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A269441

Alternating sum of 10-gonal (or decagonal) pyramidal numbers.
(history; published version)

#10 by Charles R Greathouse IV at Thu Sep 08 08:46:15 EDT 2022

PROG

(MAGMAMagma) [((-1)^n*(16*n^3+30*n^2-4*n-9)+9)/24: n in [0..40]]; // Vincenzo Librandi, Feb 27 2016

Discussion

Thu Sep 08

08:46

OEIS Server: https://oeis.org/edit/global/2944

#9 by Vaclav Kotesovec at Sat Feb 27 03:42:32 EST 2016

STATUS

proposed

approved

#8 by Vincenzo Librandi at Sat Feb 27 01:31:56 EST 2016

STATUS

editing

proposed

#7 by Vincenzo Librandi at Sat Feb 27 01:31:41 EST 2016

PROG

(MAGMA) [((-1)^n*(16*n^3+30*n^2-4*n-9)+9)/24: n in [0..40]]; // Vincenzo Librandi, Feb 27 2016

STATUS

reviewed

editing

#6 by Vaclav Kotesovec at Fri Feb 26 12:59:42 EST 2016

STATUS

proposed

reviewed

#5 by Vaclav Kotesovec at Fri Feb 26 12:58:02 EST 2016

STATUS

editing

proposed

Discussion

Fri Feb 26

12:59

Ilya Gutkovskiy: Thanks.

#4 by Vaclav Kotesovec at Fri Feb 26 12:56:04 EST 2016

FORMULA

Sum_{n>=1} 1/a(n) = -0.9251958836055717745244669... . - Vaclav Kotesovec, Feb 26 2016

STATUS

proposed

editing

Discussion

Fri Feb 26

12:58

Vaclav Kotesovec: (3/34)*(-2 - 16*Pi*Cot[Pi/16] + 32*Log[32] + 64*Cos[Pi/8]*Log[Cos[Pi/16]] + 32*Sqrt[2]*Log[Cos[Pi/8]] - 64*Cos[Pi/8]* Log[Sin[Pi/16]] - 32*Sqrt[2]*Log[Sin[Pi/8]] + 17*RootSum[3 + 26*#1 + 39*#1^2 + 16*#1^3 & , PolyGamma[0, -#1]/(13 + 39*#1 + 24*#1^2) & ] + 64*Log[Cos[(3*Pi)/16]]*Sin[Pi/8] - 64*Log[Sin[(3*Pi)/16]]*Sin[Pi/8])

#3 by Ilya Gutkovskiy at Fri Feb 26 08:36:47 EST 2016

STATUS

editing

proposed

#2 by Ilya Gutkovskiy at Fri Feb 26 08:36:22 EST 2016

NAME

allocated for Ilya GutkovskiyAlternating sum of 10-gonal (or decagonal) pyramidal numbers.

DATA

0, -1, 10, -28, 62, -113, 188, -288, 420, -585, 790, -1036, 1330, -1673, 2072, -2528, 3048, -3633, 4290, -5020, 5830, -6721, 7700, -8768, 9932, -11193, 12558, -14028, 15610, -17305, 19120, -21056, 23120, -25313, 27642, -30108, 32718, -35473, 38380, -41440, 44660

OFFSET

0,3

LINKS

OEIS Wiki, <a href="http://oeis.org/wiki/Figurate_numbers">Figurate numbers</a>

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PyramidalNumber.html">Pyramidal Number</a>

<a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (-3,-2,2,3,1).

FORMULA

G.f.: x*(1 - 7*x)/((x - 1)*(x + 1)^4).

a(n) = ((-1)^n*(16*n^3 + 30*n^2 - 4*n - 9) + 9) /24.

a(n) = Sum_{k = 0..n} (-1)^k*A007585(k).

MATHEMATICA

Table[((-1)^n (16 n^3 + 30 n^2 - 4 n - 9) + 9)/24, {n, 0, 40}]

LinearRecurrence[{-3, -2, 2, 3, 1}, {0, -1, 10, -28, 62}, 41]

CROSSREFS

Cf. A000292, A002717, A007585, A173196, A266677, A269428, A269429.

KEYWORD

allocated

easy,sign

AUTHOR

Ilya Gutkovskiy, Feb 26 2016

STATUS

approved

editing

#1 by Ilya Gutkovskiy at Fri Feb 26 08:36:22 EST 2016

NAME

allocated for Ilya Gutkovskiy

KEYWORD

allocated

STATUS

approved