A074377 -id:A074377 - cp's OEIS frontend

A214263 Expansion of f(x^1, x^7) in powers of x where f() is Ramanujan's general theta function.

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

Offset: 0

Michael Somos and Omar E. Pol, Jul 09 2012

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

Characteristic function of A074377: a(n) = 1 if and only if n is in A074377.

			G.f. = 1 + x + x^7 + x^10 + x^22 + x^27 + x^45 + x^52 + x^76 + x^85 + x^115 + ...
G.f. = q^9 + q^25 + q^121 + q^169 + q^361 + q^441 + q^729 + q^841 + q^1225 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Index entries for characteristic functions.

Cf. A047621, A074377, A115359.
A000122, A080995, A010054, A133100, A089801 have g.f. of f(x,x^k) for k=1..5.
Cf. A000122, A000700, A121373.

f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; Table[SeriesCoefficient[f[q, q^7], {q, 0, n}], {n, 0, 50}] (* G. C. Greubel, Oct 05 2017 *)

PARI
```
{a(n) = issquare(16*n + 9)};
```

Euler transform of period 16 sequence [ 1, -1, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, -1, 1, -1, ...].
G.f.: f(x, x^7) = sum_{k in Z} x^(4*k^2 - 3*k).
a(n) = A010054(2*n + 1) = A115359(2*n).
Sum_{k=1..n} a(k) ~ sqrt(n). - Amiram Eldar, Jan 13 2024

A195315 Centered 32-gonal numbers.

Original entry on oeis.org

1, 33, 97, 193, 321, 481, 673, 897, 1153, 1441, 1761, 2113, 2497, 2913, 3361, 3841, 4353, 4897, 5473, 6081, 6721, 7393, 8097, 8833, 9601, 10401, 11233, 12097, 12993, 13921, 14881, 15873, 16897, 17953, 19041, 20161, 21313, 22497, 23713, 24961, 26241, 27553, 28897, 30273

Offset: 1

Omar E. Pol, Sep 16 2011

Sequence found by reading the line from 1, in the direction 1, 33, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. Semi-axis opposite to A016802 in the same spiral.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A195146.
Cf. A003154, A069129, A069133, A069190, A195314, A195316, A195317, A195318.
Cf. A016802, A074377.

[(16*n^2-16*n+1): n in [1..50]]; // Vincenzo Librandi, Sep 19 2011

Table[16*n^2 - 16*n + 1, {n, 1, 41}] (* Amiram Eldar, Feb 11 2022 *)
LinearRecurrence[{3,-3,1},{1,33,97},50] (* Harvey P. Dale, Feb 11 2024 *)

a(n)=16*n^2-16*n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 16*n^2 - 16*n + 1.
G.f.: -x*(1 + 30*x + x^2)/(x-1)^3. - R. J. Mathar, Sep 18 2011
Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(3)*Pi/4)/(8*sqrt(3)). - Amiram Eldar, Feb 11 2022
From Elmo R. Oliveira, Nov 14 2024: (Start)
E.g.f.: exp(x)*(16*x^2 + 1) - 1.
a(n) = 2*A069129(n) - 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A181995 a(n) = if n mod 2 = 1 then n*(n - 1) else (n - 1)^2 + (n - 2)/2.

Original entry on oeis.org

0, 0, 1, 6, 10, 20, 27, 42, 52, 72, 85, 110, 126, 156, 175, 210, 232, 272, 297, 342, 370, 420, 451, 506, 540, 600, 637, 702, 742, 812, 855, 930, 976, 1056, 1105, 1190, 1242, 1332, 1387, 1482, 1540, 1640, 1701, 1806, 1870, 1980, 2047, 2162, 2232, 2352, 2425, 2550, 2626, 2756, 2835, 2970, 3052, 3192, 3277, 3422, 3510, 3660, 3751, 3906, 4000, 4160, 4257, 4422

Offset: 0

N. J. A. Sloane, Apr 05 2012

Decagonal numbers (A001107) and twice second hexagonal numbers (A002943) interleaved. - Omar E. Pol, Aug 03 2012
Similar to A074377. Members of this family are A093005, A210977, A006578, A210978, this sequence, A210981, A210982. - Omar E. Pol, Aug 09 2012
Number of kites whose vertices are the vertices a regular 2n-gon. - Halil Ibrahim Kanpak, Nov 08 2018

H. L. Abbott, D. Hanson, N. Sauer, Intersection theorems for systems of sets, J. Combinatorial Theory Ser. A 12 (1972), 381--389.MR0297579 (45 #6633).
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A001107, A002943, A006578, A074377, A093005.

