A014493 -id:A014493 - cp's OEIS frontend

A352116 Partial sums of the odd triangular numbers (A014493).

1, 4, 19, 40, 85, 140, 231, 336, 489, 660, 891, 1144, 1469, 1820, 2255, 2720, 3281, 3876, 4579, 5320, 6181, 7084, 8119, 9200, 10425, 11700, 13131, 14616, 16269, 17980, 19871, 21824, 23969, 26180, 28595, 31080, 33781, 36556, 39559, 42640, 45961, 49364, 53019, 56760, 60765

Offset: 1

David James Sycamore, Mar 05 2022

			a(1) = 1 because 1 is the first odd term in A000027.
a(2) = 1 + 3 = 4, the sum of the first two odd terms in A000027, and so on.

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A001477, A000217, A000292, A014493, A014494, A352115, A000447.

LinearRecurrence[{2, 1, -4, 1, 2, -1}, {1, 4, 19, 40, 85, 140}, 50] (* Amiram Eldar, Mar 05 2022 *)

to(n) = (2*n-1)*(2*n-1-(-1)^n)/2; \\ A014493
a(n) = sum(k=1, n, to(k)); \\ Michel Marcus, Mar 05 2022

def A352116(n): return n*((n-1)<<1)*(n+1)//3 + n*(n&1) # Chai Wah Wu, Feb 12 2023

a(n) = Sum_{k=1..n} A014493(k) = Sum_{k=1..n} (2*k-1)(2*k-1-(-1)^k)/2.
a(n) = A352115(n-1) + (-1)^(n-1)*n.
a(n) = A000447(n) - A352115(n-1).
From Stefano Spezia, Mar 05 2022: (Start)
a(n) = n*(4*n^2 - 1 - 3*(-1)^n)/6.
G.f.: x*(1 + 2*x + 10*x^2 + 2*x^3 + x^4)/((1 - x)^4*(1 + x)^2). (End)

More terms from Michel Marcus, Mar 05 2022

A074377 Generalized 10-gonal numbers: m(4m - 3) for m = 0, +- 1, +- 2, +- 3, ...

Original entry on oeis.org

0, 1, 7, 10, 22, 27, 45, 52, 76, 85, 115, 126, 162, 175, 217, 232, 280, 297, 351, 370, 430, 451, 517, 540, 612, 637, 715, 742, 826, 855, 945, 976, 1072, 1105, 1207, 1242, 1350, 1387, 1501, 1540, 1660, 1701, 1827, 1870, 2002, 2047, 2185, 2232, 2376, 2425

Offset: 0

W. Neville Holmes, Sep 04 2002

Also called generalized decagonal numbers.
Odd triangular numbers decremented and halved.
It appears that this is zero together with the partial sums of A165998. - Omar E. Pol, Sep 10 2011 [this is correct, see the g.f., Joerg Arndt, Sep 29 2013]
Also, A033954 and positive members of A001107 interleaved. - Omar E. Pol, Aug 04 2012
Also, numbers m such that 16*m+9 is a square. After 1, therefore, there are no squares in this sequence. - Bruno Berselli, Jan 07 2016
Convolution of the sequences A047522 and A059841. - Ilya Gutkovskiy, Mar 16 2017
Numbers k such that the concatenation k5625 is a square. - Bruno Berselli, Nov 07 2018
Exponents in expansion of Product_{n >= 1} (1 + x^(8*n-7))*(1 + x^(8*n-1))*(1 - x^(8*n)) = 1 + x + x^7 + x^10 + x^22 + .... - Peter Bala, Dec 10 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Neville Holmes, More Gemometric Integer Sequences. [Wayback Machine copy]
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A011848, A014493, A074378, A118277, A165998.
Cf. A001107 (10-gonal numbers).
Column 6 of A195152.
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), this sequence (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).
Cf. sequences of the form m*(m+k)/(k+1) listed in A274978. [Bruno Berselli, Jul 25 2016]

[n^2+n-1/4+(-1)^n/4+n*(-1)^n/2: n in [0..50]]; // Vincenzo Librandi, Sep 29 2013

CoefficientList[Series[x(1 +6x +x^2)/((1-x)(1-x^2)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Sep 29 2013 *)
LinearRecurrence[{1,2,-2,-1,1}, {0,1,7,10,22}, 50] (* G. C. Greubel, Nov 07 2018 *)

