A074377 -id:A074377 - cp's OEIS frontend

A143859 Ulam's spiral (WNW spoke).

1, 18, 67, 148, 261, 406, 583, 792, 1033, 1306, 1611, 1948, 2317, 2718, 3151, 3616, 4113, 4642, 5203, 5796, 6421, 7078, 7767, 8488, 9241, 10026, 10843, 11692, 12573, 13486, 14431, 15408, 16417, 17458, 18531, 19636, 20773, 21942, 23143, 24376

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 03 2008

Also sequence found by reading the segment (1, 18) together with the line from 18, in the direction 18, 67, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 05 2012

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

List([1..50], n-> ((32*n-31)^2+63)/64); # G. C. Greubel, Nov 09 2019

[16*n^2-31*n+16: n in [1..50]]; // Vincenzo Librandi, Nov 08 2014

seq( ((32*n-31)^2+63)/64, n=1..50); # G. C. Greubel, Nov 09 2019

f[n_]:= 16n^2 -31n +16; Array[f, 40] (* Vladimir Joseph Stephan Orlovsky, Sep 03 2008 *)
LinearRecurrence[{3,-3,1},{1,18,67},40] (* Harvey P. Dale, Mar 24 2012 *)
CoefficientList[Series[(1+15x+16x^2)/(1-x)^3, {x,0,40}], x] (* Vincenzo Librandi, Nov 08 2014 *)

vector(50, n, 16*n^2-31*n+16) \\ Michel Marcus, Nov 08 2014

[((32*n-31)^2+63)/64 for n in (1..50)] # G. C. Greubel, Nov 09 2019

a(n) = 16*n^2 - 31*n + 16. - R. J. Mathar, Sep 08 2008
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(1)=1, a(2)=18, a(3)=67. - Harvey P. Dale, Mar 24 2012
G.f.: x*(1 + 15*x + 16*x^2)/(1-x)^3. - Colin Barker, Aug 03 2012
E.g.f.: -16 + (16 - 15*x + 16*x^2)*exp(x). - G. C. Greubel, Nov 09 2019

A143860 Ulam's spiral (NNW spoke).

Original entry on oeis.org

1, 16, 63, 142, 253, 396, 571, 778, 1017, 1288, 1591, 1926, 2293, 2692, 3123, 3586, 4081, 4608, 5167, 5758, 6381, 7036, 7723, 8442, 9193, 9976, 10791, 11638, 12517, 13428, 14371, 15346, 16353, 17392, 18463, 19566, 20701, 21868, 23067, 24298

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 03 2008

Also, except for the first term, sequence found by reading the line from 16, in the direction 16, 63,... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 05 2012

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

List([1..40], n-> ((32*n-33)^2 +63)/64); # G. C. Greubel, Nov 09 2019

[16*n^2-33*n+18: n in [1..40]]; // Vincenzo Librandi, Jul 10 2012

seq( ((32*n-33)^2 +63)/64, n=1..40); # G. C. Greubel, Nov 09 2019

f[n_]:= 16n^2 -33n +18; Array[f, 40] (* Robert G. Wilson v, Oct 31 2011 *)
((32*Range[50]-33)^2 +63)/64 (* G. C. Greubel, Nov 09 2019 *)

a(n)=16*n^2-33*n+18 \\ Charles R Greathouse IV, Jun 17 2017

[((32*n-33)^2 +63)/64 for n in (1..40)] # G. C. Greubel, Nov 09 2019

a(n) = 16*n^2 - 33*n + 18. - R. J. Mathar, Sep 08 2008
G.f. x*(1 + 13*x + 18*x^2)/(1-x)^3. - R. J. Mathar, Oct 31 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jul 10 2012
E.g.f.: -18 + (18 - 17*x + 16*x^2)*exp(x). - G. C. Greubel, Nov 09 2019

A157446 a(n) = 16*n^2 - n.

