A036666 -id:A036666 - cp's OEIS frontend

A147874 a(n) = (5n-7)(n-1).

0, 3, 16, 39, 72, 115, 168, 231, 304, 387, 480, 583, 696, 819, 952, 1095, 1248, 1411, 1584, 1767, 1960, 2163, 2376, 2599, 2832, 3075, 3328, 3591, 3864, 4147, 4440, 4743, 5056, 5379, 5712, 6055, 6408, 6771, 7144, 7527, 7920, 8323, 8736, 9159, 9592, 10035

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 16 2008

Zero followed by partial sums of A017305.
Appears to be related to various other sequences: a(n) = A036666(2*n-2) for n>1; a(n) = A115006(2*n-3) for n>1; a(n) = A118015(5*n-6) for n>1; a(n) = A008738(5*n-7) for n>1.
Even dodecagonal numbers divided by 4. - Omar E. Pol, Aug 19 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Leo Tavares, Illustration: Triangular Badges
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A017305 (10n+3), A036666, A115006, A118015 (floor(n^2/5)), A008738 (floor((n^2+1)/5)), A294830.
Cf. A051624, A193872. - Omar E. Pol, Aug 19 2011
Cf. A014105, A002378.

List([1..50], n-> (5*n-7)*(n-1)); # G. C. Greubel, Jul 30 2019

[ 0 ] cat [ &+[ 10*k+3: k in [0..n-1] ]: n in [1..50] ]; // Klaus Brockhaus, Nov 17 2008

Magma
```
[ 5*n^2-2*n: n in [0..50] ];
```

s=0;lst={s};Do[s+=n++ +3;AppendTo[lst,s],{n,0,6!,10}];lst
Table[5n^2-12n+7,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{0,3,16},50] (* or *) PolygonalNumber[12,Range[0,100,2]]/4 (* Harvey P. Dale, Aug 08 2021 *)

{m=50; a=7; for(n=0, m, print1(a=a+10*(n-1)+3, ","))} \\ Klaus Brockhaus, Nov 17 2008

[(5*n-7)*(n-1) for n in (1..50)] # G. C. Greubel, Jul 30 2019

a(n) = Sum_{k=0..n-2} 10*k+3 = Sum_{k=0..n-2} A017305(k).
G.f.: x*(3 + 7*x)/(1-x)^3.
a(n) = 10*(n-2) + 3 + a(n-1).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = A193872(n-1)/4. - Omar E. Pol, Aug 19 2011
a(n+1) = A131242(10n+2). - Philippe Deléham, Mar 27 2013
E.g.f.: -7 + (7 - 7*x + 5*x^2)*exp(x). - G. C. Greubel, Jul 30 2019
Sum_{n>=2} 1/a(n) = A294830. - Amiram Eldar, Nov 15 2020
a(n) = A014105(n-1) + 3*A002378(n-2). - Leo Tavares, Mar 31 2025

Edited by R. J. Mathar and Klaus Brockhaus, Nov 17 2008, Nov 20 2008

A132209 a(0) = 0 and a(n) = 2n^2 + 2n - 1, for n>=1.

Original entry on oeis.org

0, 3, 11, 23, 39, 59, 83, 111, 143, 179, 219, 263, 311, 363, 419, 479, 543, 611, 683, 759, 839, 923, 1011, 1103, 1199, 1299, 1403, 1511, 1623, 1739, 1859, 1983, 2111, 2243, 2379, 2519, 2663, 2811, 2963, 3119, 3279, 3443, 3611, 3783, 3959, 4139, 4323, 4511

Offset: 0

Mohamed Bouhamida, Nov 06 2007

nonn

Previous name was: Sequence gives X values that satisfy the integer equation 2*X^3 + 3*X^2 = Y^2.

To find Y values: b(n) = (2*n^2 + 2*n - 1)*(2*n - 1).

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3, -3, 1).

Cf. A005563, A046092, A001082, A002378, A036666, A062717, A028347, A087475, A000217.

