A007588 -id:A007588 - cp's OEIS frontend

A277792 Squares that are also pentagonal pyramidal numbers.

0, 1, 196, 2601, 15376, 60025, 181476, 461041, 1032256, 2099601, 3960100, 7027801, 11861136, 19193161, 29964676, 45360225, 66846976, 96216481, 135629316, 187662601, 255360400, 342287001, 452583076, 591024721, 763085376, 975000625, 1233835876, 1547556921, 1925103376, 2376465001, 2912760900

Offset: 0

Ilya Gutkovskiy, Oct 31 2016

Intersection of A000290 and A002411.

			a(2) = 196 because 196 = 14^2 is a perfect square and 196 = 7^2*(7 + 1)/2 is the 7th pentagonal pyramidal number.

Daniel Mondot, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Pentagonal Pyramidal Number
Eric Weisstein's World of Mathematics, Square Number
Index to sequences related to polygonal numbers
Index to sequences related to pyramidal numbers
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000290, A001653, A002411, A007588, A036353, A056220, A254333.

[n^2*(2*n^2-1)^2: n in [0..30]]; // Vincenzo Librandi, Nov 01 2016

Table[n^2 (2 n^2 - 1)^2, {n, 0, 30}]
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,1,196,2601,15376,60025,181476},40] (* Harvey P. Dale, Nov 01 2024 *)

O.g.f.: x*(1 + 189*x + 1250*x^2 + 1250*x^3 + 189*x^4 + x^5)/(1 - x)^7.
E.g.f.: x*(1 + 97*x + 336*x^2 + 256*x^3 + 60*x^4 + 4*x^5)*exp(x).
a(n) = a(-n).
a(n) = n^2*(2*n^2 - 1)^2.
a(n) = A000290(A007588(n)).
a(n) = A000290(n)*A000290(A056220(n)).
Sum_{n>=1} 1/a(n) = (2*Pi^2+9*sqrt(2)*Pi*cot(Pi/sqrt(2))+3*Pi^2*csc(Pi/sqrt(2))^2-24)/12 = 1.0055779712856...

A281699 Sierpinski stellated octahedron numbers: a(n) = 2(-32^(n+1) + 2^(2n+3) + 5).

Original entry on oeis.org

14, 50, 218, 938, 3914, 16010, 64778, 260618, 1045514, 4188170, 16764938, 67084298, 268386314, 1073643530, 4294770698, 17179475978, 68718690314, 274876334090, 1099508482058, 4398040219658, 17592173461514, 70368719011850, 281474926379018, 1125899806179338, 4503599426043914, 18014398106828810

Offset: 0

Steven Beard, Jan 27 2017

Stella octangula with Sierpinski recursion.

Colin Barker, Table of n, a(n) for n = 0..1000
Wikipedia, Sierpinski triangle, see section on higher dimensional analogs.
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A007588, A027693, A052539, A052548, A067771, A178789, A233774, A279511.

Table[8 (2^(2 n + 1) + 2) - 6 (2^(n + 1) + 1), {n, 0, 25}] (* or *)
LinearRecurrence[{7, -14, 8}, {14, 50, 218}, 26] (* or *)
CoefficientList[Series[2 (7 - 24 x + 32 x^2)/((1 - x) (1 - 2 x) (1 - 4 x)), {x, 0, 25}], x] (* Michael De Vlieger, Jan 28 2017 *)

Vec(2*(7 - 24*x + 32*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Jan 28 2017

a(n) = 16*4^n - 12*2^n + 10 \\ Charles R Greathouse IV, Jan 29 2017

a(n) = 8*(2^(2*n+1)+2) - 6*(2^(n+1)+1).
From Colin Barker, Jan 28 2017: (Start)
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3) for n>2.
G.f.: 2*(7 - 24*x + 32*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)).
(End)

A342237 Table read by upward antidiagonals: T(n,k) is the number of strings of length k over an n-letter alphabet that begin with a palindrome of two or more letters; n, k >= 1.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 0, 3, 6, 1, 0, 4, 15, 14, 1, 0, 5, 28, 51, 30, 1, 0, 6, 45, 124, 165, 62, 1, 0, 7, 66, 245, 532, 507, 126, 1, 0, 8, 91, 426, 1305, 2164, 1551, 254, 1, 0, 9, 120, 679, 2706, 6605, 8788, 4683, 510, 1, 0, 10, 153, 1016, 5005, 16386, 33405, 35284, 14127, 1022, 1

