A181123 -id:A181123 - cp's OEIS frontend

A056107 Third spoke of a hexagonal spiral.

1, 4, 13, 28, 49, 76, 109, 148, 193, 244, 301, 364, 433, 508, 589, 676, 769, 868, 973, 1084, 1201, 1324, 1453, 1588, 1729, 1876, 2029, 2188, 2353, 2524, 2701, 2884, 3073, 3268, 3469, 3676, 3889, 4108, 4333, 4564, 4801, 5044, 5293, 5548, 5809, 6076, 6349

Offset: 0

Henry Bottomley, Jun 09 2000

a(n+1) is the number of lines crossing n cells of an n X n X n cube. - Lekraj Beedassy, Jul 29 2005
Equals binomial transform of [1, 3, 6, 0, 0, 0, ...]. - Gary W. Adamson, May 03 2008
Each term a(n), with n>1 represents the area of the right trapezoid with bases whose values are equal to hex number A003215(n) and A003215(n+1)and height equal to 1. The right trapezoid is formed by a rectangle with the sides equal to A003215(n) and 1 and a right triangle whose area is 3*n with the greater cathetus equal to the difference A003215(n+1)-A003215(n). - Giacomo Fecondo, Jun 11 2010
2*a(n)^2 is of the form x^4+y^4+(x+y)^4. In fact, 2*a(n)^2 = (n-1)^4+(n+1)^4+(2n)^4. - Bruno Berselli, Jul 16 2013
Numbers m such that m+(m-1)+(m-2) is a square. - César Aguilera, May 26 2015
After 4, twice each term belongs to A181123: 2*a(n) = (n+1)^3 - (n-1)^3. - Bruno Berselli, Mar 09 2016
This is a subsequence of A003136: a(n) = (n-1)^2 + (n-1)*(n+1) + (n+1)^2. - Bruno Berselli, Feb 08 2017
For n > 3, also the number of (not necessarily maximal) cliques in the n X n torus grid graph. - Eric W. Weisstein, Nov 30 2017

Edward J. Barbeau, Murray S. Klamkin and William O. J. Moser, Five Hundred Mathematical Challenges, MAA, Washington DC, 1995, Problem 444, pp. 42 and 195.
Ben Hamilton, Brainteasers and Mindbenders, Fireside, 1992, p. 107.

Nathaniel Johnston, Table of n, a(n) for n = 0..5000
Henry Bottomley, Illustration of initial terms
A. J. C. Cunningham, Factorisation of N and N' = (x^n -+ y^n) / (x -+ y) [when x-y=n], Messenger Math., 54 (1924), 17-21 [Incomplete annotated scanned copy]
Gabriele Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.
A. L. Rubinoff and Leo Moser, Solution to Problem E773, The American Mathematical Monthly, Vol. 55, No. 2 (Feb., 1948), p. 99.
Eric Weisstein's World of Mathematics, Clique.
Eric Weisstein's World of Mathematics, Torus Grid Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002648 (prime terms), A201053.
Cf. A000578, A003136, A132111, A181123.
Other spokes: A003215, A056105, A056106, A056108, A056109.
Other spirals: A054552.

List([0..40], n -> 3*n^2 + 1); # G. C. Greubel, Dec 02 2018

[3*n^2 + 1: n in [0..40]]; // G. C. Greubel, Dec 02 2018

seq(3*n^2+1, n=0..46); # Nathaniel Johnston, Jun 26 2011

Table[3 n^2 + 1, {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Jun 26 2011 *)
LinearRecurrence[{3, -3, 1}, {1, 4, 13}, 47] (* Michael De Vlieger, Feb 08 2017 *)
CoefficientList[Series[(1 + x + 4 x^2)/(1 - x)^3, {x, 0, 46}], x] (* Michael De Vlieger, Feb 08 2017 *)
1 + 3 Range[0, 20]^2 (* Eric W. Weisstein, Nov 30 2017 *)

for(n=0,1000,if(issquare(n+(n-1)+(n-2)),print1(n", "))); \\ César Aguilera, May 26 2015