[0] cat [2*n^2+2*n-1: n in [1..50]]; // Vincenzo Librandi, Sep 22 2015

Join[{0}, LinearRecurrence[{3, -3, 1}, {3, 11, 23}, 40]] (* Vincenzo Librandi, Sep 22 2015 *)

for(n=0,50, print1(if(n==0, 0, 2*n^2 + 2*n -1), ", ")) \\ G. C. Greubel, Jul 13 2017

a(n) = 2*n^2 + 2*n - 1 for n>=1.
G.f.: x*(1+x)*(3-x)/(1-x)^3. - R. J. Mathar, Nov 14 2007
E.g.f.: 1 + (2*x^2 + 4*x -1)*exp(x). - G. C. Greubel, Jul 13 2017
From Amiram Eldar, Mar 07 2021: (Start)
Sum_{n>=1} 1/a(n) = 1 + sqrt(3)*Pi*tan(sqrt(3)*Pi/2)/6.
Product_{n>=1} (1 + 1/a(n)) = -Pi*sec(sqrt(3)*Pi/2)/2.
Product_{n>=1} (1 - 1/a(n)) = cos(sqrt(5)*Pi/2)*sec(sqrt(3)*Pi/2)/2. (End)

Edited by the Associate Editors of the OEIS, Nov 15 2009
More terms from Vincenzo Librandi, Sep 22 2015
Shorter name (using formula given) from Joerg Arndt, Sep 27 2015

A062317 Numbers k such that 5*k-1 is a perfect square.

Original entry on oeis.org

1, 2, 10, 13, 29, 34, 58, 65, 97, 106, 146, 157, 205, 218, 274, 289, 353, 370, 442, 461, 541, 562, 650, 673, 769, 794, 898, 925, 1037, 1066, 1186, 1217, 1345, 1378, 1514, 1549, 1693, 1730, 1882, 1921, 2081, 2122, 2290, 2333, 2509, 2554, 2738, 2785, 2977

Offset: 1

Santi Spadaro, Jul 12 2001

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

Cf. A036666.

f[n_]:=IntegerQ[Sqrt[5*n-1]]; Select[Range[0,8! ],f[ # ]&] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{1,2,-2,-1,1},{1,2,10,13,29},50] (* Harvey P. Dale, Dec 29 2018 *)

je=[]; for(n=1,5000, if(issquare(5*n-1),je=concat(je,n))); je

{ n=0; for (m=1, 10^9, if (issquare(5*m - 1), write("b062317.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 04 2009

a(n) = ((2+5*(n-1)/2)^2 + 1)/5 if n is odd; a(n) = ((3+5*(n-2)/2)^2 + 1)/5 if n is even.
From R. J. Mathar, Jan 30 2010: (Start)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
G.f.: x*(1+x+6*x^2+x^3+x^4)/((1+x)^2*(1-x)^3). (End)
a(n) = (10*n*(n-1) + 5 - (6*n-3)*(-1)^n)/8. - Eric Simon Jacob, Jan 20 2020

More terms from Jason Earls, Jul 14 2001

A212967 Number of (w,x,y) with all terms in {0,...,n} and w < range{w,x,y}.

Original entry on oeis.org

0, 3, 10, 26, 50, 89, 140, 212, 300, 415, 550, 718, 910, 1141, 1400, 1704, 2040, 2427, 2850, 3330, 3850, 4433, 5060, 5756, 6500, 7319, 8190, 9142, 10150, 11245, 12400, 13648, 14960, 16371, 17850, 19434, 21090, 22857, 24700, 26660, 28700

Offset: 0

Clark Kimberling, Jun 02 2012

For a guide to related sequences, see A212959.

Muniru A Asiru, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A212959, A212968, A036666.

