  0,   7,   20,   45,    88,   155,  ...
  6,  19,   70,  195,   454,   931,  ...
  0,  51,  252,  873,  2424,  5775,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
N. J. A. Sloane, Transforms

Columns k=0..7 give A126869, A002426, A000984, A026375, A081671, A098409, A098410, A104454.
Rows n=0..2 give A000012, A001477, A059100.
Main diagonal gives A186925.
Cf. A292628, A307847.

Table[Function[k, n! SeriesCoefficient[Exp[k x] BesselI[0, 2 x], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/Sqrt[(1 + 2 x - k x) (1 - 2 x - k x)], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

O.g.f. of column k: 1/sqrt( (1 - (k-2)*x)*(1 - (k+2)*x) ).
E.g.f. of column k: exp(k*x)*BesselI(0,2*x).
From Seiichi Manyama, May 01 2019: (Start)
A(n,k) is the coefficient of x^n in the expansion of (1 + k*x + x^2)^n.
A(n,k) = Sum_{j=0..n} (k-2)^(n-j) * binomial(n,j) * binomial(2*j,j).
A(n,k) = Sum_{j=0..n} (k+2)^(n-j) * (-1)^j * binomial(n,j) * binomial(2*j,j).
n * A(n,k) = k * (2*n-1) * A(n-1,k) - (k^2-4) * (n-1) * A(n-2,k). (End)
A(n,k) = Sum_{j=0..floor(n/2)} k^(n-2*j) * binomial(n,2*j) * binomial(2*j,j). - Seiichi Manyama, May 04 2019
T(n,k) =  (1/Pi) * Integral_{x = -1..1} (k - 2 + 4*x^2)^n/sqrt(1 - x^2) dx  = (1/Pi) * Integral_{x = -1..1} (k + 2 - 4*x^2)^n/sqrt(1 - x^2) dx. - Peter Bala, Jan 27 2020
A(n,k) = (1/4)^n * Sum_{j=0..n} (k-2)^j * (k+2)^(n-j) * binomial(2*j,j) * binomial(2*(n-j),n-j). - Seiichi Manyama, Aug 18 2025

A086381 Numbers n such that p=n^2+2 and p+2 are primes.

Original entry on oeis.org

1, 3, 15, 33, 45, 57, 117, 147, 243, 255, 303, 375, 423, 447, 453, 477, 573, 753, 837, 897, 903, 1035, 1497, 1905, 2055, 2085, 2193, 2283, 2433, 2487, 2535, 2583, 2757, 2823, 2943, 2955, 3003, 3213, 3285, 3345, 3603, 3657, 3687, 4407, 4575, 4977, 5037, 5043, 5325, 5355, 5367, 5403, 5727

Offset: 1

Zak Seidov, Sep 07 2003

The twin primes are given by A253639 and A085554. Except for the initial term, all a(n)=3 (mod 6). - M. F. Hasler, Jan 16 2015

M. F. Hasler, Table of n, a(n) for n = 1..10000

Cf. A253639, A085554, A059100, A087475.

[n: n in [0..10000]|IsPrime(n^2+2) and IsPrime(n^2+4)] // Vincenzo Librandi, Dec 16 2010

is_A086381(x)=ispseudoprime(x^2+2)&&ispseudoprime(x^2+4)
forstep(x=1,9999,2,is_A086381(x)&&print1(x",")) \\ M. F. Hasler, Jan 16 2015

Intersection of A067201 and A007591. - M. F. Hasler, Jan 19 2015

More terms from Vincenzo Librandi, Dec 16 2010

A163255 An interspersion: the order array of A163254.

Original entry on oeis.org

1, 3, 2, 7, 5, 4, 13, 10, 8, 6, 21, 17, 14, 11, 9, 31, 26, 22, 18, 15, 12, 43, 37, 32, 27, 23, 19, 16, 57, 50, 44, 38, 33, 28, 24, 20, 73, 65, 58, 51, 45, 39, 34, 29, 25, 91, 82, 74, 66, 59, 52, 46, 40, 35, 30, 111, 101, 92, 83, 75, 67, 60, 53, 47, 41, 36

Offset: 1

Clark Kimberling, Jul 24 2009

A permutation of the natural numbers.
Except for initial terms, rows 1 to 4 are A002061, A002522, A014206, A059100 and columns 1 to 4 are A002620, A024206, A014616, A004116.
This is the interspersion of the fractal sequence A167430; i.e., row n of this array consists of the numbers k such that n=A167430(k). - Clark Kimberling, Nov 03 2009

			Corner:
1....3....7...13
2....5...10...17
4....8...14...22
To obtain A163255 from A163254, replace each term of A163254 by its rank when all the terms of A163254 are arranged in increasing order.

Clark Kimberling, Doubly interspersed sequences, double interspersions and fractal sequences, The Fibonacci Quarterly 48 (2010) 13-20.

Cf. A002061, A002522, A014206, A059100, A002620, A024206, A014616, A004116, A163253, A163254.

Cf. A167430. [From Clark Kimberling, Nov 03 2009]

A027578 Sums of five consecutive squares: a(n) = n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 + (n+4)^2.