a(n) = 3*n^2 + 1; \\ Altug Alkan, Feb 08 2017

[3*n^2 + 1 for n in range(40)] # G. C. Greubel, Dec 02 2018

a(n) = 3*n^2 + 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2.
G.f.: (1+x+4*x^2)/(1-x)^3.
a(n) = a(n-1) + 6*n - 3 for n>0.
a(n) = 2*a(n-1) - a(n-2) + 6 for n>1.
a(n) = A056105(n) + 2*n = A056106(n) + n.
a(n) = A056108(n) - n = A056109(n) - 2*n = A003215(n) - 3*n.
a(n) = (A000578(n+1) - A000578(n-1))/2. - Lekraj Beedassy, Jul 29 2005
a(n) = A132111(n+1,n-1) for n>1. - Reinhard Zumkeller, Aug 10 2007
E.g.f.: (1 + 3*x + 3*x^2)*exp(x). - G. C. Greubel, Dec 02 2018
From Amiram Eldar, Jul 15 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(3))*coth(Pi/sqrt(3)))/2.
Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(3))*csch(Pi/sqrt(3)))/2. (End)
From Amiram Eldar, Feb 05 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(3))*sinh(sqrt(2/3)*Pi).
Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(3))*csch(Pi/sqrt(3)). (End)

A014439 Differences between two positive cubes in exactly 1 way.

Original entry on oeis.org

7, 19, 26, 37, 56, 61, 63, 91, 98, 117, 124, 127, 152, 169, 189, 208, 215, 217, 218, 271, 279, 296, 316, 331, 335, 342, 386, 387, 397, 448, 469, 485, 488, 504, 511, 513, 547, 602, 604, 631, 657, 665, 702, 784, 817, 819, 866, 875, 919, 936, 973, 988, 992

Offset: 1

Glen Burch (gburch(AT)erols.com)

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000578, A038593, A181123, A034179 (more than one way), A014440 (exactly two ways), A265625 (more than two ways), A014441 (exactly three ways), A333376, A333377.

N:= 1000: # to get all terms <= N
X:= floor(sqrt(N/3)):
V:= Vector(N):
for x from 2 to X do
  if x^3 > N then
     y0:= iroot(x^3-N, 3);
     if x^3 - y0^3 > N then y0:= y0+1 fi;
  else y0:= 1 fi;
  for y from y0 to x-1 do
     V[x^3 - y^3] := V[x^3 - y^3]+1
  od
od: select(t -> V[t] = 1, [$1..N]); # Robert Israel, Dec 11 2015

r = 992; p = 3; Sort@Drop[Flatten@Select[Tally@Reap[Do[n = i^p - j^p; If[n <= r, Sow[n]], {i, Ceiling[(r/p)^(1/(p - 1))]}, {j, i}]][[2, 1]], #[[2]] == 1 &], {2, -1, 2}] (* Arkadiusz Wesolowski, Dec 10 2015 *)

Corrected and extended by Don Reble, Nov 19 2006

A014440 Differences between two positive cubes in exactly 2 ways.

Original entry on oeis.org

721, 728, 999, 5768, 5824, 5859, 7992, 8911, 9919, 10621, 12663, 12824, 19467, 19656, 23877, 25669, 26973, 27937, 28063, 34209, 35208, 35929, 41743, 43561, 46144, 46592, 46872, 49959, 53144, 63936, 68857, 68913, 71288, 77779, 79352, 80379, 84968, 90125

Offset: 1

Glen Burch (gburch(AT)erols.com)

nonn

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000578, A181123, A014439 (exactly one way), A034179 (more than one way), A265625 (more than two ways), A014441 (exactly three ways), A333376, A333377.

r = 90125; p = 3; Sort@Drop[Flatten@Select[Tally@Reap[Do[n = i^p - j^p; If[n <= r, Sow[n]], {i, Ceiling[(r/p)^(1/(p - 1))]}, {j, i}]][[2, 1]], #[[2]] == 2 &], {2, -1, 2}] (* Arkadiusz Wesolowski, Dec 10 2015 *)

Extended by Don Reble, Nov 19 2006

A014441 Differences between two positive cubes in exactly 3 ways.

Original entry on oeis.org

3367, 26936, 90909, 152551, 205352, 215488, 420875, 622232, 625177, 727272, 754299, 757701, 845208, 1147627, 1154881, 1220408, 1642816, 1723904, 2113921, 2454543, 2741256, 3056473, 3367000, 3442887, 3492125, 4481477, 4977856, 5001416, 5544504, 5818176

Offset: 1

Glen Burch (gburch(AT)erols.com)

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000578, A181123, A014439 (exactly one way), A034179 (more than one way), A014440 (exactly two ways), A265625 (more than two ways), A333376 (exactly 4 ways), A333377 (exactly 5 ways).

r = 5818176; p = 3; Sort@Drop[Flatten@Select[Tally@Reap[Do[n = i^p - j^p; If[n <= r, Sow[n]], {i, Ceiling[(r/p)^(1/(p - 1))]}, {j, i}]][[2, 1]], #[[2]] == 3 &], {2, -1, 2}] (* Arkadiusz Wesolowski, Dec 10 2015 *)

Extended by Don Reble, Nov 19 2006

A181124 Difference of two positive 5th powers.

