A074378 -id:A074378 - cp's OEIS frontend

A057570 Numbers of the form n*(7n+-1)/2.

0, 3, 4, 13, 15, 30, 33, 54, 58, 85, 90, 123, 129, 168, 175, 220, 228, 279, 288, 345, 355, 418, 429, 498, 510, 585, 598, 679, 693, 780, 795, 888, 904, 1003, 1020, 1125, 1143, 1254, 1273, 1390, 1410, 1533, 1554, 1683, 1705, 1840, 1863, 2004

Offset: 1

N. J. A. Sloane, Oct 04 2000

Also integers of the form Sum_{k = 1..n} k/7. - Alonso del Arte, Jan 20 2012
Sequence provides all integers m such that 56*m + 1 is a square. [Bruno Berselli, Oct 07 2015]
The sequence terms occur as the exponents in the expansion of Product_{n >= 1} (1 - x^(7*n)) * (1 + x^(7*n-3)) * (1 + x^(7*n-4)) = 1 + x^3 + x^4 + x^13 + x^15 + x^30 + x^33 + .... Cf. A363801. - Peter Bala, Nov 21 2024

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

Cf. A074378, A001318, A057569, A154260, A154292, A154293, A047341, A274830, A363801.

Select[Table[Plus@@Range[n]/7, {n, 0, 199}], IntegerQ] (* Alonso del Arte, Jan 20 2012 *)
CoefficientList[Series[-x (3 + x + 3 x^2) / ((1 + x)^2 (x - 1)^3), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 19 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{0,3,4,13,15},50] (* Harvey P. Dale, Sep 17 2023 *)

a(n)=(14*n*(n-1)+5*(2*n-1)*(-1)^n+5)/16 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: -x^2*(3+x+3*x^2) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Jan 25 2011
a(n) = +1*a(n-1)+2*a(n-2)-2*a(n-3)-1*a(n-4)+1*a(n-5). - Joerg Arndt, Jan 25 2011
a(n) = (14*n*(n-1)+5*(2*n-1)*(-1)^n+5)/16. - Bruno Berselli, Jan 25 2011
a(n)-a(n-2) = A047341(n-1) for n>2. - Bruno Berselli, Jan 25 2011
Sum_{n>=2} 1/a(n) = 14 - 2*cot(Pi/7)*Pi. - Amiram Eldar, Mar 17 2022

A219257 Numbers k such that 11*k+1 is a square.

Original entry on oeis.org

0, 9, 13, 40, 48, 93, 105, 168, 184, 265, 285, 384, 408, 525, 553, 688, 720, 873, 909, 1080, 1120, 1309, 1353, 1560, 1608, 1833, 1885, 2128, 2184, 2445, 2505, 2784, 2848, 3145, 3213, 3528, 3600, 3933, 4009, 4360, 4440, 4809, 4893, 5280, 5368, 5773, 5865

Offset: 1

Bruno Berselli, Nov 16 2012

Equivalently, numbers of the form m*(11*m+2), where m = 0,-1,1,-2,2,-3,3,...

Also, integer values of h*(h+2)/11.

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

Cf. numbers k such that h*k+1 is a square: A005563 (h=1), A046092 (h=2), A001082 (h=3), A002378 (h=4), A036666 (h=5), A062717 (h=6), A132354 (h=7), A000217 (h=8), A132355 (h=9), A132356 (h=10), A152749 (h=12), A219389 (h=13), A219390 (h=14), A204221 (h=15), A074378 (h=16), A219394 (h=17), A219395 (h=18), A219396 (h=19), A219190 (h=20), A219391 (h=21), A219392 (h=22), A219393 (h=23), A001318 (h=24), A219259 (h=25), A217441 (h=26), A219258 (h=27), A219191 (h=28).

Cf. A175885 (square roots of 11*a(n)+1).

