A032528 -id:A032528 - cp's OEIS frontend

A260955 Differences of the increasing arithmetic progression a^2+a, b^2+b, c^2+c, where b = 5a+2, c = 7a+3 and a >= 0.

6, 54, 150, 294, 486, 726, 1014, 1350, 1734, 2166, 2646, 3174, 3750, 4374, 5046, 5766, 6534, 7350, 8214, 9126, 10086, 11094, 12150, 13254, 14406, 15606, 16854, 18150, 19494, 20886, 22326, 23814, 25350, 26934, 28566, 30246, 31974, 33750, 35574, 37446, 39366, 41334

Offset: 0

Marco Ripà, Aug 05 2015

			By the definition, given a = 7 and b = 5*7+2 = 37, c = 7*7+3 = 52, it follows that a^2+a = 56, b^2+b = 1406, c^2+c = 2756, where 56, 1406, 2756 are in arithmetic progression. Therefore, 2756-1406 = 1406-56 = 1350 and 1350 is in the sequence (8th term).

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

Cf. A016754, A032528.

Second bisection of A033581 (the first bisection is A195824).

[24*n^2+24*n+6: n in [0..40]]; // Vincenzo Librandi, Aug 05 2015

Table[24 n^2 + 24 n + 6, {n, 0, 40}] (* Bruno Berselli, Aug 05 2015 *)
LinearRecurrence[{3, -3, 1}, {6, 54, 150}, 50] (* Vincenzo Librandi, Aug 05 2015 *)

Vec(6*(1+6*x+x^2)/(1-x)^3 + O(x^100)) \\ Colin Barker, Aug 05 2015

a(n) = 24*n^2 + 24*n + 6.
From Colin Barker, Aug 05 2015: (Start)
G.f.: 6*(1 + 6*x + x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Bruno Berselli, Aug 05 2015: (Start)
a(n) = A032528(4*n+2).
a(n)*(2*h-1)^2 = a((2*h-1)*n+h-1). For h=0, a(n) = a(-n-1); for h=7, 169*a(n) = a(13*n+6). (End)
From Elmo R. Oliveira, Dec 28 2024: (Start)
E.g.f.: 6*exp(x)*(1 + 8*x + 4*x^2).
a(n) = 6*A016754(n). (End)

A270693 Alternating sum of centered 25-gonal numbers.

Original entry on oeis.org

1, -25, 51, -100, 151, -225, 301, -400, 501, -625, 751, -900, 1051, -1225, 1401, -1600, 1801, -2025, 2251, -2500, 2751, -3025, 3301, -3600, 3901, -4225, 4551, -4900, 5251, -5625, 6001, -6400, 6801, -7225, 7651, -8100, 8551, -9025, 9501, -10000, 10501

Offset: 0

Ilya Gutkovskiy, Mar 21 2016

The absolute value alternating sum of centered k-gonal numbers gives concentric k-gonal numbers.

More generally, the ordinary generating function for the alternating sum of centered k-gonal numbers is (1 - (k - 2)*x + x^2)/((1 - x)*(1 + x)^3).

OEIS Wiki, Centered polygonal numbers
Eric Weisstein's World of Mathematics, Centered Polygonal Number
Index entries for linear recurrences with constant coefficients, signature (-2,0,2,1).

Cf. A262221 (centered 25-gonal numbers).

Cf. A032527, A032528, A077043, A077221, A195041, A195042, A195045, A195046, A195047, A195048, A195049, A195058, A195142, A195043, A195143, A195145, A195146, A195147, A195148, A195149, A195158.

[((-1)^n*(50*n^2 + 100*n + 29) - 21)/8 : n in [0..40]]; // Wesley Ivan Hurt, Mar 21 2016

A270693:=n->((-1)^n*(50*n^2 + 100*n + 29) - 21)/8: seq(A270693(n), n=0..100); # Wesley Ivan Hurt, Sep 18 2017

LinearRecurrence[{-2, 0, 2, 1}, {1, -25, 51, -100}, 41]
Table[((-1)^n (50 n^2 + 100 n + 29) - 21)/8, {n, 0, 40}]

x='x+O('x^100); Vec((1-23*x+x^2)/((1-x)*(1+x)^3)) \\ Altug Alkan, Mar 21 2016

G.f.: (1 - 23*x + x^2)/((1 - x)*(1 + x)^3).
E.g.f.: (1/8)*(-21*exp(x) + (29 - 150*x + 50*x^2)*exp(-x)).
a(n) = -2*a(n-1) + 2*a(n-3) + a(n-4).
a(n) = ((-1)^n*(50*n^2 + 100*n + 29) - 21)/8.

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

A195060 Numbers on the main diagonals and on the axes (x,y) in the square spiral whose vertices are the generalized pentagonal numbers A001318.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 9, 12, 13, 15, 18, 22, 24, 26, 30, 35, 37, 40, 45, 51, 54, 57, 63, 70, 73, 77, 84, 92, 96, 100, 108, 117, 121, 126, 135, 145, 150, 155, 165, 176, 181, 187, 198, 210, 216, 222, 234, 247, 253, 260, 273, 287, 294, 301, 315, 330

Offset: 0

Omar E. Pol, Sep 08 2011

nonn

Union of A001318, A032528 and A045943, without repetitions.

Cf. A000326, A001318, A005449, A032528, A045943.

a(2*n) = A001318(n). a(4*n+1) = A032528(n+1). a(4*n+3) = A045943(n+1).

Conjectures: a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) + a(n-8) - 2*a(n-9) + 2*a(n-10) - 2*a(n-11) + a(n-12); g.f.: -x*(1+x^2+x^3-x^4+x^5+x^7-2*x^8+x^11+2*x^9-2*x^10) / ( (1+x)*(x^4+1)*(x^2+1)^2*(x-1)^3 ). - R. J. Mathar, Oct 11 2011

cp's OEIS Frontend

A260955 Differences of the increasing arithmetic progression a^2+a, b^2+b, c^2+c, where b = 5*a+2, c = 7*a+3 and a >= 0.

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A270693 Alternating sum of centered 25-gonal numbers.

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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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A195060 Numbers on the main diagonals and on the axes (x,y) in the square spiral whose vertices are the generalized pentagonal numbers A001318.

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Author

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Comments

Crossrefs

Formula

A260955 Differences of the increasing arithmetic progression a^2+a, b^2+b, c^2+c, where b = 5a+2, c = 7a+3 and a >= 0.