List([1..45],n->Sum([1..n],k->(10*k*(k-1)+(2*k-1)*(-1)^k+1)/8)); # Muniru A Asiru, Nov 28 2018

[(n+1)*(10*n*(n+2) - 3*(-1)^n+3)/24: n in [0..50]]; // Vincenzo Librandi, Nov 29 2018

A212967:=n->(n+1)*(10*n*(n+2)-3*(-1)^n+3)/24: seq(A212967(n), n=0..100); # Wesley Ivan Hurt, Apr 28 2017

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w < (Max[w, x, y] - Min[w, x, y]), s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
Map[t[#] &, Range[0, 60]]   (* A212967 *)
Accumulate[Accumulate[Table[n + LCM[n, 2], {n, 0, 60}]]] (* Jon Maiga, Nov 28 2018 *)
LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 3, 10, 26, 50, 89}, 50] (* Vincenzo Librandi, Nov 29 2018 *)

a(n) + A212968(n) = (n + 1)^3.
a(n) = (n + 1)*(10*n*(n + 2) - 3*(-1)^n + 3)/24.
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).
G.f.: f(x)/g(x), where f(x) = x*(3 + 4*x + 3*x^2) and g(x) = ((1 - x)^4)(1 + x)^2.
a(n) = Sum_{k=1..n} A036666(k). - Jon Maiga, Nov 28 2018
E.g.f.: (exp(x)*(3 + 63*x + 60*x^2 + 10*x^3) - 3*exp(-x)*(1 - x))/24. - Franck Maminirina Ramaharo, Nov 29 2018

A132354 Integers m such that 7*m + 1 is a square.

Original entry on oeis.org

0, 5, 9, 24, 32, 57, 69, 104, 120, 165, 185, 240, 264, 329, 357, 432, 464, 549, 585, 680, 720, 825, 869, 984, 1032, 1157, 1209, 1344, 1400, 1545, 1605, 1760, 1824, 1989, 2057, 2232, 2304, 2489, 2565, 2760, 2840, 3045, 3129, 3344, 3432, 3657, 3749, 3984, 4080

Offset: 0

Mohamed Bouhamida, Nov 08 2007

Numbers of the form m*(7*m + 2) for m = 0, -1, 1, -2, 2, -3, 3, ... - Bruno Berselli, Feb 26 2018

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

Cf. A054000, A056220, A000217, A087475, A028347, A062717, A036666, A002378, A001082, A046092, A005563.

[n^2+n+3*Ceiling(n/2)^2: n in [0..50]]; // Wesley Ivan Hurt, Jul 07 2014

seq(n^2+n+3*ceil(n/2)^2, n=0..48); # Gary Detlefs, Feb 23 2010

f[n_]:=IntegerQ[Sqrt[1+7*n]]; Select[Range[0,8! ],f[ # ]&] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)

a(2*k) = k*(7*k + 2), a(2*k + 1) = 7*k^2 + 12*k + 5.
a(n) = n^2 + n + 3*ceiling(n/2)^2. - Gary Detlefs, Feb 23 2010
G.f.: -x*(5*x^2 + 4*x + 5)/((x - 1)^3*(x + 1)^2). - Colin Barker, Oct 24 2012
Sum_{n>=1} 1/a(n) = 7/4 - cot(2*Pi/7)*Pi/2. - Amiram Eldar, Mar 15 2022

More terms from Max Alekseyev, Nov 13 2009

Better definition from Max Alekseyev, Oct 24 2012

A205633 Expansion of f(x^3, x^7) in powers of x where f() is Ramanujan's two-variable theta function.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Feb 02 2012

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

			G.f. = 1 + x^3 + x^7 + x^16 + x^24 + x^39 + x^51 + x^72 + x^88 + x^115 + x^135 + ...
G.f. = q + q^16 + q^36 + q^81 + q^121 + q^196 + q^256 + q^361 + q^441 + q^576 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares. Discrete Math. 274 (2004), no. 1-3, 9-24. See D(q).
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A036666, A205988.