PARI
```
a(n)=(2*n+3-4*(n%2))*(n-n\2)
```

concat([0],Vec(x*(1 + 6*x + x^2)/((1 - x)*(1 - x^2)^2) +O(x^50))) \\ Indranil Ghosh, Mar 16 2017

def A074377(n): return (n+1>>1)*((n<<1)+(-1 if n&1 else 3)) # Chai Wah Wu, Mar 11 2025

(n(n+1)-2)/4 where n(n+1)/2 is odd.
G.f.: x*(1+6*x+x^2)/((1-x)*(1-x^2)^2). - Michael Somos, Mar 04 2003
a(2*k) = k*(4*k+3); a(2*k+1) = (2*k+1)^2+k. - Benoit Jubin, Feb 05 2009
a(n) = n^2+n-1/4+(-1)^n/4+n*(-1)^n/2. - R. J. Mathar, Oct 08 2011
Sum_{n>=1} 1/a(n) = (4 + 3*Pi)/9. - Vaclav Kotesovec, Oct 05 2016
E.g.f.: exp(x)*x^2 + (2*exp(x) - exp(-x)/2)*x - sinh(x)/2. - Ilya Gutkovskiy, Mar 16 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2) - 4/9. - Amiram Eldar, Feb 28 2022
a(n) = (n+1)*(2*n-1)/2 if n is odd and a(n) = n*(2*n+3)/2 if n is even. - Chai Wah Wu, Mar 11 2025

New name from T. D. Noe, Apr 21 2006

Formula in sequence name from Omar E. Pol, May 28 2012

A042963 Numbers congruent to 1 or 2 mod 4.

Original entry on oeis.org

1, 2, 5, 6, 9, 10, 13, 14, 17, 18, 21, 22, 25, 26, 29, 30, 33, 34, 37, 38, 41, 42, 45, 46, 49, 50, 53, 54, 57, 58, 61, 62, 65, 66, 69, 70, 73, 74, 77, 78, 81, 82, 85, 86, 89, 90, 93, 94, 97, 98, 101, 102, 105, 106, 109, 110, 113, 114, 117, 118, 121, 122, 125, 126, 129, 130, 133, 134, 137, 138

Offset: 1

N. J. A. Sloane

Complement of A014601. - Reinhard Zumkeller, Oct 04 2004
Let S(x) = (1, 2, 2, 2, ...). Then A042963 = ((S(x))^2 + S(x^2))/2 = ((1, 4, 8, 12, 16, 20, ...) + (1, 0, 2, 0, 2, 0, 2, ...))/2 = (1, 2, 5, 6, 9, 10, ...). - Gary W. Adamson, Jan 03 2011
(a(n)*(a(n) + 1 + 4*k))/2 is odd, for k >= 0. - Gionata Neri, Jul 19 2015
Equivalent to the following variation on Fermat's Diophantine m-tuple: 1 + the product of any two distinct terms is not a square; this sequence, which we'll call sequence S, is produced by the following algorithm. At the start, S is initially empty. At stage n, starting at n = 1, the algorithm checks whether there exists a number m already in the sequence, such that mn+1 is a perfect square. If such a number m is found, then n is not added to the sequence; otherwise, n is added. Then n is incremented to n + 1, and we repeat the procedure. Proof by Clark R. Lyons: We prove by strong induction that n is in the sequence S if and only if n == 1 (mod 4) or n == 2 (mod 4). Suppose now that this holds for all k < n. In case 1, either n == 1 (mod 4) or n == 2 (mod 4), and we wish to show that n does indeed enter the sequence S. That is, we wish to show that there does not exist m < n, already in the sequence at this point such that mn+1 is a square. By the inductive hypothesis m == 1 (mod 4) or m == 2 (mod 4). This means that both m and n are one of 1, 2, 5, or 6 mod 8. Using a multiplication table mod 8, we see that this implies mn+1 is congruent to one of 2, 3, 5, 6, or 7 mod 8. But we also see that mod 8, a perfect square is congruent to 0, 1, or 4. Thus mn+1 is not a perfect square, so n is added to the sequence. In case 2, n == 0 (mod 4) or n == 3 (mod 4), and we wish to show that n is not added to the sequence. That is, we wish to show that there exists m < n already in the sequence such that mn+1 is a perfect square. For this we let m = n - 2, which is positive since n >= 3. By the inductive hypothesis, since m == 1 (mod 4) or m == 2 (mod 4) and m < n, m is already in the sequence. And we have m*n + 1 = (n - 2)*n + 1 = n^2 - 2*n + 1 = (n - 1)^2, so mn+1 is indeed a perfect square, and so n is not added to the sequence. Thus n is added to the sequence if and only if n == 1 (mod 4) or n == 2 (mod 4). This completes the proof. - Robert C. Lyons, Jun 30 2016
Also the number of maximal cliques in the (n + 1) X (n + 1) black bishop graph. - Eric W. Weisstein, Dec 01 2017
Lexicographically earliest sequence of distinct positive integers such that the average of any two or more consecutive terms is never an integer. (For opposite property see A005408.) - Ivan Neretin, Dec 21 2017
Numbers whose binary reflected Gray code (A014550) ends with 1. - Amiram Eldar, May 17 2021
Also: append its negated last bit to n-1. - M. F. Hasler, Oct 17 2022