Cf. A210977, A210978, A210981, A210982.

[n*(4*n - 5 - (-1)^n)/4 : n in [0..80]]; // Wesley Ivan Hurt, Apr 11 2016

f:=n->if n mod 2 = 1 then n*(n-1) else (n-1)^2+(n-2)/2; fi;
[seq(f(n),n=0..130)];

Table[n*(4*n - 5 - (-1)^n)/4, {n, 0, 80}] (* Wesley Ivan Hurt, Apr 11 2016 *)

a(n)=n*(4*n-5-(-1)^n)/4 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: -x^2*(1 + 5*x + 2*x^2)/((1 + x)^2*(x - 1)^3). - R. J. Mathar, Apr 06 2012
a(n) = n*(4*n - 5 - (-1)^n)/4. - Luce ETIENNE, Oct 04 2014
From Wesley Ivan Hurt, Apr 11 2016: (Start)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
a(n) = Sum_{i=floor((n-1)/2)..floor(3*(n-1)/2)} i. (End)
E.g.f.: x^2*cosh(x) - x*(1 - 2*x)*sinh(x)/2. - Franck Maminirina Ramaharo, Nov 08 2018

A274579 Values of k such that 2k+1 and 5k+1 are both triangular numbers.

Original entry on oeis.org

0, 1, 7, 27, 540, 2002, 10660, 39501, 779247, 2887450, 15372280, 56960982, 1123674201, 4163701465, 22166817667, 82137697110, 1620337419162, 6004054625647, 31964535704101, 118442502272205, 2336525434757970, 8657842606482076, 46092838318496542

Offset: 1

Colin Barker, Jun 29 2016

Intersection of A074377 and A085787.

			7 is in the sequence because 2*7+1 = 15, 5*7+1 = 36, and 15 and 36 are both triangular numbers.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1442,-1442,0,0,-1,1).

Cf. A000217, A074377, A085787, A124174.

concat(0, Vec(x^2*(1+6*x+20*x^2+513*x^3+20*x^4+6*x^5+x^6)/((1-x)*(1+6*x-x^2)*(1-6*x-x^2)*(1+38*x^2+x^4)) + O(x^30)))

isok(n) = ispolygonal(2*n+1, 3) && ispolygonal(5*n+1, 3); \\ Michel Marcus, Jun 29 2016

G.f.: x^2*(1+6*x+20*x^2+513*x^3+20*x^4+6*x^5+x^6) / ((1-x)*(1+6*x-x^2)*(1-6*x-x^2)*(1+38*x^2+x^4)).

A299645 Numbers of the form m(8m + 5), where m is an integer.

Original entry on oeis.org

0, 3, 13, 22, 42, 57, 87, 108, 148, 175, 225, 258, 318, 357, 427, 472, 552, 603, 693, 750, 850, 913, 1023, 1092, 1212, 1287, 1417, 1498, 1638, 1725, 1875, 1968, 2128, 2227, 2397, 2502, 2682, 2793, 2983, 3100, 3300, 3423, 3633, 3762, 3982, 4117, 4347, 4488, 4728, 4875

Offset: 1

Bruno Berselli, Feb 26 2018

Equivalently, numbers k such that 32*k + 25 is a square. This means that 4*a(n) + 3 is a triangular number.

Interleaving of A139277 and A139272 (without 0).

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. A139272, A139277.
Subsequence of A011861, A047222.
Cf. numbers of the form m*(8*m + h): A154260 (h=1), A014494 (h=2), A274681 (h=3), A046092 (h=4), this sequence (h=5), 2*A074377 (h=6), A274979 (h=7).