Original entry on oeis.org

0, 31, 211, 242, 781, 992, 1023, 2101, 2882, 3093, 3124, 4651, 6752, 7533, 7744, 7775, 9031, 13682, 15783, 15961, 16564, 16775, 16806, 24992, 26281, 29643, 31744, 32525, 32736, 32767, 40951, 42242, 51273, 55924, 58025, 58806, 59017, 59048, 61051

Offset: 1

T. D. Noe, Oct 06 2010

nonn

Because x^5-y^5 = (x-y)(x^4+x^3*y+x^2*y^2+x*y^3+y^4), the difference of two 5th powers is a prime number only if x=y+1, in which case all the primes are in A121616. The number 7744 is the first of an infinite number of squares in this sequence.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A024352 (squares), A181123 (cubes), A147857 (4th powers), A181125-A181128 (6th to 9th powers)

nn=10^9; p=5; Union[Reap[Do[n=i^p-j^p; If[n<=nn, Sow[n]], {i,Ceiling[(nn/p)^(1/(p-1))]}, {j,i}]][[2,1]]]

A226903 Shiraishi numbers: a parametrized family of solutions c to the Diophantine equation a^3 + b^3 + c^3 = d^3 with d = c+1.

Original entry on oeis.org

5, 18, 53, 102, 197, 306, 491, 684, 989, 1290, 1745, 2178, 2813, 3402, 4247, 5016, 6101, 7074, 8429, 9630, 11285, 12738, 14723, 16452, 18797, 20826, 23561, 25914, 29069, 31770, 35375, 38448, 42533, 46002, 50597, 54486, 59621, 63954, 69659, 74460, 80765, 86058

Offset: 1

Jonathan Sondow, Jun 22 2013

Shiraishi's solutions to a^3 + b^3 + c^3 = d^3 are a = 3n^2; b = 6n^2 - 3n + 1 or 6n^2 + 3n + 1; c = 9n^3 - 6n^2 + 3n - 1 or  9n^3 + 6n^2 + 3n, respectively, for n > 0; and d = c+1. See Smith and Mikami for a derivation.
Shiraishi's formulas show that the sequence is infinite. Hence the sequences A023042 (solutions to x^3 + y^3 + z^3 = w^3), A225908 (solutions to a^3 + b^3 = c^3 - d^3), A225909 (solutions to a^3 + b^3 = (c+1)^3 - c^3) and A226902 (numbers c in A225909) are also infinite.
Shiraishi's solution b = 6n^2 +/- 3n + 1 is the centered triangular numbers A005448 except 1.

			The first two terms are a(1) = 9 - 6 + 3 - 1 = 5 and a(2) = 9 + 6 + 3 = 18. Then Shiraishi's formulas give 3^3 + 4^3 + 5^3 = 6^3 and 3^3 + 10^3 + 18^3 = 19^3.

Shiraishi Chochu (aka Shiraishi Nagatada), Shamei Sampu (Sacred Mathematics), 1826.

David Eugene Smith and Yoshio Mikami, A History of Japanese Mathematics, Open Court, Chicago, 1914; Dover reprint, 2004; pp. 233-235.
Wikipedia (French), Shiraishi Nagatada
Wikipedia (German), Shiraishi Nagatada
Index entries for sequences related to sums of cubes
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A003325, A005448, A023042, A181123, A225908, A225909, A226902.

a(2n-1) = 9n^3 - 6n^2 + 3n - 1.
a(2n) = 9n^3 + 6n^2 + 3n.
G.f.: x*(5 + 13*x + 20*x^2 + 10*x^3 + 5*x^4 + x^5) / ((1 + x)^3*(1 - x)^4). [Bruno Berselli, Jun 22 2013]
a(n) = (18*n^3 + 27*n^2 + 27*n + 1 - (3*n^2 + 3*n + 1)*(-1)^n)/16. [Bruno Berselli, Jun 22 2013]
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7) for n > 7. - Chai Wah Wu, Aug 05 2025

A333376 Differences between two positive cubes in exactly 4 ways.