David Lovler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Reinhard Zumkeller)
Eric Weisstein's World of Mathematics, Black Bishop Graph.
Eric Weisstein's World of Mathematics, Maximal Clique.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A153284 (first differences), A014848 (partial sums).
Cf. A014550, A046712 (subsequence).
Union of A016813 and A016825.

a042963 n = a042963_list !! (n-1)
a042963_list = [x | x <- [0..], mod x 4 `elem` [1,2]]
-- Reinhard Zumkeller, Feb 14 2012

[ n : n in [1..165] | n mod 4 eq 1 or n mod 4 eq 2 ]; // Vincenzo Librandi, Jan 25 2011

A046923:=n->(n mod 2) + 2n - 2; seq(A046923(n), n=1..100); # Wesley Ivan Hurt, Oct 10 2013

Select[Range[109], Or[Mod[#, 4] == 1, Mod[#, 4] == 2] &] (* Ant King, Nov 17 2010 *)
Table[(4 n - 3 - (-1)^n)/2, {n, 20}] (* Eric W. Weisstein, Dec 01 2017 *)
LinearRecurrence[{1, 1, -1}, {1, 2, 5}, 20] (* Eric W. Weisstein, Dec 01 2017 *)
CoefficientList[Series[(1 + x + 2 x^2)/((-1 + x)^2 (1 + x)), {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017 *)

a(n)=2*n-1-(n-1)%2 \\ Jianing Song, Oct 06 2018; adapted to offset by Michel Marcus, Sep 09 2022

apply( A042963(n)=n*2-2+n%2, [1..99]) \\ M. F. Hasler, Oct 17 2022

a(n) = 1 + A042948(n-1). [Corrected by Jianing Song, Oct 06 2018]
From Michael Somos, Jan 12 2000: (Start)
G.f.: x*(1 + x + 2*x^2)/((1 - x)^2*(1 + x)).
a(n) = a(n-1) + 2 + (-1)^n, a(0) = 1. (End) [This uses offset 0. - Jianing Song, Oct 06 2018]
A014493(n) = A000217(a(n)). - Reinhard Zumkeller, Oct 04 2004, Feb 14 2012
a(n) = Sum_{k=0..n} (A001045(k) mod 4). - Paul Barry, Mar 12 2004
A145768(a(n)) is odd. - Reinhard Zumkeller, Jun 05 2012
a(n) = A005843(n-1) + A059841(n-1). - Philippe Deléham, Mar 31 2009 [Corrected by Jianing Song, Oct 06 2018]
a(n) = 4*n - a(n-1) - 5 for n > 1. [Corrected by Jerzy R Borysowicz, Jun 09 2023]
From Ant King, Nov 17 2010: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3).
a(n) = (4*n - 3 - (-1)^n)/2. (End)
a(n) = (n mod 2) + 2*n - 2. - Wesley Ivan Hurt, Oct 10 2013
A163575(a(n)) = n - 1. - Reinhard Zumkeller, Jul 22 2014
E.g.f.: 2 + (2*x - 1)*sinh(x) + 2*(x - 1)*cosh(x). - Ilya Gutkovskiy, Jun 30 2016
E.g.f.: 2 + (2*x - 1)*exp(x) - cosh(x). - David Lovler, Jul 19 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/8 + log(2)/4. - Amiram Eldar, Dec 05 2021

Offset corrected by Reinhard Zumkeller, Feb 14 2012

More terms by David Lovler, Jul 19 2022

A074378 Even triangular numbers halved.