[n: n in [0..7000] | IsSquare(11*n+1)];

I:=[0,9,13,40,48]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

Select[Range[0, 7000], IntegerQ[Sqrt[11 # + 1]] &]
CoefficientList[Series[x (9 + 4 x + 9 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)

G.f.: x^2*(9+4*x+9*x^2)/((1+x)^2*(1-x)^3).
a(n) = a(-n+1) = (22*n*(n-1)+7*(-1)^n*(2*n-1)-1)/8 + 1 = (1/176)*(22*n+7*(-1)^n-15)*(22*n+7*(-1)^n-7).
Sum_{n>=2} 1/a(n) = 11/4 - cot(2*Pi/11)*Pi/2. - Amiram Eldar, Mar 15 2022

A057569 Numbers of the form k(5k+1)/2 or k(5k-1)/2.

Original entry on oeis.org

0, 2, 3, 9, 11, 21, 24, 38, 42, 60, 65, 87, 93, 119, 126, 156, 164, 198, 207, 245, 255, 297, 308, 354, 366, 416, 429, 483, 497, 555, 570, 632, 648, 714, 731, 801, 819, 893, 912, 990, 1010, 1092, 1113, 1199, 1221, 1311, 1334, 1428, 1452, 1550

Offset: 1

N. J. A. Sloane, Oct 04 2000

a(n) is the set of all m such that 40*m+1 is a perfect square. - Gary Detlefs, Feb 22 2010
Integers of the form (n^2 - n) / 10. Numbers of the form  n * (5*n - 1) / 2 where n is an integer. - Michael Somos, Jan 13 2012
Also integers of the form sum_{k=1..n} k/5. - Alonso del Arte, Jan 20 2012
These numbers appear in a theta function identity. See the Hardy-Wright reference, Theorem 356 on p. 284. See the G.f. of A113428. - Wolfdieter Lang, Oct 28 2016

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, p. 284.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A074378, A001318, A057570, A154260. - Vladimir Joseph Stephan Orlovsky, Jan 06 2009

Cf. A001622, A113428.

[(10*(n^2-n)+12*(-1)^n*(n div 2))/16: n in [1..60]]; // Vincenzo Librandi, Oct 29 2016

Select[Table[Plus@@Range[n]/5, {n, 0, 199}], IntegerQ] (* Alonso del Arte, Jan 20 2012 *)
LinearRecurrence[{1,2,-2,-1,1},{0,2,3,9,11},50] (* Harvey P. Dale, Jul 05 2021 *)

{a(n) = (10 * (n^2 - n) + 12 * (-1)^n * (n\2)) / 16}; \\ Michael Somos, Jan 13 2012

Vec(x^2*(2*x^2+x+2) / ((1-x)^3*(1+x)^2) + O(x^60)) \\ Colin Barker, Jun 13 2017

A005475 UNION A005476. G.f.: x^2*(2x^2+x+2)/((1-x)^3*(1+x)^2). a(n) = A132356(n+1)/4. - R. J. Mathar, Apr 07 2008
a(n) = (A090771(n)^2 -1)/40. - Gary Detlefs, Feb 22 2010
|A113428(n)| is the characteristic function of the numbers a(n).
a(n) = a(1 - n) for all n in Z. - Michael Somos, Jan 13 2012
From Colin Barker, Jun 13 2017: (Start)
a(n) = n*(5*n - 2)/8 for n even.
a(n) = (5*n - 3)*(n - 1)/8 for n odd.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>5.
(End)
From Amiram Eldar, Mar 17 2022: (Start)
Sum_{n>=2} 1/a(n) = 10 - 2*sqrt(1+2/sqrt(5))*Pi.
Sum_{n>=2} (-1)^n/a(n) = 2*sqrt(5)*log(phi) - 5*(2-log(5)), where phi is the golden ratio (A001622). (End)

A014494 Even triangular numbers.