List([1..50], n -> (8*n*(n-1)-(2*n-1)*(-1)^n-1)/4);

[div((8n*(n-1)-(2n-1)*(-1)^n-1), 4) for n in 1:50] # Peter Luschny, Feb 27 2018

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

seq((exp(I*Pi*x)*(1-2*x)+8*(x-1)*x-1)/4, x=1..50); # Peter Luschny, Feb 27 2018

Table[(8 n (n - 1) - (2 n - 1) (-1)^n - 1)/4, {n, 1, 50}]

makelist((8*n*(n-1)-(2*n-1)*(-1)^n-1)/4, n, 1, 50);

vector(50, n, nn; (8*n*(n-1)-(2*n-1)*(-1)^n-1)/4)

concat(0, Vec(x^2*(3 + 10*x + 3*x^2)/((1 - x)^3*(1 + x)^2) + O(x^60))) \\ Colin Barker, Feb 27 2018

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

def A299645(n): return (n>>1)*((n<<2)+(1 if n&1 else -5)) # Chai Wah Wu, Mar 11 2025

[(8*n*(n-1)-(2*n-1)*(-1)^n-1)/4 for n in (1..50)]

O.g.f.: x^2*(3 + 10*x + 3*x^2)/((1 - x)^3*(1 + x)^2).
E.g.f.: (1 + 2*x - (1 - 8*x^2)*exp(2*x))*exp(-x)/4.
a(n) = a(-n+1) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = (8*n*(n - 1) - (2*n - 1)*(-1)^n - 1)/4 = (2*n + (-1)^n - 1)*(4*n - 3*(-1)^n - 2)/4. Therefore, 3 and 13 are the only prime numbers in this sequence.
a(n) + a(n+1) = 4*n^2 for even n, otherwise a(n) + a(n+1) = 4*n^2 - 1.
From Amiram Eldar, Mar 18 2022: (Start)
Sum_{n>=2} 1/a(n) = 8/25 + (sqrt(2)-1)*Pi/5.
Sum_{n>=2} (-1)^n/a(n) = 8*log(2)/5 - sqrt(2)*log(2*sqrt(2)+3)/5 - 8/25. (End)
a(n) = (n-1)*(4*n+1)/2 if n is odd and a(n) = n*(4*n-5)/2 if n is even. - Chai Wah Wu, Mar 11 2025

A152750 Eight times hexagonal numbers: a(n) = 8n(2*n-1).

Original entry on oeis.org

0, 8, 48, 120, 224, 360, 528, 728, 960, 1224, 1520, 1848, 2208, 2600, 3024, 3480, 3968, 4488, 5040, 5624, 6240, 6888, 7568, 8280, 9024, 9800, 10608, 11448, 12320, 13224, 14160, 15128, 16128, 17160, 18224, 19320, 20448, 21608, 22800, 24024, 25280, 26568, 27888, 29240

Offset: 0

Omar E. Pol, Dec 12 2008

Equals Engel expansion of cosh(1/2), except first member (see A067239).
Also sequence found by reading the line from 0, in the direction 0, 8, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Sep 18 2011
a(n) = the sum of the edges of a rectangular prism having edges 2*(n-1)*n, n^2 - (n-1)^2, and n^2 + (n-1)^2. - J. M. Bergot, Apr 24 2014

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

Cf. A000384, A002939, A067239, A074377, A085250.

[ 8*n*(2*n-1) : n in [0..50] ]; // Wesley Ivan Hurt, Jun 09 2014

A152750:=n->8*n*(2*n-1); seq(A152750(n), n=0..50); # Wesley Ivan Hurt, Jun 09 2014

Table[8*n*(2*n - 1), {n, 0, 50}] (* Wesley Ivan Hurt, Jun 09 2014 *)

concat(0, Vec(8*x*(1+3*x)/(1-x)^3 + O(x^50))) \\ Colin Barker, Sep 25 2016

a(n) = 16*n^2 - 8*n = 8*A000384(n) = 4*A002939(n) = 2*A085250(n).
a(n) = A067239(n), for n > 0.
a(n) = a(n-1) + 32*n - 24 (with a(0)=0). - Vincenzo Librandi, Nov 26 2010
From Colin Barker, Sep 25 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
G.f.: 8*x*(1+3*x)/(1-x)^3. (End)
Sum_{n>=1} 1/a(n) = log(2)/4. - Vaclav Kotesovec, Sep 25 2016
E.g.f.: 8*exp(x)*x*(1 + 2*x). - Elmo R. Oliveira, Dec 15 2024
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/16 - log(2)/8. - Amiram Eldar, May 05 2025

A157474 a(n) = 16n^2 + n.