Original entry on oeis.org

0, 3, 5, 14, 18, 33, 39, 60, 68, 95, 105, 138, 150, 189, 203, 248, 264, 315, 333, 390, 410, 473, 495, 564, 588, 663, 689, 770, 798, 885, 915, 1008, 1040, 1139, 1173, 1278, 1314, 1425, 1463, 1580, 1620, 1743, 1785, 1914, 1958, 2093, 2139, 2280, 2328, 2475

Offset: 0

W. Neville Holmes, Sep 04 2002

Set of integers k such that k + (1 + 2 + 3 + 4 + ... + x) = 3*k, where x is sufficiently large. For example, 203 is a term because 203 + (1 + 2 + 3 + 4 + ... +28) = 609 and 609 = 3*203. - Gil Broussard, Sep 01 2008
Set of all m such that 16*m+1 is a perfect square. - Gary Detlefs, Feb 21 2010
Integers of the form Sum_{k=0..n} k/2. - Arkadiusz Wesolowski, Feb 07 2012
Numbers of the form h*(4*h + 1) for h = 0, -1, 1, -2, 2, -3, 3, ... - Bruno Berselli, Feb 26 2018
Numbers whose distance to nearest square equals their distance to nearest oblong; that is, numbers k such that A053188(k) = A053615(k). - Lamine Ngom, Oct 27 2020
The sequence terms are the exponents in the expansion of Product_{n >= 1} (1 - q^(8*n))*(1 + q^(8*n-3))*(1 + q^(8*n-5)) = 1 + q^3 + q^5 + q^14 + q^18 + .... - Peter Bala, Dec 30 2024

David A. Corneth, Table of n, a(n) for n = 0..9999
Neville Holmes, More Geometric Integer Sequences
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A011848, A014493, A014494, A074377, A035294, A274757.
Cf. A010709, A047522. [Vincenzo Librandi, Feb 14 2009]
Cf. A266883 (numbers n such that 16*n-15 is a square).
Cf. A016687, A153799.
Cf. A053615, A053188.

f:=func; [0] cat [f(n*m): m in [-1,1], n in [1..25]]; // Bruno Berselli, Nov 13 2012

Maple

a:=n->(2*n+1)*floor((n+1)/2): seq(a(n),n=0..50); # Muniru A Asiru, Feb 01 2019

Mathematica

1/2 * Select[PolygonalNumber@ Range[0, 100], EvenQ] (* Michael De Vlieger, Jun 01 2017, Version 10.4 *) Select[Accumulate[Range[0,100]],EvenQ]/2 (* Harvey P. Dale, Feb 15 2025 *)

PARI

a(n)=(2*n+1)*(n-n\2)

Formula

Sum_{n>=0} q^a(n) = (Prod_{n>0} (1-q^n))*(Sum_{n>=0} A035294(n)*q^n).

a(n) = n*(n + 1)/4 where n*(n + 1)/2 is even.

G.f.: x*(3 + 2*x + 3*x^2)/((1 - x)*(1 - x^2)^2).

From Benoit Jubin, Feb 05 2009: (Start)

a(n) = (2*n + 1)*floor((n + 1)/2).

a(2*k) = k*(4*k+1); a(2*k+1) = (k+1)*(4*k+3). (End)

a(2*n) = A007742(n), a(2*n-1) = A033991(n). - Arkadiusz Wesolowski, Jul 20 2012

a(n) = (4*n + 1 - (-1)^n)*(4*n + 3 - (-1)^n)/4^2. - Peter Bala, Jan 21 2019

a(n) = (2*n+1)*(n+1)*(1+(-1)^(n+1))/4 + (2*n+1)*(n)*(1+(-1)^n)/4. - Eric Simon Jacob, Jan 16 2020

From Amiram Eldar, Jul 03 2020: (Start)

Sum_{n>=1} 1/a(n) = 4 - Pi (A153799).