Original entry on oeis.org

0, 6, 10, 28, 36, 66, 78, 120, 136, 190, 210, 276, 300, 378, 406, 496, 528, 630, 666, 780, 820, 946, 990, 1128, 1176, 1326, 1378, 1540, 1596, 1770, 1830, 2016, 2080, 2278, 2346, 2556, 2628, 2850, 2926, 3160, 3240, 3486, 3570, 3828, 3916, 4186, 4278, 4560

Offset: 0

Mohammad K. Azarian

Even numbers of the form n*(n+1)/2.
Even generalized hexagonal numbers. - Omar E. Pol, Apr 24 2016
The sequence terms occur as the exponents in the expansion of (1 - q^6) * Product_{n >= 1} (1 - q^(16*n-6))*(1 - q^(16*n))*(1 - q^(16*n+6)) = Sum_{n in Z} (-1)^n * q^(2*n*(4*n+1)) = 1 - q^6 - q^10 + q^28 + q^36 - q^66 - q^78 + + - - . - Peter Bala, Dec 23 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000217, A000796, A014493, A056575, A074378, A193867.

See also similar sequences listed in A299645.

[1/2*(2*n+1)*(2*n+1-(-1)^n): n in [0..50]]; // Vincenzo Librandi, Aug 18 2011

Table[2Ceiling[n/2]*(2n + 1), {n, 0, 47}] (* Robert G. Wilson v, Nov 05 2004 *)
1/2 (2#+1)(2#+1-(-1)^#) &/@Range[0,47] (* Ant King, Nov 18 2010 *)
Select[1/2 #(#+1) &/@Range[0,95],EvenQ] (* Ant King, Nov 18 2010 *)

a(n)=(2*n+1)*(2*n+1-(-1)^n)/2 \\ Charles R Greathouse IV, Oct 07 2015

def A014494(n): return (2*n+1)*(n+n%2) # Chai Wah Wu, Mar 11 2022

From Ant King, Nov 18 2010: (Start)
a(n) = (2*n+1)*(2*n+1-(-1)^n)/2.
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5). (End)
G.f.: -2*x*(3*x^2+2*x+3)/((x+1)^2*(x-1)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
a(n) = A000217(A014601(n)). - Reinhard Zumkeller, Oct 04 2004
a(n) = A014493(n+1)-(2n+1)*(-1)^n. - R. J. Mathar, Sep 15 2009
a(n) = A193867(n+1) - 1. - Omar E. Pol, Aug 17 2011
Sum_{n>=1} 1/a(n) = 2 - Pi/2. - Robert Bilinski, Jan 20 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*log(2)-2. - Amiram Eldar, Mar 06 2022
E.g.f.: x*(5 + 2*x)*cosh(x) + (1 + x)*(1 + 2*x)*sinh(x). - Stefano Spezia, Dec 24 2024

More terms from Erich Friedman

A132356 a(2k) = k(10k+2), a(2k+1) = 10k^2 + 18k + 8, with k >= 0.

Original entry on oeis.org

0, 8, 12, 36, 44, 84, 96, 152, 168, 240, 260, 348, 372, 476, 504, 624, 656, 792, 828, 980, 1020, 1188, 1232, 1416, 1464, 1664, 1716, 1932, 1988, 2220, 2280, 2528, 2592, 2856, 2924, 3204, 3276, 3572, 3648, 3960, 4040, 4368, 4452, 4796, 4884, 5244, 5336, 5712

Offset: 0

Mohamed Bouhamida, Nov 08 2007

X values of solutions to the equation 10*X^3 + X^2 = Y^2.
Polygonal number connection: 2*H_n + 6S_n, where H_n is the n-th hexagonal number and S_n is the n-th square number. This is the base formula that is expanded upon to achieve the full series. See contributing formula below. - William A. Tedeschi, Sep 12 2010
Equivalently, numbers of the form 2*h*(5*h+1), where h = 0, -1, 1, -2, 2, -3, 3, -4, 4, ... . - Bruno Berselli, Feb 02 2017