Original entry on oeis.org

17, 66, 147, 260, 405, 582, 791, 1032, 1305, 1610, 1947, 2316, 2717, 3150, 3615, 4112, 4641, 5202, 5795, 6420, 7077, 7766, 8487, 9240, 10025, 10842, 11691, 12572, 13485, 14430, 15407, 16416, 17457, 18530, 19635, 20772, 21941, 23142, 24375, 25640

Offset: 1

Vincenzo Librandi, Mar 01 2009

The identity (2048*n^2+128*n+1)^2 - (16*n^2+n)*(512*n+16)^2 = 1 can be written as A157476(n)^2 - a(n)*A157475(n)^2 = 1 (see also second comment in A157476).

Sequence found by reading the line from 17, in the direction 17, 66,... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A157475, A157476.

[16*n^2 + n: n in [1..40]]; // Vincenzo Librandi, Jan 01 2015

Table[16n^2+n,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{17,66,147},50] (* Harvey P. Dale, Nov 08 2011 *)
CoefficientList[Series[(17 + 14 x + 3 x^2 - 3 x^3 + x^4) / (1-x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Jan 01 2015 *)

a(n)=16*n^2+n \\ Charles R Greathouse IV, Feb 09 2012

a(n) = A173511(2*n). - Reinhard Zumkeller, Feb 20 2010
a(1)=17, a(2)=66, a(3)=147, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Nov 08 2011
G.f.: x*(17 + 14*x + 3*x^2 - 3*x^3 + x^4)/(1-x)^3. - Vincenzo Librandi, Jan 01 2015

Comment rewritten by Bruno Berselli, Aug 22 2011

A274603 Numbers n such that 2n+1 and 3n+1 are both triangular numbers.

Original entry on oeis.org

45, 4455, 436590, 42781410, 4192141635, 410787098865, 40252943547180, 3944377680524820, 386508759747885225, 37873914077612227275, 3711257070846250387770, 363665319028854925774230, 35635490007756936475486815, 3491914355441150919671933685

Offset: 1

Altug Alkan, Jun 30 2016

Inspired by A274579.
a(n+1) / a(n) goes to 49 + 20*sqrt(6) when n goes to infinity.
Intersection of A045943 and A074377. - Colin Barker, Jun 30 2016

			45 is a term because 2*45 + 1 = 91 and 3*45 + 1 = 136 are both triangular numbers.

Colin Barker, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

Cf. A000217, A045943, A074377, A274579.

isok(n) = ispolygonal(2*n+1, 3) && ispolygonal(3*n+1, 3);

Vec(45*x/((1-x)*(1-98*x+x^2)) + O(x^20)) \\ Colin Barker, Jun 30 2016

From Colin Barker, Jun 30 2016: (Start)
a(n) = 99*a(n-1)-99*a(n-2)+a(n-3) for n>3.
G.f.: 45*x / ((1-x)*(1-98*x+x^2)).
(End)

More terms from Colin Barker, Jun 30 2016

A274680 Values of n such that 2n+1 and 4n+1 are both triangular numbers.

Original entry on oeis.org

0, 16065, 545751, 21394547226, 726784809030, 28491418065071115, 967869505172593485, 37942420317086720855700, 1288925370210688376036076, 50528452330120333959563160501, 1716479960463788790499334882595, 67289447366315927998308608003134830

Offset: 1

Colin Barker, Jul 02 2016

			16065 is in the sequence because 2*16065+1 = 32131, 4*16065+1 = 64261, and 32131 and 64261 are both triangular numbers.

Colin Barker, Table of n, a(n) for n = 1..325
Index entries for linear recurrences with constant coefficients, signature (1,1331714,-1331714,-1,1).

Cf. A124174 (2*n+1 and 9*n+1), A274579 (2*n+1 and 5*n+1), A274603 (2*n+1 and 3*n+1).