Sum_{n>=1} (-1)^(n+1)/a(n) = 6*log(2) - 4 (See A016687). (End)

a(n) = A014494(n)/2 = A274757(n)/3 = A266883(n) - 1. - Hugo Pfoertner, Dec 31 2024

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

A067589 Numbers k such that A067588(k) is an odd number.

Original entry on oeis.org

1, 5, 7, 15, 35, 51, 57, 77, 117, 145, 155, 187, 247, 287, 301, 345, 425, 477, 495, 551, 651, 715, 737, 805, 925, 1001, 1027, 1107, 1247, 1335, 1365, 1457, 1617, 1717, 1751, 1855, 2035, 2147, 2185, 2301, 2501, 2625, 2667, 2795, 3015, 3151, 3197, 3337

Offset: 1

Naohiro Nomoto, Jan 31 2002

The terms are exactly the odd pentagonal numbers; that is, they are all the odd numbers of the form k*(3*k-1)/2 where k is an integer. - James Sellers, Jun 09 2007
Apparently groups of two odd pentagonal numbers (A000326, A014632) followed by two odd 2nd pentagonal numbers (A005449), which leads to the conjectured generating function x*(x^2+4*x+1)*(x^4-2*x^3+4*x^2-2*x+1)/((x^2+1)^2*(1-x)^3). - R. J. Mathar, Jul 26 2009
Odd generalized pentagonal numbers. - Omar E. Pol, Aug 19 2011
From Peter Bala, Jan 10 2025: (Start)
The sequence terms are the exponents in the expansion of Sum_{n >= 0} x^(2*n+1)/(Product_{k = 1..2*n+1} 1 + x^(2*k+1)) = x + x^5 - x^7 - x^15 + x^35 + x^51 - x^57 - x^77 + + - - ... (follows from Berndt et al., Theorem 3.3). Cf. A193828.
For positive integer m, define b_m(n) = Sum_{k = 1..n} k^(2*m+1)*A000009(k)*A000009(n-k). We conjecture that
i) for odd n, b(n)/ n is an integer
ii) b(2*n)/n is an integer, which is odd iff n is a member of this sequence.
Cf. A067567. (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
B. C. Berndt, B. Kim, and A. J. Yee, Ramanujan's lost notebook: Combinatorial proofs of identities associated with Heine's transformation or partial theta functions, J. Comb. Thy. Ser. A, 117 (2010), 957-973.

Cf. A067567, A067588, A000009.
Cf. A001318, A193828.
Cf. A014493, A128880, A154293.

With[{nn=50},Sort[Select[Table[(n(3n-1))/2,{n,-nn,nn}],OddQ]]] (* Harvey P. Dale, Feb 16 2014 *)

Sum_{n>=1} 1/a(n) = Pi/2. - Amiram Eldar, Aug 18 2022

Corrected by T. D. Noe, Oct 25 2006

A167683 Hankel transform of A007325.

Original entry on oeis.org

1, 0, -1, 1, 0, 0, -1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 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, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0

Offset: 0

Paul Barry, Nov 09 2009

sign

Nonzero elements appear to be indexed by abs(A131723). -1 elements appear to be indexed by 2*A014493(n).

A193868 Even central polygonal numbers.

Original entry on oeis.org

2, 4, 16, 22, 46, 56, 92, 106, 154, 172, 232, 254, 326, 352, 436, 466, 562, 596, 704, 742, 862, 904, 1036, 1082, 1226, 1276, 1432, 1486, 1654, 1712, 1892, 1954, 2146, 2212, 2416, 2486, 2702, 2776, 3004, 3082, 3322, 3404, 3656, 3742, 4006, 4096, 4372

Offset: 1

Omar E. Pol, Aug 15 2011

Odd triangular numbers plus 1.

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

Cf. A000124, A014493, A174114, A193867.