Original entry on oeis.org

15, 62, 141, 252, 395, 570, 777, 1016, 1287, 1590, 1925, 2292, 2691, 3122, 3585, 4080, 4607, 5166, 5757, 6380, 7035, 7722, 8441, 9192, 9975, 10790, 11637, 12516, 13427, 14370, 15345, 16352, 17391, 18462, 19565, 20700, 21867, 23066, 24297, 25560

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 A157448(n)^2 - a(n)*A157447(n)^2 = 1. - Vincenzo Librandi, Jan 26 2012
This is the case s=4 of the identity (8*n^2*s^4 - 8*n*s^2 + 1)^2 - (n^2*s^2 - n)*(8*n*s^3 - 4*s)^2 = 1. - Bruno Berselli, Jan 26 2012
Sequence found by reading the line from 15, in the direction 15, 62, ... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012
The continued fraction expansion of sqrt(a(n)) is [4n-1; {1, 6, 1, 8n-2}]. For n=1, this collapses to [3; {1, 6}]. - Magus K. Chu, Sep 22 2022

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A157447, A157448.

I:=[15, 62, 141]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jan 26 2012

LinearRecurrence[{3,-3,1},{15,62,141},40] (* Vincenzo Librandi, Jan 26 2012 *)

for(n=1, 22, print1(16*n^2 - n", ")); \\ Vincenzo Librandi, Jan 26 2012

G.f.: x*(15 + 17*x)/(1-x)^3. - Vincenzo Librandi, Jan 26 2012

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jan 26 2012

A143861 Ulam's spiral (NNE spoke).

Original entry on oeis.org

1, 14, 59, 136, 245, 386, 559, 764, 1001, 1270, 1571, 1904, 2269, 2666, 3095, 3556, 4049, 4574, 5131, 5720, 6341, 6994, 7679, 8396, 9145, 9926, 10739, 11584, 12461, 13370, 14311, 15284, 16289, 17326, 18395, 19496, 20629, 21794, 22991, 24220

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 03 2008

Stanislaw M. Ulam was doodling during the presentation of a "long and very boring paper" at a scientific meeting in 1963. The spiral is its result. Note that conforming to trigonometric conventions, the spiral begins on the abscissa and rotates counterclockwise. Other spirals, orientations, direction of rotation and initial values exist, even in the OEIS.

Also sequence found by reading the segment (1, 14) together with the line from 14, in the direction 14, 59, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 05 2012

Chris K. Caldwell & G. L. Honaker, Jr., Prime Curios! The Dictionary of Prime Number Trivia, CreateSpace, Sept 2009, pp. 2-3.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Martin Gardner, Mathematical Recreations: The Remarkable Lore of the Prime Number, Scientific American 210 3: 120 - 128.
Hermetic Systems, Prime Number Spiral
OEIS wiki, Ulam spiral
Ivars Peterson's MathTrek, Prime Spirals, Science News, May 3 2002.
Robert Sacks, Number Spiral
Scientific American, Cover page of the March 1964
Eric Weisstein's World of Mathematics, Prime Spiral
Wikipedia, Ulam spiral
Wikipedia, Boxing the compass
Robert G. Wilson v, Ulam's spiral
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016754, A033638, A033951, A053755, A054552, A054554, A054556, A054567, A054569, A073337, A143838, A143839, A143854, A143855, A143856, A143859, A143860.

List([1..40], n-> ((32*n-35)^2 +55)/64); # G. C. Greubel, Nov 09 2019

[((32*n-35)^2 +55)/64: n in [1..40]]; // G. C. Greubel, Nov 09 2019

seq( ((32*n-35)^2 +55)/64, n=1..40); # G. C. Greubel, Nov 09 2019

(* From Robert G. Wilson v, Oct 29 2011 *)
f[n_]:= 16n^2 -35n +20; Array[f, 40]
LinearRecurrence[{3,-3,1}, {1,14,59}, 40]
FoldList[#1 + #2 &, 1, 32Range@ 10 - 19] (* End *)
((32*Range[40] -35)^2 +55)/64 (* G. C. Greubel, Nov 09 2019 *)

a(n)=16*n^2-35*n+20 \\ Charles R Greathouse IV, Oct 29 2011

[((32*n-35)^2 +55)/64 for n in (1..40)] # G. C. Greubel, Nov 09 2019

a(n) = 16*n^2 - 35*n + 20. - R. J. Mathar, Sep 08 2008
G.f.: x*(1 + 11*x + 20*x^2)/(1-x)^3. - Colin Barker, Aug 03 2012
E.g.f.: -20 + (20 - 19*x + 16*x^2)*exp(x). - G. C. Greubel, Nov 09 2019

A158443 a(n) = 16*n^2 - 4.