Offset: 1

Peter Kagey, Mar 06 2021

			Table begins:
n\k | 1  2   3    4      5       6        7         8
----+------------------------------------------------
  1 | 0  1   1    1      1       1        1         1
  2 | 0  2   6   14     30      62      126       254
  3 | 0  3  15   51    165     507     1551      4683
  4 | 0  4  28  124    532    2164     8788     35284
  5 | 0  5  45  245   1305    6605    33405    167405
  6 | 0  6  66  426   2706   16386    99186    595986
  7 | 0  7  91  679   5005   35287   248731   1742839
  8 | 0  8 120 1016   8520   68552   551496   4415048

Peter Kagey, Antidiagonals n = 1..100, flattened

Rows: A000918 (n=2), A248122 (n=3), A249629 (n=4), A249638 (n=5), A249639 (n=6), A249640 (n=7), A249641 (n=8), A249642 (n=9), A249643 (n=10).

Columns: A000384 (k=3), A007588 (k=4).

T(n,1)    = 0.
T(n,2k)   = n*T(n,2k-1) + n^k - T(n,k).
T(n,2k+1) = n*T(n,2k) + n^(k+1) - T(n,k+1).

A347753 Number of polyhedra formed when a row of n adjacent cubes are internally cut by all the planes defined by any three of their vertices.

Original entry on oeis.org

96, 2968, 42384, 319416

Offset: 1

Scott R. Shannon, Sep 12 2021

For a row of n adjacent cubes create all possible planes defined by connecting any three of their vertices. For example, in the case of a single cube this results in fourteen planes; six planes between the pairs of parallel edges connected to each end of the face diagonals, and eight planes from connecting the three vertices adjacent to each corner vertex. Use all the resulting planes to cut the entire solid into individual smaller polyhedra. The sequence lists the numbers of resulting polyhedra for n adjacent cubes.

See A347918 for the number of k-faced polyhedra for each value of n.

			a(1) = 96. A single cube, with eight vertices, has 14 internal cutting planes resulting in 96 polyhedra. See A333539 and A338571.
a(2) = 2968. Two adjacent cubes, with twelve vertices, have 51 internal cutting planes resulting in 2968 polyhedra.
a(3) = 42384. Three adjacent cubes, with sixteen vertices, have 124 internal cutting planes resulting in 42384 polyhedra.
a(4) = 319416. Four adjacent cubes, with twenty vertices, have 245 internal cutting planes resulting in 319416 polyhedra.

Scott R. Shannon, The 245 cutting planes on the surface of 4 adjacent cubes.
Scott R. Shannon, The surface of the 4 adjacent cubes after cutting. The 4-, 5-, 6-, 7-, 8-, and 9-faced polyhedra created by the planes are colored red, orange, yellow, green, blue, and indigo, respectively. The 10-, 11-, and 12-faced polyhedra are not visible on the surface. See also A347918.
Scott R. Shannon, The 4 adjacent cubes after cutting exploded. Each of the 319416 polyhedra is moved away from the center of the solid a distance proportional to the average distance of its vertices from the center.

Cf. A347918 (number of k-faced polyhedra), A333539 (n-dimensional cube), A338571 (Platonic solids), A338783 (n-prism), A338809 (n-bipyramid), A007588.

a(1) = A333539(3).

Conjectured formula for the number of internal cutting planes for n adjacent cubes is A007588(n+1).

A152912 Primes p such that 2*p^2-1 is not prime.

Original entry on oeis.org

5, 19, 23, 29, 31, 37, 47, 53, 61, 67, 71, 79, 83, 89, 97, 101, 103, 107, 131, 139, 149, 151, 163, 167, 173, 191, 193, 223, 227, 229, 233, 239, 241, 257, 269, 271, 281, 283, 307, 313, 317, 331, 337, 347, 349, 359, 373, 383, 389, 397, 401, 421, 431, 439, 443

Offset: 1

Vincenzo Librandi, Dec 15 2008

Primes not in A106483. Primes p such that 2p^2-1 is not in A092057. - R. J. Mathar, Dec 19 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A092057, A106483, A007588.

[p: p in PrimesUpTo(500)|not IsPrime(2*p^2-1)]; // Vincenzo Librandi, Aug 30 2012

a := proc (n) if isprime(n) = true and isprime(2*n^2-1) = false then n else end if end proc: seq(a(n), n = 1 .. 500); # Emeric Deutsch, Jan 02 2009

Select[Prime[Range[100]], !PrimeQ[2 #^2 - 1] &] (* Vincenzo Librandi, Aug 30 2012 *)

Definition clarified by R. J. Mathar, Dec 19 2008

Extended by Emeric Deutsch, Jan 02 2009

A229384 Positive integer solutions y1, x1, y2, x2 to Ljunggren's equation x^2 + 1 = 2y^4.