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

Cf. A054000, A056220, A087475, A028347, A046092, A132209.
Cf. numbers m such that k*m+1 is a square: A005563 (k=1), A046092 (k=2), A001082 (k=3), A002378 (k=4), A036666 (k=5), A062717 (k=6), A132354 (k=7), A000217 (k=8), A132355 (k=9), A219257 (k=11), A152749 (k=12), A219389 (k=13), A219390 (k=14), A204221 (k=15), A074378 (k=16), A219394 (k=17), A219395 (k=18), A219396 (k=19), A219190 (k=20), A219391 (k=21), A219392 (k=22), A219393 (k=23), A001318 (k=24), A219259 (k=25), A217441 (k=26), A219258 (k=27), A219191 (k=28).
Cf. A220082 (numbers k such that 10*k-1 is a square).

CoefficientList[Series[4*x*(2*x^2 + x + 2)/((1 - x)^3*(1 + x)^2), {x, 0, 50}], x] (* G. C. Greubel, Jun 12 2017 *)
LinearRecurrence[{1,2,-2,-1,1},{0,8,12,36,44},50] (* Harvey P. Dale, Dec 15 2023 *)

my(x='x+O('x^50)); concat([0], Vec(4*x*(2*x^2+x+2)/((1-x)^3*(1+x)^2))) \\ G. C. Greubel, Jun 12 2017

a(n) = n^2 + n + 6*((n+1)\2)^2 \\ Charles R Greathouse IV, Sep 11 2022

G.f.: 4*x*(2*x^2+x+2)/((1-x)^3*(1+x)^2). - R. J. Mathar, Apr 07 2008
a(n) = 10*x^2 - 2*x, where x = floor(n/2)*(-1)^n for n >= 1. - William A. Tedeschi, Sep 12 2010
a(n) = ((2*n+1-(-1)^n)*(10*(2*n+1)-2*(-1)^n))/16. - Luce ETIENNE, Sep 13 2014
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 4. - Chai Wah Wu, May 24 2016
Sum_{n>=1} 1/a(n) = 5/2 - sqrt(1+2/sqrt(5))*Pi/2. - Amiram Eldar, Mar 15 2022
a(n) = n^2 + n + 6*ceiling(n/2)^2. - Ridouane Oudra, Aug 06 2022

More terms from Max Alekseyev, Nov 13 2009

A154260 Numbers of the form m(4m +- 1)/2.

Original entry on oeis.org

0, 7, 9, 30, 34, 69, 75, 124, 132, 195, 205, 282, 294, 385, 399, 504, 520, 639, 657, 790, 810, 957, 979, 1140, 1164, 1339, 1365, 1554, 1582, 1785, 1815, 2032, 2064, 2295, 2329, 2574, 2610, 2869, 2907, 3180, 3220, 3507, 3549, 3850, 3894, 4209, 4255, 4584

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 06 2009

Also integers of the form Sum_{k = 1..j} k/4 = j*(j + 1)/8. - Alonso del Arte, Jan 20 2012
Numbers h such that 32*h + 1 is a square. - Bruno Berselli, Mar 30 2014
The sequence terms are the exponents in the expansion of Product_{n >= 1} (1 - q^(16*n))*(1 - q^(16*n-7))*(1 - q^(16*n-9)) = 1 - q^7 - q^9 + q^30 + q^34 - q^69 - q^75 + + - - .... - Peter Bala, Dec 24 2024

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

Cf. A000217, A001318, A074378, A057569, A057570.