Original entry on oeis.org

12, 60, 140, 252, 396, 572, 780, 1020, 1292, 1596, 1932, 2300, 2700, 3132, 3596, 4092, 4620, 5180, 5772, 6396, 7052, 7740, 8460, 9212, 9996, 10812, 11660, 12540, 13452, 14396, 15372, 16380, 17420, 18492, 19596, 20732, 21900, 23100, 24332, 25596, 26892, 28220, 29580

Offset: 1

Vincenzo Librandi, Mar 19 2009

The identity (8*n^2 - 1)^2 - (16*n^2 - 4) *(2*n)^2 = 1 can be written as A157914(n)^2 - a(n)*A005843(n)^2 = 1.

Sequence found by reading the line from 12, in the direction 12, 60, ... 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
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000466, A005843, A074377, A157914.

I:=[12, 60, 140]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]];

16Range[60]^2-4  (* Harvey P. Dale, Mar 18 2011 *)

PARI
```
a(n) = 16*n^2 - 4.
```

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f: 4*x*(3+6*x-x^2)/(1-x)^3.
From Amiram Eldar, Mar 05 2023: (Start)
Sum_{n>=1} 1/a(n) = 1/8.
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi-2)/16. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: 4*(exp(x)*(4*x^2 + 4*x - 1) + 1).
a(n) = 4*A000466(n). (End)

A194801 Square array read by antidiagonals: T(n,k) = k((n+1)k-n+1)/2, k = 0, +- 1, +- 2,..., n >= 0.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 1, 3, 0, 1, 2, 4, 1, 0, 1, 3, 5, 4, 6, 0, 1, 4, 6, 7, 9, 3, 0, 1, 5, 7, 10, 12, 9, 10, 0, 1, 6, 8, 13, 15, 15, 16, 6, 0, 1, 7, 9, 16, 18, 21, 22, 16, 15, 0, 1, 8, 10, 19, 21, 27, 28, 26, 25, 10, 0, 1, 9, 11, 22, 24, 33, 34, 36, 35

Offset: 0

Omar E. Pol, Feb 05 2012

Note that a single formula gives several types of numbers. Row 0 lists 0 together the Molien series for 3-dimensional group [2,k]+ = 22k. Row 1 lists, except first zero, the squares repeated. If n >= 2, row n lists the generalized (n+3)-gonal numbers, for example: row 2 lists the generalized pentagonal numbers A001318. See some other examples in the cross-references section.

			Array begins:
(A008795): 0, 1,  0,  3,  1,  6,  3, 10,   6,  15,  10...
(A008794): 0, 1,  1,  4,  4,  9,  9, 16,  16,  25,  25...
A001318:   0, 1,  2,  5,  7, 12, 15, 22,  26,  35,  40...
A000217:   0, 1,  3,  6, 10, 15, 21, 28,  36,  45,  55...
A085787:   0, 1,  4,  7, 13, 18, 27, 34,  46,  55,  70...
A001082:   0, 1,  5,  8, 16, 21, 33, 40,  56,  65,  85...
A118277:   0, 1,  6,  9, 19, 24, 39, 46,  66,  75, 100...
A074377:   0, 1,  7, 10, 22, 27, 45, 52,  76,  85, 115...
A195160:   0, 1,  8, 11, 25, 30, 51, 58,  86,  95, 130...
A195162:   0, 1,  9, 12, 28, 33, 57, 64,  96, 105, 145...
A195313:   0, 1, 10, 13, 31, 36, 63, 70, 106, 115, 160...
A195818:   0, 1, 11, 14, 34, 39, 69, 76, 116, 125, 175...