Original entry on oeis.org

1, 1, 13, 239

Offset: 1

Jonathan Sondow, Sep 30 2013

See the Wikipedia links for other references.

The only square stella octangula numbers are A007588(1) = (a(1)*a(2))^2 = 1 and A007588(169) = (a(3)*a(4))^2 = 9653449.

			239^2 + 1 = 57122 = 2*13^4.

W. Ljunggren, Zur Theorie der Gleichung x^2 + 1 = Dy^4, Avh. Norske Vid. Akad. Oslo. I. 1942 (5): 27.

Wikipedia, Stella octangula number
Wikipedia, Wilhelm Ljunggren

Cf. A007588.

A281999 Half of the height of the right trapezoidal gnomon (of the derivative of Y=X^5).

Original entry on oeis.org

1, 30, 181, 600, 1501, 3150, 5881, 10080, 16201, 24750, 36301, 51480, 70981, 95550, 126001, 163200, 208081, 261630, 324901, 399000, 485101, 584430, 698281, 828000, 975001, 1140750, 1326781, 1534680, 1766101, 2022750, 2306401, 2618880, 2962081, 3337950, 3748501, 4195800

Offset: 1

Stefano Maruelli, Feb 05 2017

The curves Y = X^m are characterized by the fact that the first derivative Y'= m*X^(m-1) (and all the following derivatives) are squarable in the integers by rectangular columns called gnomons with base=1 and height M_m = X^m - (X-1)^m. Calling Y' = X^m - (X-1)^m the first "integer" derivative, considering the case m=5, {a(n)} represents the values of half of the maximum (right) height of the trapezoidal gnomons. The formula is: a(n) = (n^5 - (n-1)^5) - a(n-1). The broken line given by joining the points (n; 2*a(n)); define a series of trapezoidal areas (gnomons) that have the same area below the curve Y'=5*X^4. It means that the recursive sum of the trapezoidal gnomon's area, (a(n) + a(n-1))*1, from 1 to n, gives n^5.

The general formula, changing the exponent for all the Y = X^m curves, gives infinitely many new sequences: b(m,k) = m^k - (m-1)^k - b(m-1,k). The same can be done for all the following derivatives. For the smallest exponents k of Y = X^k the sequences are known: for k=3 the sequence is A032528, for k=4 the sequence is A007588, and k=5 corresponds to this sequence.

			For n=2, a(2) = (2^5 - 1^5) - (1) = 30.

Colin Barker, Table of n, a(n) for n = 1..1000
Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, Enumeration of multi-rooted plane trees, arXiv:2301.09765 [math.CO], 2023.
Stefano Maruelli, Trapezoidal gnomon roof, case n=5.
Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

LinearRecurrence[{4,-5,0,5,-4,1},{1,30,181,600,1501,3150},40] (* Harvey P. Dale, May 03 2024 *)

Vec(x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4)/((1 + x)*(1 - x)^5) + O(x^30)) \\ Colin Barker, Feb 27 2017

G.f.: x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4)/((1 + x)*(1 - x)^5).
a(n) = (5*(n^2 - 1)*n^2 - (-1)^n + 1)/2.
a(n) = (n^5-(n-1)^5) - a(n-1).
a(n) = 4*a(n-1) - 5*a(n-2) + 5*a(n-4) - 4*a(n-5) + a(n-6) for n>6. - Colin Barker, Feb 27 2017

cp's OEIS Frontend

A277792 Squares that are also pentagonal pyramidal numbers.

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A281699 Sierpinski stellated octahedron numbers: a(n) = 2*(-3*2^(n+1) + 2^(2n+3) + 5).

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A342237 Table read by upward antidiagonals: T(n,k) is the number of strings of length k over an n-letter alphabet that begin with a palindrome of two or more letters; n, k >= 1.

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A347753 Number of polyhedra formed when a row of n adjacent cubes are internally cut by all the planes defined by any three of their vertices.

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A152912 Primes p such that 2*p^2-1 is not prime.

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A229384 Positive integer solutions y1, x1, y2, x2 to Ljunggren's equation x^2 + 1 = 2y^4.

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A281999 Half of the height of the right trapezoidal gnomon (of the derivative of Y=X^5).

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Mathematica

PARI

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A281699 Sierpinski stellated octahedron numbers: a(n) = 2(-32^(n+1) + 2^(2n+3) + 5).