Cf. similar sequences listed in A219257 and A299645.

k:=8; f:=func; [0] cat [f(n*m): m in [-1, 1], n in [1..25]]; // Bruno Berselli, Nov 14 2012

Select[Union[Flatten[Table[{n (4n - 1)/2, n (4n + 1)/2}, {n, 0, 199}]]], IntegerQ] (* Alonso del Arte, Jan 20 2012 *)

print1(0);forstep(n=2,1e2,2,print1(", "n*(4*n-1)/2", "n*(4*n+1)/2)) \\ Charles R Greathouse IV, Jan 20 2012

print1(s=0);for(n=1,1e3,s+=n/4;if(denominator(s)==1,print1(s", "))) \\ Charles R Greathouse IV, Jan 20 2012

def A154260(n): return (n>>1)*((n<<2)+(-3 if n&1 else -1)) # Chai Wah Wu, Mar 11 2025

From R. J. Mathar, Jan 07 2009: (Start)
A139274 UNION A139275.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
G.f.: x^2*(7 + 2x + 7x^2)/((1+x)^2*(1-x)^3). (End)
From G. C. Greubel, Sep 08 2016: (Start)
a(n) = (1/4)*(8*n^2 + 6*(-1)^n*n - 8*n - 3*(-1)^n + 3).
E.g.f.: (1/4)*( (3 + 8*x^2)*exp(x) - 3*(1 + 2*x)*exp(-x) ). (End)
From Amiram Eldar, Mar 17 2022: (Start)
Sum_{n>=2} 1/a(n) = 8 - (sqrt(2)+1)*Pi.
Sum_{n>=2} (-1)^n/a(n) = 2*sqrt(2)*log(sqrt(2)+1) - 8*(1-log(2)). (End)
a(n) = (n-1)*(4*n-3)/2 if n is odd and a(n) = n*(4*n-1)/2 if n is even. - Chai Wah Wu, Mar 11 2025

A165717 Integers of the form k*(5+k)/4.

Original entry on oeis.org

6, 9, 21, 26, 44, 51, 75, 84, 114, 125, 161, 174, 216, 231, 279, 296, 350, 369, 429, 450, 516, 539, 611, 636, 714, 741, 825, 854, 944, 975, 1071, 1104, 1206, 1241, 1349, 1386, 1500, 1539, 1659, 1700, 1826, 1869, 2001, 2046, 2184, 2231, 2375, 2424, 2574, 2625

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 24 2009

Integers of the form k+k*(k+1)/4 = k+A000217(k)/2; for k see A014601, for A000217(k)/2 see A074378.
Are all terms composite?
Yes, because a(2*k) = k*(4*k+5) and a(2*k-1) = (k+1)*(4*k-1). - Bruno Berselli, Apr 07 2013
Numbers m such that 16*m + 25 is a square. - Vincenzo Librandi, Apr 07 2013

			For k =1,2,3,.. the value of k*(k+5)/4 is 3/2, 7/2, 6, 9, 25/2, 33/2, 21, 26, 63/2, 75/2, 44, 51,.. and the integer values define the sequence.

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

Cf. A000217, A014601, A074378.

[n: n in [1..3000] | IsSquare(16*n+25)]; // Vincenzo Librandi, Apr 07 2013

q=2;s=0;lst={};Do[s+=((n+q)/q);If[IntegerQ[s],AppendTo[lst,s]],{n,6!}];lst
Select[Table[k*(5+k)/4,{k,100}],IntegerQ] (* or *) LinearRecurrence[ {1,2,-2,-1,1},{6,9,21,26,44},60] (* Harvey P. Dale, Aug 11 2011 *)
Select[Range[1, 3000], IntegerQ[Sqrt[16 # + 25]]&] (* Vincenzo Librandi, Apr 07 2013 *)

From R. J. Mathar, Sep 25 2009: (Start)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
G.f.: x*(-6-3*x+x^3)/( (1+x)^2 * (x-1)^3 ). (End)
Sum_{n>=1} 1/a(n) = 29/25 - Pi/5. - Amiram Eldar, Jul 26 2024

Definition simplified by R. J. Mathar, Sep 25 2009

A214264 Expansion of f(x^3, x^5) in powers of x where f() is Ramanujan's two-variable theta function.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Michael Somos, Jul 09 2012

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			1 + x^3 + x^5 + x^14 + x^18 + x^33 + x^39 + x^60 + x^68 + x^95 + x^105 +
q + q^49 + q^81 + q^225 + q^289 + q^529 + q^625 + q^961 + q^1089 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A010054, A047522, A074378.