Rows (0-11): 0 together with A008795, (truncated A008794), A001318, A000217, A085787, A001082, A118277, A074377, A195160, A195162, A195313, A195818
Columns (0-9): A000004, A000012, A001477, (truncated A000027), A016777, (truncated A008585), A016945, (truncated A016957), A017341, (truncated A017329).
Cf. A139600.

A204675 a(n) = 16n^2 + 2n + 1.

Original entry on oeis.org

1, 19, 69, 151, 265, 411, 589, 799, 1041, 1315, 1621, 1959, 2329, 2731, 3165, 3631, 4129, 4659, 5221, 5815, 6441, 7099, 7789, 8511, 9265, 10051, 10869, 11719, 12601, 13515, 14461, 15439, 16449, 17491, 18565, 19671, 20809, 21979, 23181, 24415, 25681, 26979

Offset: 0

Reinhard Zumkeller, Jan 18 2012

Central terms of the triangle A033293.

Also sequence found by reading the line from 1, in the direction 1, 19, ... 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 = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A017077, A033293, A074377, A136392.

Haskell
```
a204675 n = 2 * n * (8 * n + 1) + 1
```

I:=[1, 19, 69]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Mar 19 2012

CoefficientList[Series[(1+x)*(1+15*x)/(1-x)^3,{x,0,50}],x] (* or *) LinearRecurrence[{3, -3, 1}, {1, 19, 69}, 50] (* Vincenzo Librandi, Mar 19 2012 *)

a(n)=16*n^2+2*n+1 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: (1+x)*(1+15*x)/(1-x)^3. - Bruno Berselli, Jan 18 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Wesley Ivan Hurt, Jun 09 2023
E.g.f.: exp(x)*(1 + 2*x*(9 + 8*x)). - Elmo R. Oliveira, Oct 18 2024

A274756 Values of n such that 2n+1 and 6n+1 are both triangular numbers.

Original entry on oeis.org

0, 945, 13167, 35578242, 495540990, 1338951572595, 18649189618605, 50390103447476100, 701843601611053692, 1896381151803363988917, 26413182084381205040235, 71368408216577696911440390, 994033693861758668873164410, 2685878672926303893761783662455

Offset: 1

Colin Barker, Jul 04 2016

Intersection of A074377 and A274757.

			945 is in the sequence because 2*945+1 = 1891, 6*945+1 = 5671, and 1891 and 5671 are both triangular numbers.

Colin Barker, Table of n, a(n) for n = 1..400
Index entries for linear recurrences with constant coefficients, signature (1,37634,-37634,-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), A274680 (2*n+1 and 4*n+1).

isok(n) = ispolygonal(2*n+1, 3) && ispolygonal(6*n+1, 3)

concat(0, Vec(63*x^2*(15+194*x+15*x^2)/((1-x)*(1-194*x+x^2)*(1+194*x+x^2)) + O(x^20)))

G.f.: 63*x^2*(15+194*x+15*x^2) / ((1-x)*(1-194*x+x^2)*(1+194*x+x^2)).

a(n) = a(n-1)+37634*a(n-2)-37634*a(n-3)-a(n-4)+a(n-5). - Wesley Ivan Hurt, Apr 24 2021

A317300 Sequence obtained by taking the general formula for generalized k-gonal numbers: m((k - 2)m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 0.

Original entry on oeis.org

0, 1, -3, 0, -8, -3, -15, -8, -24, -15, -35, -24, -48, -35, -63, -48, -80, -63, -99, -80, -120, -99, -143, -120, -168, -143, -195, -168, -224, -195, -255, -224, -288, -255, -323, -288, -360, -323, -399, -360, -440, -399, -483, -440, -528, -483, -575, -528, -624, -575, -675, -624, -728, -675, -783

Offset: 0

Omar E. Pol, Jul 29 2018

Taking the same formula with k = 1 we have A317301.
Taking the same formula with k = 2 we have A001057 (canonical enumeration of integers).
Taking the same formula with k = 3 we have 0 together with A008795 (Molien series for 3-dimensional representation of dihedral group D_6 of order 6).
Taking the same formula with k = 4 we have A008794 (squares repeated) except the initial zero.
Taking the same formula with k >= 5 we have the generalized k-gonal numbers (see Crossrefs section).