Original entry on oeis.org

4118877, 32951016, 52324993, 94287375, 111209679, 214435711, 263608128, 418599944, 442245349, 514859625, 754299000, 889677432, 995635368, 1080305856, 1147627000, 1715485688, 2108865024, 2545759125, 3002661333, 3348799552, 3537962792, 3701994688, 4118877000, 4304670552

Offset: 1

Giovanni Resta, Mar 17 2020

			4118877 = 162^3 - 51^3 = 165^3 - 72^3 = 178^3 - 115^3 = 678^3 - 675^3.

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A014439, A014440, A014441, A181123, A034179, A265625, A333377, A098110.

Subsequence of A265625.

a(1) = A098110(4).

A333377 Differences between two positive cubes in exactly 5 ways.

Original entry on oeis.org

1412774811, 11302198488, 38144919897, 90417587904, 105443078832, 176596851375, 305159359176, 370544908608, 484581760173, 723340703232, 843544630656, 1029912837219, 1238805803151, 1412774811000, 1808088149952, 1880403273441, 2441274873408, 2846963128464, 2863636114248

Offset: 1

Giovanni Resta, Mar 17 2020

			1412774811 = 1134^3 - 357^3 = 1155^3 - 504^3 = 1246^3 - 805^3 = 2115^3 - 2004^3 = 4746^3 - 4725^3.

Giovanni Resta, Table of n, a(n) for n = 1..150

Cf. A014439, A014440, A014441, A181123, A034179, A265625, A333376, A098110.

a(1) = A098110(5).

A034179 Difference between two positive cubes in more than one way.

Original entry on oeis.org

721, 728, 999, 3367, 5768, 5824, 5859, 7992, 8911, 9919, 10621, 12663, 12824, 19467, 19656, 23877, 25669, 26936, 26973, 27937, 28063, 34209, 35208, 35929, 41743, 43561, 46144, 46592, 46872, 49959, 53144, 63936, 68857, 68913, 71288, 77779

Offset: 1

Erich Friedman

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A014439, A014440, A014441, A333376, A333377, A181123, A265625.

fQ[n_] := Block[{r = Reduce[0 < y < x && n == x^3 - y^3, {x, y}, Integers]}, If[r === False || Head[r] === And, False, Length[r] >= 2]]; Select[ Range[77780], If[ fQ[#], Print[#]; True, False] &] (* Jean-François Alcover, Apr 11 2011 *)
With[{nn=50},Take[Sort[Transpose[Select[Tally[#[[2]]-#[[1]]&/@Subsets[ Range[ nn*20]^3,{2}]],#[[2]]>1&]][[1]]],nn]] (* Harvey P. Dale, Mar 09 2016 *)

Extended by Ray Chandler, Nov 29 2008

Offset corrected by Giovanni Resta, Mar 17 2020

A265625 Differences between two positive cubes in more than two ways.

Original entry on oeis.org

3367, 26936, 90909, 152551, 205352, 215488, 420875, 622232, 625177, 727272, 754299, 757701, 845208, 1147627, 1154881, 1220408, 1642816, 1723904, 2113921, 2454543, 2741256, 3056473, 3367000, 3442887, 3492125, 4118877, 4481477, 4977856, 5001416, 5544504

Offset: 1

Arkadiusz Wesolowski, Dec 10 2015

nonn

			3367 = 15^3 - 2^3 = 16^3 - 9^3 = 34^3 - 33^3.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000578, A181123, A014439 (exactly one way), A034179 (more than one way), A014440 (exactly two ways), A014441 (exactly three ways), A333376, A333377.

r = 5544504; p = 3; Rest@Sort@Drop[Flatten@Select[Tally@Reap[Do[n = i^p - j^p; If[n <= r, Sow[n]], {i, Ceiling[(r/p)^(1/(p - 1))]}, {j, i}]][[2, 1]], #[[2]] > 2 &], {2, -1, 2}]

cp's OEIS Frontend

A056107 Third spoke of a hexagonal spiral.

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A014439 Differences between two positive cubes in exactly 1 way.

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A014440 Differences between two positive cubes in exactly 2 ways.

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A014441 Differences between two positive cubes in exactly 3 ways.

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A181124 Difference of two positive 5th powers.

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A226903 Shiraishi numbers: a parametrized family of solutions c to the Diophantine equation a^3 + b^3 + c^3 = d^3 with d = c+1.

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A333376 Differences between two positive cubes in exactly 4 ways.

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A333377 Differences between two positive cubes in exactly 5 ways.

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