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

Table[(3 + (-1)^n - 2 (2 + (-1)^n) n + 4 n^2)/2, {n, 50}] (* or *)
Select[PolygonalNumber@ Range@ 100, OddQ] + 1 (* Version 10.4, or *)
Table[If[EvenQ@ n, 2 n^2 - 3 n + 2, 2 n^2 - n + 1], {n, 50}] (* or *)
Rest@ CoefficientList[Series[-2 x (1 + x + 4 x^2 + x^3 + x^4)/((1 + x)^2 (x - 1)^3), {x, 0, 50}], x] (* Michael De Vlieger, Jun 30 2016 *)
LinearRecurrence[{1,2,-2,-1,1},{2,4,16,22,46},50] (* Harvey P. Dale, Sep 13 2020 *)

a(n)=(2*n-1)*(2*n-1-(-1)^n)/2+1 \\ Charles R Greathouse IV, Jun 11 2015

Vec(2*x*(1+x+4*x^2+x^3+x^4)/((1-x)^3*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 27 2016

a(n) = A000124(A042963(n-1)).
a(n) = 1 + A014493(n).
a(n) = 2*A174114(n).
G.f.: -2*x*(1+x+4*x^2+x^3+x^4) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Aug 25 2011
From Colin Barker, Jan 27 2016: (Start)
a(n) = (3+(-1)^n-2*(2+(-1)^n)*n+4*n^2)/2.
a(n) = 2*n^2-3*n+2 for n even.
a(n) = 2*n^2-n+1 for n odd. (End)
Sum_{n>=1} 1/a(n) = 2*Pi*sinh(sqrt(7)*Pi/4)/(sqrt(7)*(sqrt(2) + 2*cosh(sqrt(7)*Pi/4))). - Amiram Eldar, May 11 2025

A034954 Odd triangular numbers with prime indices.

Original entry on oeis.org

3, 15, 91, 153, 435, 703, 861, 1431, 1891, 2701, 4005, 4753, 5151, 5995, 6441, 9453, 11175, 12403, 15051, 16471, 18721, 19503, 26335, 27261, 29161, 33153, 36315, 38503, 39621, 43071, 49141, 50403, 56953, 61075, 62481, 69751, 75855, 79003

Offset: 1

Patrick De Geest, Oct 15 1998

nonn

For prime indices p such that p(p+1)/2 is 'odd' see A002313 (primes of form 4n+1).

G. C. Greubel, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Triangular Number

Cf. A000217, A002313.

Cf. A014493, A034953, A034955.

Select[Table[Prime[n] (Prime[n] + 1)/2, {n, 500}], OddQ[#] &] (* G. C. Greubel, Nov 03 2017 *)
With[{nn=400},Select[Thread[{Accumulate[Range[nn]],Range[nn]}],OddQ[ #[[1]]] && PrimeQ[#[[2]]]&]][[All,1]] (* Harvey P. Dale, Mar 06 2019 *)

a(n) = A000217(A002313(n)). - Michel Marcus, Nov 04 2017

A034955 Even triangular numbers with prime indices.

Original entry on oeis.org

6, 28, 66, 190, 276, 496, 946, 1128, 1770, 2278, 2556, 3160, 3486, 5356, 5778, 8128, 8646, 9730, 11476, 13366, 14028, 16110, 18336, 19900, 22366, 24976, 25878, 28680, 31626, 34716, 36856, 40186, 47278, 48516, 54946, 60378, 64620, 67528, 72010, 73536, 87990

Offset: 1

Patrick De Geest, Oct 15 1998

nonn

For prime indices p such that p(p+1)/2 is even, see A002145 (Primes of form 4n+3).

G. C. Greubel, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Triangular Number

Cf. A000217, A002145.

Cf. A014493, A034953, A034954.

Select[Table[Prime[n] (Prime[n] + 1)/2, {n, 500}], EvenQ[#] &] (* G. C. Greubel, Nov 03 2017 *)

from sympy import primerange
def aupto(lim):
    pitri = (p*(p+1)//2 for p in primerange(2, int((2*lim)**.5)+1))
    return [t for t in pitri if t%2 == 0]
print(aupto(90000)) # Michael S. Branicky, Jun 28 2021

a(n) = A000217(A002145(n)). - Michel Marcus, Nov 04 2017

cp's OEIS Frontend

A352116 Partial sums of the odd triangular numbers (A014493).

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A074377 Generalized 10-gonal numbers: m*(4*m - 3) for m = 0, +- 1, +- 2, +- 3, ...

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A042963 Numbers congruent to 1 or 2 mod 4.

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A074378 Even triangular numbers halved.

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A014494 Even triangular numbers.

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A067589 Numbers k such that A067588(k) is an odd number.

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A074377 Generalized 10-gonal numbers: m(4m - 3) for m = 0, +- 1, +- 2, +- 3, ...