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

Row 0 of A303301.
Cf. A001057, A008795, A008794, A174474, A317301.
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (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).

concat(0, Vec(x*(1 - 4*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^50))) \\ Colin Barker, Aug 01 2018

a(n) = -A174474(n+1).
From Colin Barker, Aug 01 2018: (Start)
G.f.: x*(1 - 4*x + 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>4.
a(n) = -n*(n + 4) / 4 for n even.
a(n) = -(n - 3)*(n + 1) / 4 for n odd.
(End)

A317301 Sequence obtained by taking the general formula for generalized k-gonal numbers: m((k - 2)m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 1.

Original entry on oeis.org

0, 1, -2, 1, -5, 0, -9, -2, -14, -5, -20, -9, -27, -14, -35, -20, -44, -27, -54, -35, -65, -44, -77, -54, -90, -65, -104, -77, -119, -90, -135, -104, -152, -119, -170, -135, -189, -152, -209, -170, -230, -189, -252, -209, -275, -230, -299, -252, -324, -275, -350, -299, -377, -324, -405, -350, -434

Offset: 0

Omar E. Pol, Jul 29 2018

Taking the same formula with k = 0 we have A317300.
Taking the same formula with k = 2 we have A001057 (canonical enumeration of integers).
Taking the same formula with k = 3 we have 0 together with A008795 (Molien series for 3-dimensional representation of dihedral group D_6 of order 6).
Taking the same formula with k = 4 we have A008794 (squares repeated) except the initial zero.
Taking the same formula with k >= 5 we have the generalized k-gonal numbers (see Crossrefs section).

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

Row 1 of A303301.
Cf. A000096, A001057, A008794, A008795, A317300.
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (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).

/* By definition: */ k:=1; [0] cat [m*i*((k-2)*m*i-k+4)/2: i in [1,-1], m in [1..30]]; // Bruno Berselli, Jul 30 2018

Table[(-2 n (n + 1) - 5 (2 n + 1) (-1)^n + 5)/16, {n, 0, 60}] (* Bruno Berselli, Jul 30 2018 *)

concat(0, Vec(x*(1 - 3*x + x^2)/((1 + x)^2*(1 - x)^3) + O(x^50))) \\ Colin Barker, Aug 01 2018

From Bruno Berselli, Jul 30 2018: (Start)
O.g.f.: x*(1 - 3*x + x^2)/((1 + x)^2*(1 - x)^3).
E.g.f.: (-5*(1 + 2*x) + (5 - 2*x^2)*exp(2*x))*exp(-x)/16.
a(n) = a(-n+1) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = (-2*n*(n + 1) - 5*(2*n + 1)*(-1)^n + 5)/16. Therefore:
a(n) = -n*(n + 6)/8 for even n;
a(n) = -(n - 5)*(n + 1)/8 for odd n. Also:
a(n) = a(n-5) for odd n > 3.
2*(2*n - 1)*a(n) + 2*(2*n + 1)*a(n-1) + n*(n^2 - 3) = 0. (End)

cp's OEIS Frontend

A143859 Ulam's spiral (WNW spoke).

Views

Author

Keywords

Comments

Links

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A143860 Ulam's spiral (NNW spoke).

Views

Author

Keywords

Comments

Links

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A157446 a(n) = 16*n^2 - n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A143861 Ulam's spiral (NNE spoke).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A158443 a(n) = 16*n^2 - 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A194801 Square array read by antidiagonals: T(n,k) = k*((n+1)*k-n+1)/2, k = 0, +- 1, +- 2,..., n >= 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

A204675 a(n) = 16*n^2 + 2*n + 1.

Views

Author

Keywords

A194801 Square array read by antidiagonals: T(n,k) = k((n+1)k-n+1)/2, k = 0, +- 1, +- 2,..., n >= 0.

A204675 a(n) = 16n^2 + 2n + 1.

A274756 Values of n such that 2n+1 and 6n+1 are both triangular numbers.

A317300 Sequence obtained by taking the general formula for generalized k-gonal numbers: m((k - 2)m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 0.

A317301 Sequence obtained by taking the general formula for generalized k-gonal numbers: m((k - 2)m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 1.