Cf. A000122, A000700, A121373.

f[x_, y_]:= QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A214264[n_] := SeriesCoefficient[f[x^3, x^5], {x, 0, n}]; Table[A214264[n], {n, 0, 50}] (* G. C. Greubel, Dec 03 2017 *)

PARI
```
{a(n) = issquare( 16*n + 1)}
```

Euler transform of period 16 sequence [ 0, 0, 1, 0, 1, -1, 0, -1, 0, -1, 1, 0, 1, 0, 0, -1, ...].
G.f.: Sum_{k} x^(((8*k + 1)^2 - 1) / 16).
Characteristic function of A074378. a(n) = 1 if and only if n is in A074378.
a(n) = A010054(2*n).
Sum_{k=1..n} a(k) ~ sqrt(n). - Amiram Eldar, Jan 13 2024

A219191 Numbers of the form k(7k+1), where k = 0,-1,1,-2,2,-3,3,...

Original entry on oeis.org

0, 6, 8, 26, 30, 60, 66, 108, 116, 170, 180, 246, 258, 336, 350, 440, 456, 558, 576, 690, 710, 836, 858, 996, 1020, 1170, 1196, 1358, 1386, 1560, 1590, 1776, 1808, 2006, 2040, 2250, 2286, 2508, 2546, 2780, 2820, 3066, 3108, 3366, 3410, 3680, 3726, 4008

Offset: 1

Bruno Berselli, Nov 14 2012

Equivalently, numbers m such that 28*m+1 is a square.
Also, integer values of h*(h+1)/7.
Let F(r) = Product_{n >= 1} 1 - q^(14*n-r). The sequence terms are the exponents in the expansion of F(0)*F(6)*F(8) = 1 - q^6 - q^8 + q^26 + q^30 - q^60 - q^66 + + - - ... (by the triple product identity).- Peter Bala, Dec 25 2024

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

Cf. numbers of the form k*(i*k+1) with k in A001057: i=0, A001057; i=1, A110660; i=2, A000217; i=3, A152749; i=4, A074378; i=5, A219190; i=6, A036498; i=7, this sequence; i=8, A154260.
Cf. A113801 (square roots of 28*a(n)+1, see the comment).
Cf. similar sequences listed in A219257.
Subsequence of A011860.

k:=7; f:=func; [0] cat [f(n*m): m in [-1,1], n in [1..25]];

I:=[0,6,8,26,30]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

A := proc (q) local n; for n from 0 to q do if type(sqrt(28*n+1), integer) then print(n) fi; od; end: A(4100); # Peter Bala, Dec 25 2024

Rest[Flatten[{# (7 # - 1), # (7 # + 1)} & /@ Range[0, 25]]]
CoefficientList[Series[2 x (3 + x + 3 x^2) / ((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{0,6,8,26,30},50] (* Harvey P. Dale, Sep 14 2022 *)

G.f.: 2*x^2*(3+x+3*x^2)/((1+x)^2*(1-x)^3).
a(n) = a(-n+1) = (14*n*(n-1)+5*(-1)^n*(2*n-1)+5)/8.
a(n) = 2*A057570(n) = (1/7)*A047335(n)*A047274(n+1).
Sum_{n>=2} 1/a(n) = 7 - cot(Pi/7)*Pi. - Amiram Eldar, Mar 17 2022

A266883 Numbers of the form m(4m+1)+1, where m = 0,-1,1,-2,2,-3,3,...