Rest@ CoefficientList[Series[459 x^2 (35 + 1154 x + 35 x^2)/((1 - x) (1 - 1154 x + x^2) (1 + 1154 x + x^2)), {x, 0, 12}], x] (* Michael De Vlieger, Jul 02 2016 *)

isok(n) = ispolygonal(2*n+1, 3) && ispolygonal(4*n+1, 3)

concat(0, Vec(459*x^2*(35+1154*x+35*x^2)/((1-x)*(1-1154*x+x^2)*(1+1154*x+x^2)) + O(x^20)))

Intersection of A074377 and A274681.

G.f.: 459*x^2*(35+1154*x+35*x^2) / ((1-x)*(1-1154*x+x^2)*(1+1154*x+x^2)).

A274681 Numbers k such that 4*k + 1 is a triangular number.

Original entry on oeis.org

0, 5, 11, 26, 38, 63, 81, 116, 140, 185, 215, 270, 306, 371, 413, 488, 536, 621, 675, 770, 830, 935, 1001, 1116, 1188, 1313, 1391, 1526, 1610, 1755, 1845, 2000, 2096, 2261, 2363, 2538, 2646, 2831, 2945, 3140, 3260, 3465, 3591, 3806, 3938, 4163, 4301, 4536

Offset: 1

Colin Barker, Jul 02 2016

Also, numbers of the form m*(8*m + 3) for m = 0, -1, 1, -2, 2, -3, 3, ... - Bruno Berselli, Feb 26 2018

			5 is in the sequence since 4*5 + 1 = 21 is a triangular number (21 = 1 + 2 + 3 + 4 + 5 + 6). - _Michael B. Porter_, Jul 03 2016

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. A000217, A000096 (n+1), A074377 (2*n+1), A045943 (3*n+1), A085787 (5*n+1).
Cf. A057029.
Cf. similar sequences listed in A299645.

[(1-(-1)^n+2*(-4+(-1)^n)*n+8*n^2)/4: n in [1..80]]; // Wesley Ivan Hurt, Jul 02 2016

A274681:=n->(1-(-1)^n+2*(-4+(-1)^n)*n+8*n^2)/4: seq(A274681(n), n=1..100); # Wesley Ivan Hurt, Jul 02 2016

Rest@ CoefficientList[Series[x^2 (5 + 6 x + 5 x^2)/((1 - x)^3 (1 + x)^2), {x, 0, 48}], x] (* Michael De Vlieger, Jul 02 2016 *)
Select[Range[0,5000],OddQ[Sqrt[8(4#+1)+1]]&] (* or *) LinearRecurrence[ {1,2,-2,-1,1},{0,5,11,26,38},50] (* Harvey P. Dale, Apr 21 2018 *)

PARI
```
isok(n) = ispolygonal(4*n+1, 3)
```

select(n->ispolygonal(4*n+1, 3), vector(10000, n, n-1))

concat(0, Vec(x^2*(5+6*x+5*x^2)/((1-x)^3*(1+x)^2) + O(x^100)))

def A274681(n): return (n>>1)*((n<<2)+(-1 if n&1 else -3)) # Chai Wah Wu, Mar 11 2025

G.f.: x^2*(5 + 6*x + 5*x^2) / ((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>5.
a(n) = A057029(n) - 1.
a(n) = (1 - (-1)^n + 2*(-4 + (-1)^n)*n + 8*n^2)/4.
a(n) = (4*n^2 - 3*n)/2 for n even, a(n) = (4*n^2 - 5*n + 1)/2 for n odd.

cp's OEIS Frontend

A214263 Expansion of f(x^1, x^7) in powers of x where f() is Ramanujan's general theta function.

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A195315 Centered 32-gonal numbers.

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A181995 a(n) = if n mod 2 = 1 then n*(n - 1) else (n - 1)^2 + (n - 2)/2.

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A274579 Values of k such that 2*k+1 and 5*k+1 are both triangular numbers.

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A299645 Numbers of the form m*(8*m + 5), where m is an integer.

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A152750 Eight times hexagonal numbers: a(n) = 8*n*(2*n-1).

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A274579 Values of k such that 2k+1 and 5k+1 are both triangular numbers.

A299645 Numbers of the form m(8m + 5), where m is an integer.

A152750 Eight times hexagonal numbers: a(n) = 8n(2*n-1).

A274603 Numbers n such that 2n+1 and 3n+1 are both triangular numbers.

A274680 Values of n such that 2n+1 and 4n+1 are both triangular numbers.