Original entry on oeis.org

30, 55, 90, 135, 190, 255, 330, 415, 510, 615, 730, 855, 990, 1135, 1290, 1455, 1630, 1815, 2010, 2215, 2430, 2655, 2890, 3135, 3390, 3655, 3930, 4215, 4510, 4815, 5130, 5455, 5790, 6135, 6490, 6855, 7230, 7615, 8010, 8415, 8830, 9255, 9690, 10135, 10590, 11055

Offset: 0

Patrick De Geest

a(n) is defined for n < 0 and a(-n) = a(n-4) for any n; a(-3) = a(-1) = 15, a(-2) = 10. - Jean-Christophe Hervé, Nov 11 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Patrick De Geest, Palindromic Sums of Squares of Consecutive Integers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Index entries for two-way infinite sequences.

Cf. A000290, A001844, A027575, A027865, A120328, A260637, A276026.
Cf. A027580.
Subsequence of A174069, A174070, A174071.

[n^2+(n+1)^2+(n+2)^2+(n+3)^2+(n+4)^2: n in [0..50] ]; // Vincenzo Librandi, Jun 17 2011

A027578:=n->5*(n+2)^2+10: seq(A027578(n), n=0..50); # Wesley Ivan Hurt, Nov 12 2015

Table[5 (n + 2)^2 + 10, {n, 0, 50}] (* Bruno Berselli, Jul 29 2015 *)
Total/@Partition[Range[0,50]^2,5,1] (* or *) LinearRecurrence[{3,-3,1},{30,55,90},50] (* Harvey P. Dale, Mar 06 2018 *)

vector(100, n, n--; n^2+(n+1)^2+(n+2)^2+(n+3)^2+(n+4)^2) \\ Altug Alkan, Nov 11 2015

[i^2+(i+1)^2+(i+2)^2+(i+3)^2+(i+4)^2 for i in range(0,50)] # Zerinvary Lajos, Jul 03 2008

a(n) = 5*A059100(n+2).
From Colin Barker, Mar 29 2012: (Start)
G.f.: 5*(6-7*x+3*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. (End)
a(n) = 5*(n + 2)^2 + 10. a(n) is never square. - Bruno Berselli, Jul 29 2015
E.g.f.: 5*(6 + 5*x + x^2)*exp(x). - G. C. Greubel, Aug 24 2022
From Amiram Eldar, Sep 15 2022: (Start)
Sum_{n>=0} 1/a(n) = coth(sqrt(2)*Pi)*Pi/(10*sqrt(2)) - 7/60.
Sum_{n>=0} (-1)^n/a(n) = cosech(sqrt(2)*Pi)*Pi/(10*sqrt(2)) + 1/60. (End)

A064801 Take 1, skip 2, take 2, skip 3, take 3, etc.

Original entry on oeis.org

1, 4, 5, 9, 10, 11, 16, 17, 18, 19, 25, 26, 27, 28, 29, 36, 37, 38, 39, 40, 41, 49, 50, 51, 52, 53, 54, 55, 64, 65, 66, 67, 68, 69, 70, 71, 81, 82, 83, 84, 85, 86, 87, 88, 89, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 121, 122, 123, 124, 125, 126, 127, 128

Offset: 1

Robert G. Wilson v, Oct 21 2001

A253607(a(n)) < 0. - Reinhard Zumkeller, Jan 05 2015
Integers m such that A000196(m) = A079643(m). - Firas Melaih, Dec 10 2020
Also possible values of floor(x*floor(x)) for real x >= 1. - Jianing Song, Feb 16 2021

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A007606, A004202, A048859.
Cf. A061885 (complement), A253607.
Cf. A136272.

a064801 n = a064801_list !! (n-1)
a064801_list = f 1 [1..] where
   f k xs = us ++ f (k + 1) (drop (k + 1) vs)
            where (us, vs) = splitAt k xs
-- Reinhard Zumkeller, May 16 2014

seq(`if`(floor(sqrt(k)) * (floor(sqrt(k)) + 1) > k, k, NULL), k = 0..2034); # a(1)..a(1000), Rainer Rosenthal, Jul 19 2024

a = Table[n, {n, 0, 200} ]; b = {}; Do[a = Drop[a, {1, n} ]; b = Append[b, Take[a, {1, n} ]]; a = Drop[a, {1, n} ], {n, 1, 14} ]; Flatten[b]
Flatten[Table[Range[n^2,n^2+n-1],{n,12}]] (* Harvey P. Dale, Dec 18 2015 *)

{ n=0; for (m=1, 10^9, s=m^2; a=0; for (k=0, m - 1, a=s+k; write("b064801.txt", n++, " ", a); if (n==1000, return)) ) } \\ Harry J. Smith, Sep 26 2009

from math import isqrt  # after Rainer Rosenthal
def isA(k: int): return k < ((s:=isqrt(k)) * (s + 1))
print([k for k in range(129) if isA(k)]) # Peter Luschny, Jul 19 2024

a(n) = A004202(n) - 1.
Can be interpreted as a table read by rows: T(n,k) = n^2 + k, 0 <= k < n. T(n,k) = 0 iff k > A000196(n); T(n,0) = A000290(n); T(n,1) = A002522(n) for n > 1; T(n,2) = A010000(n) = A059100(n) for n > 2; T(n, n-3) = A014209(n-1) for n > 2; T(n, n-2) = A028552(n) for n > 1; T(n, n-1) = A028387(n-1); T(2*n+1, n) = A001107(n+1). - Reinhard Zumkeller, Nov 18 2003
Numbers k such that floor(sqrt(k)) * (floor(sqrt(k)) + 1) > k. - Rainer Rosenthal, Jul 19 2024

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