Original entry on oeis.org

1, 4, 6, 15, 19, 34, 40, 61, 69, 96, 106, 139, 151, 190, 204, 249, 265, 316, 334, 391, 411, 474, 496, 565, 589, 664, 690, 771, 799, 886, 916, 1009, 1041, 1140, 1174, 1279, 1315, 1426, 1464, 1581, 1621, 1744, 1786, 1915, 1959, 2094, 2140, 2281, 2329, 2476, 2526

Offset: 0

Bruno Berselli, Jan 05 2016

Also, numbers m such that 16*m-15 is a square. Therefore, the terms 1 and 4 are the only squares in this sequence.

Conjecture: the sequence terms are the exponents in the expansion of Sum_{n >= 1} q^n * (Product_{k >= 2*n-1} 1 - q^k) = q + q^4 + q^6 + q^15 + q^19 + q^34 + .... Cf. A174114. - Peter Bala, May 10 2025

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

Cf. A001057, A049486, A127365, A130472.
Cf. A002061: m*(4*m+2)+1 for m = 0,0,-1,1,-2,2,-3,3, ...
Cf. A174114: m*(4*m+3)+1 for m = 0,-1,1,-2,2,-3,3,-4,4, ...
Cf. A054556: m*(4*m+1)+1 for nonpositive m.
Cf. A054567: m*(4*m+1)+1 for nonnegative m.
Cf. A074378: numbers m such that 16*m+1 is a square.

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

I:=[1,4,6,15,19]; [n le 5 select I[n] else Self(n-1) + 2*Self(n-2) -2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..60]]; // Vincenzo Librandi, Jan 06 2016

Table[n (n + 1) + 1 - ((2 n + 1) (-1)^n - 1)/4, {n, 0, 50}]
LinearRecurrence[{1, 2, -2, -1, 1}, {1, 4, 6, 15, 19}, 60] (* Vincenzo Librandi, Jan 06 2016 *)

vector(50, n, n--; n*(n+1)+1-((2*n+1)*(-1)^n-1)/4)

Vec((1+3*x+3*x^3+x^4)/((1+x)^2*(1-x)^3) + O(x^100)) \\ Altug Alkan, Jan 06 2016

[n*(n+1)+1-((2*n+1)*(-1)**n-1)/4 for n in range(60)]

[n*(n+1)+1-((2*n+1)*(-1)^n-1)/4 for n in range(50)]

O.g.f.: (1 + 3*x + 3*x^3 + x^4)/((1 + x)^2*(1 - x)^3).
E.g.f.: (5 + 8*x + 4*x^2)*exp(x)/4 -(1 - 2*x)*exp(-x)/4.
a(n) = a(-n-1) = n*(n + 1) + 1 - ((2*n + 1)*(-1)^n - 1)/4 = (2*n + 1)*floor((n + 1)/2) + 1.
a(n) = A002061(n+1) + A001057(n) = A074378(n)+1.
a(n+1) + a(n+2) = A049486(n+3).

cp's OEIS Frontend

A057570 Numbers of the form n*(7n+-1)/2.

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A219257 Numbers k such that 11*k+1 is a square.

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A057569 Numbers of the form k*(5*k+1)/2 or k*(5*k-1)/2.

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A014494 Even triangular numbers.

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A132356 a(2*k) = k*(10*k+2), a(2*k+1) = 10*k^2 + 18*k + 8, with k >= 0.

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A154260 Numbers of the form m*(4*m +- 1)/2.

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A165717 Integers of the form k*(5+k)/4.

A057569 Numbers of the form k(5k+1)/2 or k(5k-1)/2.

A132356 a(2k) = k(10k+2), a(2k+1) = 10k^2 + 18k + 8, with k >= 0.

A154260 Numbers of the form m(4m +- 1)/2.

A219191 Numbers of the form k(7k+1), where k = 0,-1,1,-2,2,-3,3,...

A266883 Numbers of the form m(4m+1)+1, where m = 0,-1,1,-2,2,-3,3,...