A205633[n_] := SeriesCoefficient[QPochhammer[-q^3, q^10]* QPochhammer[-q^7, q^10]*QPochhammer[q^10, q^10], {q, 0, n}]; Table[A205633[n], {n,0,50}] (* G. C. Greubel, Jun 16 2017 *)

{a(n) = issquare(5*n + 1)}; /* Michael Somos, Sep 22 2012 */

Euler transform of period 20 sequence [ 0, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, 0, 1, -1, 0, 0, 1, 0, 0, -1, ...]. - Michael Somos, Sep 22 2012
Characteristic function of A036666: a(n) = 1 if and only if n is in A036666.
G.f.: Sum_{k in Z} x^(5*k^2 + 2*k).
a(3*n + 2) = a(4*n + 1) = a(4*n + 2) = 0. a(4*n + 3) = A205988(n). - Michael Somos, Sep 22 2012

A298950 Numbers k such that 5*k - 4 is a square.

Original entry on oeis.org

1, 4, 8, 17, 25, 40, 52, 73, 89, 116, 136, 169, 193, 232, 260, 305, 337, 388, 424, 481, 521, 584, 628, 697, 745, 820, 872, 953, 1009, 1096, 1156, 1249, 1313, 1412, 1480, 1585, 1657, 1768, 1844, 1961, 2041, 2164, 2248, 2377, 2465, 2600, 2692, 2833, 2929, 3076, 3176, 3329, 3433

Offset: 1

Bruno Berselli, Jan 30 2018

a(n) is a member of A140612. Proof: a(n) = n^2 + (n/2-1)^2 for even n, otherwise a(n) = (n-1)^2 + ((n+1)/2)^2; also, a(n) + 1 = (n-1)^2 + (n/2+1)^2 for even n, otherwise a(n) + 1 = n^2 + ((n-3)/2)^2. Therefore, both a(n) and a(n) + 1 belong to A001481.
Primes in sequence are listed in A245042.
Squares in sequence are listed in A081068.

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

Cf. A195162: numbers k such that 5*k + 4 is a square.
Subsequence of A001481, A020668, A036404, A140612.
Cf. A036666, A081068, A106833 (first differences), A245042.

List([1..60], n -> (10*n*(n-1)+(2*n-1)*(-1)^n+9)/8);

[(10*n*(n-1)+(2*n-1)*(-1)^n+9)/8: n in [1..60]];

Table[(10 n (n - 1) + (2 n - 1) (-1)^n + 9)/8, {n, 1, 60}]
LinearRecurrence[{1,2,-2,-1,1},{1,4,8,17,25},60] (* Harvey P. Dale, Sep 16 2022 *)

makelist((10*n*(n-1)+(2*n-1)*(-1)^n+9)/8, n, 1, 60);

Vec((1+x^2)*(1+3*x+x^2)/((1-x)^3*(1+x)^2)+O(x^60))

vector(60, n, nn; (10*n*(n-1)+(2*n-1)*(-1)^n+9)/8)

[(10*n*(n-1)+(2*n-1)*(-1)**n+9)/8 for n in range(1, 60)]

[(10*n*(n-1)+(2*n-1)*(-1)^n+9)/8 for n in (1..60)]

G.f.: x*(1 + x^2)*(1 + 3*x + x^2)/((1 - x)^3*(1 + x)^2).
a(n) = a(1-n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = (10*n*(n-1) + (2*n-1)*(-1)^n + 9)/8.
a(n) = A036666(n) + 1.

cp's OEIS Frontend

A147874 a(n) = (5*n-7)*(n-1).

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A132209 a(0) = 0 and a(n) = 2*n^2 + 2*n - 1, for n>=1.

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A062317 Numbers k such that 5*k-1 is a perfect square.

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Mathematica

PARI

PARI

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A212967 Number of (w,x,y) with all terms in {0,...,n} and w < range{w,x,y}.

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GAP

Magma

Maple

Mathematica

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A132354 Integers m such that 7*m + 1 is a square.

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Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

Extensions

A205633 Expansion of f(x^3, x^7) in powers of x where f() is Ramanujan's two-variable theta function.

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Mathematica

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Formula

A147874 a(n) = (5n-7)(n-1).

A132209 a(0) = 0 and a(n) = 2n^2 + 2n - 1, for n>=1.